# Pulled, pushed or failed: the demographic impact of a gene drive can change
the nature of its spatial spread

Léna Kläy Corresponding author<EMAIL_ADDRESS>Institute of
Ecology and Environmental Sciences Paris (IEES Paris), Sorbonne Université,
CNRS, IRD, INRAE, Université Paris Est Creteil, Université de Paris, Paris
Cedex 5, France. Léo Girardin  Institut Camille Jordan, UMR 5208 CNRS and
Universite Claude Bernard Lyon 1, France  Vincent Calvez  Institut Camille
Jordan, UMR 5208 CNRS and Universite Claude Bernard Lyon 1, France  Florence
Débarre  Institute of Ecology and Environmental Sciences Paris (IEES Paris),
Sorbonne Université, CNRS, IRD, INRAE, Université Paris Est Creteil,
Université de Paris, Paris Cedex 5, France.

## Abstract

Understanding the temporal spread of gene drive alleles – alleles that bias
their own transmission – through modeling is essential before any field
experiments. In this paper, we present a deterministic reaction-diffusion
model describing the interplay between demographic and allelic dynamics, in a
one-dimensional spatial context. We focused on the traveling wave solutions,
and more specifically, on the speed of gene drive invasion (if successful). We
considered various timings of gene conversion (in the zygote or in the
germline) and different probabilities of gene conversion (instead of assuming
100$\%$ conversion as done in a previous work). We compared the types of
propagation when the intrinsic growth rate of the population takes extreme
values, either very large or very low. When it is infinitely large, the wave
can be either successful or not, and, if successful, it can be either pulled
or pushed, in agreement with previous studies (extended here to the case of
partial conversion). In contrast, it cannot be pushed when the intrinsic
growth rate is vanishing. In this case, analytical results are obtained
through an insightful connection with an epidemiological SI model. We
conducted extensive numerical simulations to bridge the gap between the two
regimes of large and low growth rate. We conjecture that, if it is pulled in
the two extreme regimes, then the wave is always pulled, and the wave speed is
independent of the growth rate. This occurs for instance when the fitness cost
is small enough, or when there is stable coexistence of the drive and the
wild-type in the population after successful drive invasion. Our model helps
delineate the conditions under which demographic dynamics can affect the
spread of a gene drive.

###### Contents

  1. 1 Introduction
  2. 2 Methodology
    1. 2.1 Models
    2. 2.2 Setting of the problem
    3. 2.3 Glossary
  3. 3 Results
    1. 3.1 Model with perfect conversion in the zygote
      1. 3.1.1 Preliminary statements on the model
      2. 3.1.2 $r=+\infty$
      3. 3.1.3 $r=0$
      4. 3.1.4 Comparison between the outcomes when $r=+\infty$ and $r=0$
    2. 3.2 Models with partial conversion
      1. 3.2.1 Conversion occurring in the zygote
      2. 3.2.2 Conversion occurring in the germline
      3. 3.2.3 Conclusion
  4. 4 Discussion
  5. 5 Acknowledgements
  6. A Model with partial conversion: growth term details
    1. A.1 Conversion occurring in the zygote
    2. A.2 Conversion occurring in the germline
  7. B System rewritten with variables $(n,p_{{}_{\mathrm{D}}})$
    1. B.1 Model with perfect conversion
    2. B.2 Model with partial conversion
      1. B.2.1 Conversion in the zygote
      2. B.2.2 Conversion in the germline
  8. C Proofs for model (11) with perfect conversion in the zygote
    1. C.1 Numerical evidence for the continuity when $r\rightarrow 0$
    2. C.2 Proof of the statements in Tables 3 and 4 when perfect conversion occurs in the zygote
      1. C.2.1 Gene drive clearance for $s\in(0.5,1)$ when $r=0$
      2. C.2.2 Numerical approximation of $s$ threshold value for the pulled/pushed wave when $r=+\infty$
  9. D Critical traveling wave for an SI similar model.
    1. D.1 Existence of critical traveling wave solutions
    2. D.2 Proof of the theorem
      1. D.2.1 Construction of sub- and super-solutions
      2. D.2.2 Existence and positivity of a critical traveling wave solution
  10. E Study of the reaction term when $r=+\infty$ in section 3.2
    1. E.1 Conversion occurring in the zygote
      1. E.1.1 Monostable drive invasion
      2. E.1.2 Monostable wild-type invasion
      3. E.1.3 Monostable coexistence state
    2. E.2 Conversion occurring in the germline
      1. E.2.1 Monostable drive invasion
      2. E.2.2 Monostable wild-type invasion
      3. E.2.3 Monostable coexistence state
  11. F Heatmap supplementary materials
    1. F.1 Effect of fitness disadvantage ($s$) and dominance coefficient ($h$) on drive dynamics, for $r=+\infty$.
    2. F.2 Heatmap lines
      1. F.2.1 Pure drive line
      2. F.2.2 Composite persistence line

##  1 Introduction

A highly accurate, cost-effective and easy-to-use technology, the CRISPR-Cas
genome editing system has been favoring the development of promising
innovations [25]. Among them, CRISPR-Cas9 gene drive [1], which aims to spread
a trait of interest in a wild type population in a relatively short number of
generations [26]. Application fields are numerous, and include i) the
eradication of insect-borne diseases [10, 19, 26]; ii) the elimination of
herbicide and pesticide resistance in pest populations [31]; iii) the control
of destructive invasive species [18, 21]; iv) the conservation of biodiversity
by spreading beneficial traits in endangered species [17, 35].

Targeting sexually reproducing species, CRISPR-Cas9 gene drive biases the
transmission of an allele from a parent to its offspring. This biased
inheritance occurs through gene conversion (also called “homing” [15]): in a
heterozygous cell, the gene drive cassette present on one chromosome induces a
double-strand break at a specific target site on the homologous chromosome,
and the repair process duplicates the cassette. Overall, this process
increases the chances of transmitting the gene drive cassette compared to its
wild-part counterpart, and the mechanism repeats through the generations. Gene
conversion can potentially take place at different timings of the life cycle:
from very early on, in the zygote, meaning that potentially every single cell
of the individual could become homozygous for the gene drive, to, in the
germline, where only the gametes are converted.

Gene drives can be classified into two main categories depending on the
purpose of their use [16, 20]. A “replacement drive” is aimed at spreading a
genetic modification in order to introduce an important and durable feature in
the natural population. Population size is then not significantly affected and
the drive construct may in principle persist indefinitely in the environment.
A “suppression drive” on the other hand is meant to reduce population size by
spreading a detrimental trait, such as a sex ratio distorter [29] or by
altering fertility [26], for example. The term “eradication drive” can be used
for the extreme case where population extinction is the aim.

As with any new tool, it is essential to balance risks (safety) and benefits
(efficacy) of the technique before running any field trials. Experiments
currently conducted in laboratories provide small- to medium-scale
information; mathematical models can help to extend these empirical results
and identify the features that are the most important in determining the
dynamics at larger scales [13].

Early gene drive models [11, 15, 40] used classical population genetics
frameworks, and considered discrete non-overlapping generations in a well-
mixed population. These simplifications helped to draw general conclusions,
but it is important to challenge them. First of all, most of the species
targeted in the context of gene drive do not have synchronous generations (for
instance mosquitoes [19, 23, 10, 26], flies [18], mice [21]). Secondly, the
assumption of a single well-mixed collection of individuals living across a
uniform space is usually not realistic. In fact, most of the natural
landscapes are heterogeneous. Individuals are also more likely to interact
with others that are in closer proximity, which might result in local genetic
variations. Finally, releases of transgenic individuals are limited in range,
which is another factor of spatial heterogeneity.

Taking into account spatio-temporal dynamics of the population size is another
key step towards more realistic models. For the sake of simplicity, most early
models focused on allele frequencies and considered a constant population
density. However in the context of gene drive, the introduction of maladapted
transgenic individuals can lead to the reduction (or even extinction) of the
population [16]. When considering a spatially structured population,
variations in population density naturally generate a demographic flux from
denser to less dense areas. This demographic flux is s directed in opposition
to the spread of the drive allele. It was previously shown [20] that the
advantage conferred by gene conversion may nevertheless counteract the
demographic effect linked to the fitness cost.

The main goal of this paper is to clarify the impact of variations in
population density over the course of drive propagation over space.

We study partial differential equations which follow the propagation of the
drive in space and time. We explore numerically and analytically two models: a
first model based on perfect conversion in the zygote, already introduced in
[20] in a spatially structured population, corresponding to an idealized case
where gene conversion always succeeds; second, a more realistic model with
partial conversion and presence of heterozygous individuals, already studied
in [35] in a well-mixed, non spatial population. In order to investigate the
possible spreading of gene drives through space after local introduction, we
focus on the description of traveling waves solutions, that is, particular
solutions which are stationary in a frame moving at constant speed. Our
analysis goes beyond [20] by several means: we extend it to the case of
partial conversion, and we systematically analyze the case where the
demographic effects are the strongest, in the regime of vanishing growth rate.
The latter is possible through an insightful connection with an
epidemiological SI model.

##  2 Methodology

###  2.1 Models

We present our model step-by-step. For a genetically and spatially homogeneous
population, we consider the following (non-dimensionalized) equation:

$\partial_{t}n(t)=\Big{(}1+r\left(1-n(t)\right)\Big{)}f\
n(t)-n(t)\quad\quad(\forall t>0).$ (1)

Fecundity is density-dependent, and parametrized by the fitness $f$ and the
rate $r$ at which the ($f-$dependent) carrying capacity is restored. When
$f=1$, the carrying capacity is $1$, and we recover the logistic equation
$\partial_{t}n(t)=r\left(1-n(t)\right)n(t)$. Other modeling options are
discussed in [20].

Then, we add genetic diversity in the population. We still denote by $n$ the
total density, and by $n_{i}$ the density of individuals with genotype $i$.
The population we consider is diploid, sexually reproducing, and the fitness
$f_{i}$ depends on the genotype. The dynamics are given by the following
equations:

$\partial_{t}n_{i}(t)=\Big{(}1+r(1-n(t))\Big{)}f_{i}\
n(t)\underbrace{\sum\limits_{l,k}\pi_{l,k}^{i}\ \dfrac{n_{l}(t)}{n(t)}\
\dfrac{n_{k}(t)}{n(t)}}_{\text{Mating term}}-n_{i}(t)\quad\quad(\forall t>0)\
(\forall i).$ (2)

The mating term takes into account the probability for each couple of parents
$l$,$k$ to have offspring of type $i$ ($\pi_{l,k}^{i}$), multiplied by the
probability of a mating event $l$,$k$ ($\frac{n_{l}(t)n_{k}(t)}{n(t)^{2}}$),
assuming random mating.

Last but not least, we consider a spatially structured population. We assume
that the movement of individuals is described by a diffusion term with equal
diffusion coefficients, normalized to 1. Since we focus on traveling wave
solutions, we restrict our analysis to a one-dimensional space. We obtain the
following equations:

$\partial_{t}n_{i}(t,x)-\partial_{xx}^{2}n_{i}(t,x)=\Big{(}1+r(1-n(t,x))\Big{)}f_{i}\
n(t)\sum\limits_{l,k}\pi_{l,k}^{i}\dfrac{n_{l}(t)}{n(t)}\
\dfrac{n_{k}(t)}{n(t)}-n_{i}(t,x)\quad\quad(\forall t>0)\ (\forall
x\in\mathbb{R})\ (\forall i).$ (3)

There are two possible alleles at the locus that we consider: the wild-type
allele ($W$) and the drive allele ($D$). We have three genotypes: wild-type
homozygotes ($i=WW$), drive homozygotes ($i=DD$) and heterozygotes ($i=DW$).
Wild-type homozygotes have fitness $f_{D}=1$, drive homozygotes have fitness
$f_{D}=1-s$, where $s$ is the fitness cost of the drive, and drive
heterozygotes have fitness $f_{H}=1-sh$, where $h$ is the dominance parameter
(see Table 1).

| Density | Adult genotype | Fitness
---|---|---|---
Drive Homozygote | $n_{{}_{\mathrm{DD}}}$ | D D | $1-s$
Heterozygote | $n_{{}_{\mathrm{DW}}}$ | W D | $1-sh$
Wild-type Homozygote | $n_{{}_{\mathrm{WW}}}$ | W W | $1$
Table 1:  Population characteristics (D: Drive allele, W: Wild-type allele).

All along the paper, we assume $s\in(0,1)$, corresponding to a fitness cost
carried by the drive alleles. Furthermore, we assume that the fitness of
heterozygotes cannot be greater than the fitness of either homozygote
($h\in[0,1]$).

Gene conversion turns a heterozygous cell into a drive homozygous cell. To
determine the probability $\pi_{l,k}^{i}$ (probability for a couple $l$,$k$ to
have offspring of type $i$), we need to take into account both the probability
$c\in[0,1]$ with which gene conversion occurs in heterozygotes, and the stage
of the life cycle at which it occurs: either in the zygote, or in the
germline. This last feature modifies significantly the probabilities: for
example, a couple $W$, $D$ of gametes has a probability $1-c$ to lead to
heterozygous offspring if conversion occurs in the zygote, whereas this
probability becomes one if conversion occurs in the germline. We detail all
$\pi_{l,k}^{i}$ values in Appendix A. For the sake of clarity, we now omit
variables in the notation ($n_{i}=n_{i}(t,x)$).

The parameters are summarized in Table 2.

Parameters | Range values | Description
---|---|---
$r$ | $(0,+\infty)$ | Intrinsic growth rate
$c$ | $[0,1]$ | Conversion rate
$s$ | $(0,1)$ | Fitness cost of drive homozygotes
$h$ | $[0,1]$ | Drive dominance
Table 2:  Model parameters.

In this article, we will analyse the three following versions/variations of
model (3):

Partial conversion occurring in the zygote:

$\left\\{\small\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{DD}}}-\partial_{xx}^{2}n_{{}_{\mathrm{DD}}}=(1-s)(r\
(1-n)+1)\ \dfrac{c\ n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}+2\ c\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}+(\frac{1}{2}\ c+\frac{1}{4})\
n_{{}_{\mathrm{DW}}}^{2}+(c+1)\
n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}+n_{{}_{\mathrm{DD}}}^{2}}{n}-n_{{}_{\mathrm{DD}}},\\\
\\\
\partial_{t}n_{{}_{\mathrm{DW}}}-\partial_{xx}^{2}n_{{}_{\mathrm{DW}}}=(1-sh)(r\
(1-n)+1)\ \ (1-c)\ \dfrac{n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}+2\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}+\frac{1}{2}\
n_{{}_{\mathrm{DW}}}^{2}+n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}}{n}-n_{{}_{\mathrm{DW}}},\\\
\\\ \partial_{t}n_{{}_{\mathrm{WW}}}-\partial_{xx}^{2}n_{{}_{\mathrm{WW}}}=(r\
(1-n)+1)\
\dfrac{n_{{}_{\mathrm{WW}}}^{2}+n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}+\frac{1}{4}\
n_{{}_{\mathrm{DW}}}^{2}}{n}-n_{{}_{\mathrm{WW}}}.\end{array}\right.$ (4)

Partial conversion occurring in the germline:

$\left\\{\small\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{DD}}}-\partial_{xx}^{2}n_{{}_{\mathrm{DD}}}=(1-s)(r\
(1-n)+1)\ \dfrac{\frac{1}{4}\ (1+c)^{2}\ n_{{}_{\mathrm{DW}}}^{2}+(1+c)\
n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}+n_{{}_{\mathrm{DD}}}^{2}}{n}-n_{{}_{\mathrm{DD}}},\\\
\\\
\partial_{t}n_{{}_{\mathrm{DW}}}-\partial_{xx}^{2}n_{{}_{\mathrm{DW}}}=(1-sh)(r\
(1-n)+1)\ \dfrac{(1+c)\ n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}+2\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}+\frac{1}{2}\ (1-c^{2})\
n_{{}_{\mathrm{DW}}}^{2}+(1-c)\
n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}}{n}-n_{{}_{\mathrm{DW}}},\\\ \\\
\partial_{t}n_{{}_{\mathrm{WW}}}-\partial_{xx}^{2}n_{{}_{\mathrm{WW}}}=(r\
(1-n)+1)\ \dfrac{n_{{}_{\mathrm{WW}}}^{2}+(1-c)\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}+\frac{1}{4}\ (1-c)^{2}\
n_{{}_{\mathrm{DW}}}^{2}}{n}-n_{{}_{\mathrm{WW}}}.\end{array}\right.$ (5)

Perfect conversion occurring in the zygote (no heterozygotes):

For a perfect conversion occurring in the zygote ($c=1$), model (4) reduces to
the following set of two equations, which was introduced in [20]:

$\left\\{\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{DD}}}-\partial_{xx}^{2}n_{{}_{\mathrm{DD}}}&=(1-s)\
\Big{(}r\ (1-n_{{}_{\mathrm{DD}}}-n_{{}_{\mathrm{WW}}})+1\Big{)}\
\dfrac{n_{{}_{\mathrm{DD}}}^{2}+2\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}-n_{{}_{\mathrm{DD}}}=F_{D}(n_{{}_{\mathrm{DD}}},n_{{}_{\mathrm{WW}}})\\\
\\\
\partial_{t}n_{{}_{\mathrm{WW}}}-\partial_{xx}^{2}n_{{}_{\mathrm{WW}}}&=\Big{(}r\
(1-n_{{}_{\mathrm{DD}}}-n_{{}_{\mathrm{WW}}})+1\Big{)}\
\dfrac{n_{{}_{\mathrm{WW}}}^{2}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}-n_{{}_{\mathrm{WW}}}=F_{W}(n_{{}_{\mathrm{DD}}},n_{{}_{\mathrm{WW}}}).\end{array}\right.$
(6)

This last model only follows the two homozygous genotypes, drive and wild-
type. Due to perfect gene conversion ($c=1$), no heterozygous individuals are
ever produced: heterozygous eggs are all transformed into homozygotes. Further
assuming that there are no heterozygotes initially, we only need to follow the
densities of homozygotes.

Note that system (6) can also be obtained from model (5) by assuming perfect
conversion in the germline ($c=1$) and drive dominance ($h=1$). In this case,
heterozygotes and drive homozygotes have the same fitnesses, and both only
produce gametes with the drive allele. We can then group them together and
follow their density $n_{{}_{\mathrm{DW}}}+n_{{}_{\mathrm{DD}}}$, whose
dynamics are given by the first line of (6).

###  2.2 Setting of the problem

#### Traveling waves

We seek stationary solutions in a reference frame moving at speed $v$, where
$v$ is some unknown:

$\left\\{\begin{array}[]{ll}n_{{}_{\mathrm{DD}}}(t,x)=n_{{}_{\mathrm{DD}}}(x-vt)\quad\quad(\forall
t>0)\ (\forall x\in\mathbb{R}),\\\
n_{{}_{\mathrm{DW}}}(t,x)=n_{{}_{\mathrm{DW}}}(x-vt)\quad\quad(\forall t>0)\
(\forall x\in\mathbb{R}),\\\
n_{{}_{\mathrm{WW}}}(t,x)=n_{{}_{\mathrm{WW}}}(x-vt)\quad\ \ (\forall t>0)\
(\forall x\in\mathbb{R}).\\\ \end{array}\right.$ (7)

Traveling wave solutions contain important information for the biological
interpretation of the results, such as the speed of invasion $v$, the genetic
composition of the expanding population, or the final equilibrium. In this
paper, we focus our study on this mathematical object and detail below the
vocabulary we use. Key to our analysis are the notions of monostable or
bistable systems, and whether the traveling wave is pulled or pushed. There
may be confusion around these concepts in the literature, so we clarify their
definitions below.

#### Numerical simulations

We complement our mathematical analysis with numerical simulations of the
Cauchy problem, with initial conditions for each genotype specified as in
Figure 1. The outcomes of the simulations are heatmaps of the expansion speed
over a wide range of paramaters.

Initial conditions for numerical simulations are as follows: the left half of
the domain is full of drive ($n_{{}_{\mathrm{DD}}}=1$), and the right half is
full of wild-type ($n_{{}_{\mathrm{WW}}}=1$) (see Figure 1).

Figure 1: Initial conditions for numerical simulations. The left half of the
domain is full of drive ($n_{{}_{\mathrm{DD}}}=n_{{}_{D}}=1$), and the right
half is full of wild-type ($n_{{}_{\mathrm{WW}}}=n_{{}_{W}}=1$).

The code is available on [Github]. We ran our simulations in Python 3.6, with
the Spyder environment. Heavy heatmaps 4, 5, 6(a), 6(b); 7 have been computed
thanks to the INRAE Migale bioinformatics facility (doi:
10.15454/1.5572390655343293E12). We are grateful to them for providing these
computing resources.

###  2.3 Glossary

#### Allelic densities and frequencies

For our analysis, it is convenient to introduce the allelic (half-) densities
$(n_{{}_{\mathrm{D}}},n_{{}_{\mathrm{W}}})$. The precise definition depends on
the model, and more specifically on the timing of conversion. In fact, we have
$n_{{}_{\mathrm{D}}}=n_{{}_{\mathrm{DD}}}+\alpha\ n_{{}_{\mathrm{DW}}}$ and
$n_{{}_{\mathrm{W}}}=n_{{}_{\mathrm{WW}}}+(1-\alpha)\ n_{{}_{\mathrm{DW}}}$,
with $\alpha=\frac{1}{2}$ when conversion occurs in the zygote, and
$\alpha=\frac{1+c}{2}$ when conversion occurs in the germline (see section
3.2). Depending on the regime of parameters, it may be more appropriate to
study the allelic frequencies
$p_{{}_{\mathrm{D}}}=\frac{n_{{}_{\mathrm{D}}}}{n_{{}_{\mathrm{D}}}+n_{{}_{\mathrm{W}}}}$,
$p_{{}_{\mathrm{W}}}=\frac{n_{{}_{\mathrm{W}}}}{n_{{}_{\mathrm{D}}}+n_{{}_{\mathrm{W}}}}$.

#### Classification of the dynamics

It can happen that the dynamics lead to the decay of the drive allele
uniformly in space. In this case, there cannot exist a traveling wave for the
drive population: we use the term gene drive clearance to describe this
situation. Then, the problem boils down to the standard Fisher-KPP traveling
wave problem for the expansion of the wild-type in the absence of a drive (see
[20]).

When traveling waves do exist, we distinguish between two cases depending on
the sign of the speed. When $v>0$, the wave moves to the right: it is a drive
invasion. When $v<0$, the wave moves to the left: it is a wild-type invasion.
In some specific cases, drive and wild-type invasions can happen
simultaneously: the waves decompose into two sub-traveling wave solutions over
half of the domain. They move in opposite directions and lead to the
coexistence of both alleles in-between.

In case of drive invasion, we distinguish several cases depending on the state
of the population in the wake of the front(s): i) eradication drives are those
for which the population vanishes in the wake of the front(s); ii) suppression
drives are those for which population persists in the wake of the front(s). In
the latter case, two scenarios are possible: persistence of drive homozygotes
only; persistence of all genotypes.

#### Monostable / Bistable system

To illustrate useful concepts in the theory of reaction-diffusion equations,
we consider the following standard equation of population genetics [33]
describing the dynamics of the frequency $p$ of an allele of interest:

$\partial_{t}p-\partial_{xx}^{2}p=p\ (1-p)\ \sigma(p)\quad\quad\text{with}\
p\in[0,1],$ (8)

where $\sigma(p)$ is the selection term, which we consider frequency-dependent
(i.e., function of $p$).

If $\sigma$ is of constant sign, say $\sigma>0$, this equation is referred to
as a monostable case. Then, the solution converges locally to the unique
stable equilibrium $p=1$ (or $p=0$ if $\sigma<0$). If $\sigma$ is changing
signs once in $(0,1)$, being negative below some threshold, and positive
above, it is referred to as a bistable case. In the latter case, the solution
converges locally to one of the two stable equilibria $p=0$ or $p=1$,
depending on the initial condition. Moreover, each equilibria has a basin of
attraction and there is a threshold effect – hence the name “threshold-
dependent drives” in the gene drive literature to describe this kind of case
(for example in reference [38]).

In both cases, there exist traveling waves connecting the two equilibria $p=0$
and $p=1$. A straightforward integration by parts shows that, whatever the
stability, the sign of the wave speed satisfies

$\operatorname{sign}(v)=\operatorname{sign}\left(\int_{0}^{1}p(1-p)\sigma(p)dp\right).$
(9)

In monostable cases with $\sigma>0$, this sign is positive; in bistable cases,
however, it depends on the details of the frequency-dependence $\sigma$.
Moreover, under some circumstances (bistable case, or degenerate monostable
case), the invasion outcome for the Cauchy problem can be changed by modifying
the inoculum size. Even if traveling waves exist such that $p=1$ is invading
$p=0$, small initial conditions may not succeed in propagating in space, see
the discussion in [38, 39, 30].

#### Pulled and pushed waves

Usually, a wave is said to be pulled if the wave speed coincides with the
minimal speed of the linearized problem at low density (resp. low frequency).
This occurs when the population at low density (resp. low frequency) has
sufficient reproductive success to determine the dynamics of the full
invasion.

Conversely, a wave is said to be pushed if the wave speed is strictly larger
than the minimal speed of the linearized problem. In contrast with pulled
waves, the whole population contributes to the dynamics of invasion.

A bistable wave is clearly pushed [22]. However, a monostable wave can be
either pulled or pushed, see [3, 24, 9] and discussion therein. Nonetheless, a
monostable wave is necessarily pulled if the per-capita growth rate is maximal
at low density (resp. low frequency). In the particular case of the scalar
problem (8), this criterion simply writes:

$\sigma(0)\geq(1-p)\ \sigma(p)\quad\quad(\forall p\in[0,1]).$ (10)

##  3 Results

In part 3.1, we study the model with perfect conversion in the zygote (6) and
compare the qualitative behavior of the solution when $r=0$ and $r=+\infty$.
In part 3.2, we proceed the same way on models with partial conversion (4) and
(5), obtaining more general results.

###  3.1 Model with perfect conversion in the zygote

####  3.1.1 Preliminary statements on the model

We introduce a few general results on model (6) when $r>0$, which will be
useful in the study.

For our analysis, it will be convenient to rewrite model (6) so that it
follows the frequency of the drive
$p_{{}_{\mathrm{D}}}=\frac{n_{{}_{\mathrm{D}}}}{n_{{}_{\mathrm{D}}}+n_{{}_{\mathrm{W}}}}=\frac{n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}$
(because $n_{{}_{\mathrm{DW}}}=0$) and total population density
$n=n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}$ (details in B.1):

$\left\\{\begin{array}[]{ll}\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}&=2\
\partial_{x}(\log n)\ \partial_{x}p_{{}_{\mathrm{D}}}+\left(r\ (1-n)+1\right)\
s\ p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}})\
\left(p_{{}_{\mathrm{D}}}-\dfrac{2s-1}{s}\right),\\\
\partial_{t}n-\partial_{xx}^{2}n&=\left(r\
(1-n)+1\right)\left(1-s+s(1-p_{{}_{\mathrm{D}}})^{2}\right)\
n-n.\end{array}\right.$ (11)

When $s\leq\frac{1}{2}$, system (11) is monostable: the only stable state is
$(n_{{}_{\mathrm{DD}}}=n_{{}_{\mathrm{DD}}}^{*},n_{{}_{\mathrm{WW}}}=0)$ with
$n_{{}_{\mathrm{DD}}}^{*}=\min(0,1-\frac{s}{r(1-s)})$ [20], leading to a drive
invasion if any. We introduce the minimal speed of problem (6) linearized at
low drive density, i.e. the speed of any pulled wave in case of a drive
invasion:

$2\sqrt{\partial_{n_{{}_{\mathrm{DD}}}}F_{D}(0,1)}=2\sqrt{1-2s}.$ (12)

When $s>\frac{1}{2}$, system (11) is bistable. Consequently traveling waves
are either semi-trivial ($n_{{}_{\mathrm{DD}}}=0$ identically, standard
Fisher-KPP problem for $n_{{}_{\mathrm{WW}}}$) or pushed.

System (11) differs from standard equations often used in populations genetics
as it contains an advection term $2\ \partial_{x}(\log n)\
\partial_{x}p_{{}_{\mathrm{D}}}$. This term represents a demographic flux from
denser to less dense areas, due to variations in population density. It is
opposed to the spread of the (costly) drive allele (see Figure 2 [20]).

####  3.1.2 $r=+\infty$

The limit of system (11) when $r\rightarrow+\infty$ has already been
determined in [20]. Using the Strugarek-Vauchelet rescaling [37], the
following limit equation is obtained, which was also previously introduced in
[38]:

$\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}=\dfrac{s\
p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}})\
\left(p_{{}_{\mathrm{D}}}-\dfrac{2s-1}{s}\right)}{1-s+s(1-p_{{}_{\mathrm{D}}})^{2}}.$
(13)

Interestingly, equation (13) is independent of the population density $n$ and
it does not contain the advection term $2\ \partial_{x}(\log n)\
\partial_{x}p_{{}_{\mathrm{D}}}$. This is due to the fact that the population
size $n(t,x)$ remains spatially homogeneous after the introduction of drive
individuals, when $r\to+\infty$. Intuitively, so many offspring are produced
at each generation that the carrying capacity is instantaneously restored, and
losing a fraction $s$ of these offspring by selection has no consequence.
Therefore the variations in population density ($n$), and consequently the
demographic flux, are negligible.

Equation (13) has a single parameter, the fitness cost of the drive $s$. The
numerical value of the threshold for the transition from positive to negative
speed ($\approx 0.70$) was already known [38, 20], and can be computed to
arbitrary precision by the formula (9). The numerical value of the threshold
for the transition from pulled to pushed ($\approx 0.35$ up to two digits) was
computed by a continuation method following [6, 24]

s value |  $0<s\lesssim 0.35$ |  $0.35\lesssim s<1/2$ |  $1/2<s\lesssim 0.70$ |  $0.70\lesssim s<1$
---|---|---|---|---
Stability | Monostable | Bistable
Speed |  $v=2\sqrt{1-2s}$ |  $v>2\sqrt{1-2s}$ |  $v>0$ |  $v<0$
Wave |  Pulled wave | Pushed wave |  Pushed wave
| Drive invasion |  Wild-type invasion
Invasion |  |
Table 3: Traveling wave study for Model (13). All statements in the table are
proved in Appendix C.2.

Note that equation (13) shows qualitative similarities with a common equation
in the population genetics literature [7] (and which is actually an
approximation of (13) under weak selection, i.e. $s\to 0$):

$\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}=s\
p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}})\
\left(p_{{}_{\mathrm{D}}}-\dfrac{2s-1}{s}\right).$ (14)

Quantitatively, the thresholds are $\frac{2}{5}$ instead of $0.35$, and
$\frac{2}{3}$ instead of $0.70$ (analytical values) [22].

####  3.1.3 $r=0$

When the intrinsic growth rate $r$ is finite, it is expected that the final
population density after the invasion of the drive (if any) is strictly below
$1$, because of the fitness cost. The smaller $r$, the lower the final size.
The spatial effect of demography on gene drive expansion is expected to be
maximal as $r$ vanishes, when the population can hardly restore its carrying
capacity, leading to a high amplitude of the population size gradient $2\
\partial_{x}(\log n)$. In this section we focus on the limit $r=0$, which
maximizes the demographic impact of the fitness cost on drive propagation.

In a purely wild-type population, the case $r=0$ corresponds to a number of
births balancing exactly the number of deaths. As soon as the drive allele is
introduced, this balance is locally broken, yielding a net decrease in the
population size. Then, the drive can either propagate by leaving empty space
behind, or disappear. The same conclusion holds as long as $r<\frac{s}{1-s}$,
see Section 3.1.1.

We checked numerically that the wave speed is continuous in the limit $r\to
0$. Therefore, each conclusion on the case $r=0$ sheds some light on the case
of small $r$ (see heatmap in Appendix C.1).

As discussed above, we cannot just consider a single equation on the drive
frequency $p_{{}_{\mathrm{D}}}$ when $r$ is finite because of the demographic
contribution $2\partial_{x}\log{n}$. Interestingly, in the case $r=0$, the
demographic system (6) reduces to the following pair of equations:

$\left\\{\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{WW}}}-\partial_{xx}^{2}n_{{}_{\mathrm{WW}}}&=\
\dfrac{-\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}},\\\
\\\ \partial_{t}n_{{}_{\mathrm{DD}}}-\partial_{xx}^{2}n_{{}_{\mathrm{DD}}}&=\
(1-s)\
\dfrac{n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}-s\
n_{{}_{\mathrm{DD}}}.\end{array}\right.$ (15)

Noticeably, the previous system shares some features with density-dependant
epidemiological SI models. In particular, the dynamics of
$n_{{}_{\mathrm{WW}}}$ is always decreasing. The dynamics of
$n_{{}_{\mathrm{DD}}}$ is the balance of creation and linear decay. By
changing notations $n_{{}_{\mathrm{WW}}}\leftrightarrow S$ (susceptible
individuals), and $n_{{}_{\mathrm{DD}}}\leftrightarrow I$ (infected
individuals), (15) can be recast as follows:

$\left\\{\begin{array}[]{ll}\partial_{t}S-\partial_{xx}^{2}S&=\ -\beta_{1}\
\dfrac{\ S\ I}{S+I},\\\ \\\ \partial_{t}I-\partial_{xx}^{2}I&=\ \beta_{2}\
\dfrac{S\ I}{S+I}-\gamma I.\\\ \end{array}\right.$ (16)

with $\beta_{1}=1$, $\beta_{2}=(1-s)$ (transmission parameters), and
$\gamma=s$ (disease clearance). Usually, in SI models, individuals of type $S$
are all transformed into individuals of type $I$ at infection, hence
$\beta_{1}=\beta_{2}$. In our case, these two rates are distinct because of
the fitness cost of the drive. The existence of traveling waves for model (15)
with $\beta_{1}=\beta_{2}$ has been studied recently in the literature [41].
Here, we extend the results of reference [41] to a more general case
$0<\beta_{1}$ and $0<\beta_{2}$. This leads to the characterization in Table 4
and Appendix D.

s value |  $0<s<1/2$ |  $1/2<s<1$
---|---|---
Stability |  Monostable |  Degenerate case
Speed |  $v=2\sqrt{1-2s}$ |  No wave
Wave |  Pulled wave |
|  Drive invasion |  Gene drive clearance
Invasion |  |
Table 4: Traveling waves study for Model (15). All statements in the table are
proved in Appendix C.2.

In contrast to the results obtained when $r=+\infty$, when $r=0$ there is only
one threshold value of $s$ determining the outcome of the model (Table 4).
When $0<s<1/2$, the system is monostable, the drive necessarily invades.
Moreover, the wave is pulled and travels at speed $v=2\sqrt{1-2s}$ (12). When
$1/2<s<1$, the problem is degenerate: there exists a family of steady states,
corresponding to homogeneous $n_{{}_{\mathrm{WW}}}\in[0,1]$ and
$n_{{}_{\mathrm{DD}}}=0$. It is a case of gene drive clearance, as
$n_{{}_{\mathrm{DD}}}$ converges to zero uniformly in space (at rate at least
$1-2s$). However, the final density of wild-type is not clearly determined, as
it boils down to diffusion only in the large time asymptotics (details in
Appendix C.2.1). Note that this conclusion holds in a well-mixed population
(without spatial consideration): the drive decays uniformly and the final
density of wild type depends on the initial data.

####  3.1.4 Comparison between the outcomes when $r=+\infty$ and $r=0$

The differences between the two regimes are strongest for intermediate values
of $s$. When $1/2<s\lesssim 0.70$, the drive can spread when the demographic
consequences are negligible ($r=+\infty$). However, such a costly drive cannot
invade when the intrinsic growth rate $r$ is very low ($r=0$). When
$0.35\lesssim s<1/2$, the drive wave advances for both $r=+\infty$ and $r=0$.
However, it is of different nature: the wave is pulled when $r=0$, while it is
pushed when $r=\infty$.

By providing analytical results for $r=0$, our study is complementary to [20],
where the invasion outcome was described numerically in [20, Figure 3.A]
together with a series of analytical estimates of the sign of the speed.

###  3.2 Models with partial conversion

In this section, we extend the study to models (4) and (5) with partial
conversion. The models are reformulated in terms of allelic densities
($n_{{}_{\mathrm{D}}}$ and $n_{{}_{\mathrm{W}}}$) rather than genotype
densities ($n_{{}_{\mathrm{DD}}}$, $n_{{}_{\mathrm{DW}}}$ and
$n_{{}_{\mathrm{WW}}}$). This reformulation enables reducing the number of
equations from three to two equations. Even if we are not able to determine
the genotypic composition of the population (which individual genotype a
gamete comes from, either homozygote or heterozygote), the spreading
properties are equivalent. Interestingly, the same roadmap as in the full
conversion model can be followed. Again, we focus on the two extreme regimes
$r=+\infty$ and $r=0$.

####  3.2.1 Conversion occurring in the zygote

When conversion occurs in the zygote, we can deduce the following system from
model (4), with $n_{{}_{\mathrm{D}}}=n_{{}_{\mathrm{DD}}}+\frac{1}{2}\
n_{{}_{\mathrm{DW}}}$ and
$n_{{}_{\mathrm{W}}}=n_{{}_{\mathrm{WW}}}+\frac{1}{2}\ n_{{}_{\mathrm{DW}}}$:

$\left\\{\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{D}}}-\partial_{xx}^{2}n_{{}_{\mathrm{D}}}\
=\ n_{{}_{\mathrm{D}}}\ \left[\dfrac{r\ (1-n)+1}{n}\left[\
(1-s)(n_{{}_{\mathrm{D}}}+2\ c\ n_{{}_{\mathrm{W}}})+(1-sh)\ (1-c)\
n_{{}_{\mathrm{W}}}\right]-1\right]=F^{z}_{D}(n_{{}_{\mathrm{D}}},n_{{}_{\mathrm{W}}}),\\\
\\\ \partial_{t}n_{{}_{\mathrm{W}}}-\partial_{xx}^{2}n_{{}_{\mathrm{W}}}=\
n_{{}_{\mathrm{W}}}\ \left[\dfrac{r\ (1-n)+1}{n}\left[\
n_{{}_{\mathrm{W}}}+(1-sh)\ (1-c)\
n_{{}_{\mathrm{D}}}\right]-1\right]=F^{z}_{W}(n_{{}_{\mathrm{D}}},n_{{}_{\mathrm{W}}}).\end{array}\right.$
(17)

The density $n_{{}_{\mathrm{W}}}$ (resp. $n_{{}_{\mathrm{D}}}$) corresponds to
one half of the wild-type (resp. drive) allele density at the time of zygote
formation. When conversion happens in the zygote, heterozygous individuals are
the result of conversion failures and produce one half of each type of gamete,
drive or wild-type.

Preliminary statements on the model

This model brings more variety in terms of traveling waves than the previous
one (6). Cases of monostable wild-type invasion can occur, as well as cases of
monostable coexistence. We introduce first all the possible minimal speeds of
the problem linearized at low densities and detail later under which
parameters they arise.

The minimal speed of the problem linearized at low drive density, i.e. the
speed of any pulled monostable wave with a positive speed is given by:

$2\sqrt{\partial_{n_{{}_{\mathrm{D}}}}F^{z}_{D}(0,1)}=2\sqrt{2c\
(1-s)+(1-sh)(1-c)-1}=2\sqrt{c(1-2s)-sh(1-c)},$ (18)

provided that the quantity is non-negative. The minimal speed of the problem
linearized at low wild-type density depends on the stable steady state of a
population only bearing drive alleles (i.e. a population of drive
homozygotes):
$n_{{}_{\mathrm{D}}}^{*}=n_{{}_{\mathrm{DD}}}^{*}=\min(0,1-\frac{s}{r(1-s)})$
[20]. If $n_{{}_{\mathrm{D}}}^{*}=1-\frac{s}{r(1-s)}$, this minimal speed is
given by:

$-2\sqrt{\partial_{n_{{}_{\mathrm{W}}}}F^{z}_{W}(n_{{}_{\mathrm{D}}}^{*},0)}=-2\sqrt{\frac{(1-sh)(1-c)}{1-s}-1}.$
(19)

If $n_{{}_{\mathrm{D}}}^{*}=0$, the minimal speed of the problem linearized at
low wild-type density is given by the classical Fisher-KPP formula:

$-2\sqrt{\partial_{n_{{}_{\mathrm{W}}}}F^{z}_{W}(0,0)}=-2\sqrt{r}.$ (20)

Note that (20) is the only minimal speed depending on parameter $r$: it
corresponds to the case of gene drive clearance, the only configuration where
the drive allele disappears uniformly in space.

For our analysis, it will be convenient to rewrite model (17) so that it
follows the frequency of the drive
$p_{{}_{\mathrm{D}}}=\frac{n_{{}_{\mathrm{D}}}}{n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}}$
and the total population density
$n=n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DW}}}+n_{{}_{\mathrm{DD}}}=n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}$
(details in B.2.1):

$\left\\{\small\begin{array}[]{ll}\partial_{t}n-\partial_{xx}^{2}n\
=&\big{(}r\ (1-n)+1\big{)}\ \Big{(}(1-s)\ p_{{}_{\mathrm{D}}}^{2}+2\
p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}})\ [c\ (1-s)+(1-c)\
(1-sh)]+(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n,\\\ \\\
\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\ =&2\
\partial_{x}\log(n)\ \partial_{x}p_{{}_{\mathrm{D}}}\\\ &+\big{(}r\
(1-n)+1\big{)}\Big{(}\big{[}1-2(1-c)(1-h)\big{]}\ s\
p_{{}_{\mathrm{D}}}-s[1-(1-c)(1-h)]+c(1-s)\Big{)}(1-p_{{}_{\mathrm{D}}})p_{{}_{\mathrm{D}}}.\end{array}\right.$
(21)

$r=+\infty$

Similarly as in Section 3.1.2, we compute formally the limiting equation on
(21) when $r\rightarrow+\infty$:

$\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
=\dfrac{\Big{(}\big{[}1-2(1-c)(1-h)\big{]}\ s\
p_{{}_{\mathrm{D}}}-s[1-(1-c)(1-h)]+c(1-s)\Big{)}\ (1-p_{{}_{\mathrm{D}}})\
p_{{}_{\mathrm{D}}}}{(1-s)\ p_{{}_{\mathrm{D}}}^{2}+2\ p_{{}_{\mathrm{D}}}\
(1-p_{{}_{\mathrm{D}}})\ [c\ (1-s)+(1-c)\
(1-sh)]+(1-p_{{}_{\mathrm{D}}})^{2}}.$ (22)

Note that as in Section 3.1, this equation does not depend on $n$. We
introduce:

$\mathscr{A}_{z}:=s\ \big{[}2(1-c)(1-h)-1\big{]},\quad\quad
s_{1}:=\dfrac{c}{1-h(1-c)},\quad\quad s_{2,z}:=\dfrac{c}{2c+h(1-c)}.$ (23)

where $z$ stands for zygote. Note that $\mathscr{A}_{z}>0\iff s_{1}<s_{2,z}$.

We distinguish between two cases, depending on the sign of $\mathscr{A}_{z}$.
If $\mathscr{A}_{z}>0$, which can only happen if we have both $c<\frac{1}{2}$
and $h<\frac{1}{2}$, the system is always monostable. We observe a drive
invasion for $s<s_{1}$, a coexistence state for $s_{1}<s<s_{2,z}$ and a wild-
type invasion for $s_{2,z}<s$. Criterion (10) is always verified (see Appendix
E.1), consequently every traveling wave (or sub-traveling wave in case of
coexistence) is pulled, moving at speed (18) or (19). These statements are
summarized in Table 5.

s value |  $s<s_{1}$ |  $s_{1}<s<s_{2,z}$ |  $s_{2,z}<s$
---|---|---|---
Stability | Monostable
Speed |  $v=v_{lin+}$ |  $v=v_{lin+}$ and $v=v_{lin-}$ |  $v=v_{lin-}$
Wave | Pulled Wave
|  Drive invasion |  Coexistence |  Wild-type invasion
Invasion |  |  |
Table 5: Traveling wave study for model (22) when $\mathscr{A}_{z}>0$, with
$v_{lin+}=2\sqrt{c(1-2s)-sh(1-c)}$ (18) and
$v_{lin-}=-2\sqrt{\frac{(1-sh)(1-c)}{1-s}-1}$ (19).

Now we turn to the case $\mathscr{A}_{z}<0$. The system is monostable for
$s<s_{2,z}$ (drive invasion), bistable for $s_{2,z}<s<s_{1}$ and monostable
for $s_{1}<s$ (wild-type invasion). In case of monostable drive invasion, we
define a set $\mathscr{S}_{z}$ of $s$ values:

$\mathscr{S}_{z}:=\Big{\\{}s\in(0,1)|\Big{(}1-2s[1-(1-h)(1-c)]\Big{)}\Big{(}(-2c-h+ch)s+c\Big{)}+s\
\big{[}2(1-c)(1-h)-1\big{]}>0\Big{\\}}.$ (24)

For all $s$ in $\mathscr{S}_{z}$ and $\mathscr{A}_{z}<0$, criterion (10) is
verified and consequently, there exists a pulled monostable traveling wave
with positive speed (see Appendix E.1). Note that such $s$ values are
necessarily strictly below $s_{2,z}$, condition for a monostable drive
invasion. In case of wild-type invasion, criterion (10) is never verified (see
Appendix E.1). These statements are summarized in Table 6.

s value |  $s\in\mathscr{S}_{z}$ |  $s\in(0,s_{2,z})\setminus\mathscr{S}_{z}$ |  $s_{2,z}<s<s_{1}$ |  $s_{1}<s<1$
---|---|---|---|---
Stability | Monostable |  Bistable |  Monostable
Speed |  $v=v_{lin+}$ |  $v\geq v_{lin+}$ |  |  $v<0$
Wave |  Pulled wave |  |  Pushed wave |
| Drive invasion |  |  Wild-type invasion
Invasion |  |  Drive or Wild-type invasion |
Table 6: Traveling wave study for model (22) when $\mathscr{A}_{z}<0$, with
$v_{lin+}=2\sqrt{c(1-2s)-sh(1-c)}$ (18).

$r=0$

Using the relation
$n=n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DW}}}+n_{{}_{\mathrm{DD}}}=n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}$,
system (17) can be rewritten as follows when $r=0$:

$\left\\{\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{D}}}-\partial_{xx}^{2}n_{{}_{\mathrm{D}}}\
=\ \Big{(}c\ (1-s)+s\ (1-c)\ (1-h)\Big{)}\ \dfrac{n_{{}_{\mathrm{D}}}\
n_{{}_{\mathrm{W}}}}{n_{{}_{\mathrm{D}}}+n_{{}_{\mathrm{W}}}}-s\
n_{{}_{\mathrm{D}}},\\\ \\\
\partial_{t}n_{{}_{\mathrm{W}}}-\partial_{xx}^{2}n_{{}_{\mathrm{W}}}=\
-\Big{(}1-(1-sh)\ (1-c)\Big{)}\ \dfrac{n_{{}_{\mathrm{D}}}\
n_{{}_{\mathrm{W}}}}{n_{{}_{\mathrm{D}}}+n_{{}_{\mathrm{W}}}}.\end{array}\right.$
(25)

We apply the results of Appendix D with $\beta_{1}=1-(1-sh)\ (1-c)$ and
$\beta_{2}=c\ (1-s)+s\ (1-c)\ (1-h)$. There exists a monostable and pulled
drive invasion wave if:

$\beta_{2}>\gamma\ \ \iff\ \ s<s_{2,z}=\dfrac{c}{2c+h(1-c)}.$ (26)

On the other hand when $\beta_{2}<\gamma$, the reaction term of
$n_{{}_{\mathrm{D}}}$ in (25) is strictly negative. As before, the density
$n_{{}_{\mathrm{D}}}$ converges to zero uniformly in space at rate at least
$\beta_{2}-\gamma$ (gene drive clearance) and the final density of wild-type
is not clearly determined: the problem boils down to diffusion only in the
large time asymptotics (details in Appendix C.2.1). These statements are
summarized in Table 7.

s value |  $0<s<s_{2,z}$ |  $s_{2,z}<s<1$
---|---|---
Stability |  Monostable |  Degenerate case
Speed |  $v=v_{lin+}$ |  No wave
Wave |  Pulled wave |
|  Drive invasion |  Gene drive clearance
Invasion |  |
Table 7: Traveling wave study for model (25), with
$v_{lin+}=2\sqrt{c(1-2s)-sh(1-c)}$ (18).

Note that, when $\mathscr{A}_{z}>0$, a condition for having a pulled wave with
positive speed for both $r=0$ and $r=+\infty$ is $s<s_{2,z}$. When
$\mathscr{A}_{z}<0$, a condition for having a pulled wave with positive speed
for both $r=0$ and $r=+\infty$ is $s\in\mathscr{S}_{z}\subseteq[0,s_{2,z}]$.
This suggests that, under those conditions, whatever the value of the
demographic parameter $r$ is, the drive invasion wave is always pulled and
consequently, travels at a speed which does not depend on $r$ either (speed
given by (18)). We verify this intuition numerically (vertical level lines) in
the following section.

Numerical illustrations

$\underline{\mathscr{A}_{z}>0}$

In a first example we choose $c=0.25$ and $h=0.1$ such that
$\mathscr{A}_{z}>0$. The $s$ threshold values are $s_{1}\approx 0.27$ and
$s_{2,z}\approx 0.43$. As discussed in the previous sections, when
$s<s_{2,z}$, all waves are pulled (sub-)traveling waves for $r=+\infty$ and
$r=0$. Note that $s>s_{1}$ is the condition for the existence of pulled
(sub-)traveling waves with negative speed only when $r=+\infty$.

We show the value of the speed (18) and (19) of the pulled waves as a function
of $s$ when $r=+\infty$, with $c=0.25$ and $h=0.1$ (in Figure 2).

Figure 2: Speed of the drive invasion (18) in orange and speed of the wild-
type invasion (19) in blue, as a function of $s$ when $r=+\infty$, with
$c=0.25$ and $h=0.1$.

Note that for $0.27\approx s_{1}<s<s_{2,z}\approx 0.43$, we have both a drive
and a wild-type invasion, leading to a stable coexistence state. This case is
illustrated in Figure 3 with $s=0.35$.

(a) $t=0$

(b) $t=1000$

(c) $t=2000$

Figure 3: Allele densities as a function of space, at different times, for
$s=0.35$, $c=0.25$, $h=0.1$ and $r=3$.

We now compute numerically the values of the wave speed for intermediate
values of $r$ (in Figure 4). In case of coexistence, for $s_{1}<s<s_{2,z}$, we
choose to show only the positive speed value.

Figure 4: Heatmap representing the speed of the waves for $c=0.25$, $h=0.1$
when conversion occurs in the zygote. When the drive invades the population,
the speed is positive (in yellow-orange). On the contrary, when the wild-type
invades the population, the speed is negative (in blue). When both drive and
wild types invade (coexistence), only the speed of the drive is shown in the
heatmap, resulting in an apparent discontinuity at $s=s_{2,z}$. As
$\mathscr{A}_{z}>0$, the system is always monostable for $r=+\infty$: when
$s<s_{1}$ the drive always invades; when $s_{1}<s<s_{2,z}$ the final state is
a coexistence state; when $s>s_{2,z}$ the wild-type invades or there is gene
drive clearance. The turquoise horizontal lines at the bottom and at the top
of the heatmap indicate the theoretical values of $s$ such that there exists a
pulled wave with positive speed, respectively for $r=+\infty$ and $r=0$. Below
the pure drive persistence line (light green), a well-mixed population
containing only drive homozygous individuals will necessarily get extinct.
Below the composite persistence line (dark green), it is the whole population
that gets extinct (calculations for both lines available in Appendix F.2). The
gray zone corresponds to the gene drive clearance area. Outside the gray zone,
the level lines are apparently vertical, meaning that the wave speed would be
independent of $r$. This is in agreement with the fact that the values of the
speed coincide when $r=+\infty$ and $r=0$ for $s<s_{2,z}$. If correct, the
value of the speed can be found in Figure 2.

For a better understanding of Figure 4, we detail the effect of fitness
disadvantage $s$ and dominance coefficient $h$ on drive dynamics for
$r=+\infty$ and $c=0.25$, without spatial structure, in Appendix F.1 (in
Figure 13(a)).

In Figure 4, the speed value for $s<s_{2,z}$ seems not to depend on the
demographic parameter $r$: whatever the final equilibrium is, going from
population extinction to full replacement of the wild-type genotypes by drive
genotypes, the invasion occurs at the same speed. This is in agreement with
the fact that the values of the speed coincide when $r=+\infty$ and $r=0$ for
$s<s_{2,z}$. If correct, the value of the speed can be found in Figure 2.

$\underline{\mathscr{A}_{z}<0}$

In a second example, we choose $c=0.75$ and $h=0.1$ such that
$\mathscr{A}_{z}<0$. The $s$ threshold values are $s_{1}\approx 0.77$ and
$s_{2,z}\approx 0.49$. As discussed in the previous section, when
$s\in\mathscr{S}_{z}$ (in our case $s\lesssim 0.38$), all waves are pulled
traveling waves. However, the latter criterion is not a sufficient condition.
It is expected that waves are indeed pulled beyond this approximate value of
$0.38$. However, this would require to use numerical continuation methods as
in Section 3.1.2. We computed numerically the speed values for intermediate
values of $r$, as shown in Figure 5. We believe that the wave speed is
independent of the demographic parameter $r$ when the wave is pulled (visual
observation for $s\lesssim 0.38$).

Figure 5: Heatmap representing the speed of the wave for $c=0.75$, $h=0.1$
when conversion occurs in the zygote. When the drive invades the population,
the speed is positive (in yellow-orange-red). On the contrary, when the wild-
type invades the population, the speed is negative (in blue). We have
$\mathscr{A}_{z}<0$, therefore when $r=+\infty$: when $s<s_{2,z}$ the system
is monostable and the drive always invades; when $s_{2,z}<s<s_{1}$ the system
is bistable and the final state depends on the initial condition; when
$s>s_{1}$ the system is monostable and the wild-type invades or there is gene
drive clearance. The turquoise horizontal lines at the bottom and at the top
of the heatmap indicate the theoretical values of $s$ such that there exists a
pulled wave with positive speed, respectively for $r=+\infty$ and $r=0$. Below
the pure drive persistence line (light green), a well-mixed population
containing only drive homozygous individuals will necessarily get extinct
(calculations for this line available in Appendix F.2). For
$s\in\mathscr{S}_{z}$, i.e. $s\lesssim 0.38$,

the level lines are apparently vertical: this is in agreement with the fact
that the values of the speed coincide when $r=+\infty$ and $r=0$ in this area.

For a better understanding of Figure 5, we detail the effect of fitness
disadvantage $s$ and dominance coefficient $h$ on drive dynamics for
$r=+\infty$ and $c=0.75$, without spatial structure, in Appendix F.1 (in
Figure 13(b)).

####  3.2.2 Conversion occurring in the germline

When conversion occurs in the germline, we can deduce the following system
from model (5), with $n_{{}_{\mathrm{D}}}=n_{{}_{\mathrm{DD}}}+(1+c)\
\frac{1}{2}\ n_{{}_{\mathrm{DW}}}$ and
$n_{{}_{\mathrm{W}}}=n_{{}_{\mathrm{WW}}}+(1-c)\ \frac{1}{2}\
n_{{}_{\mathrm{DW}}}$:

$\left\\{\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{D}}}-\partial_{xx}^{2}n_{{}_{\mathrm{D}}}\
=\ n_{{}_{\mathrm{D}}}\ \Big{[}\dfrac{r\
(1-n)+1}{n}\Big{[}(1-s)n_{{}_{\mathrm{D}}}+(1-sh)\ (1+c)\
n_{{}_{\mathrm{W}}}\Big{]}-1\Big{]}=F^{g}_{D}(n_{{}_{\mathrm{D}}},n_{{}_{\mathrm{W}}}),\\\
\\\ \partial_{t}n_{{}_{\mathrm{W}}}-\partial_{xx}^{2}n_{{}_{\mathrm{W}}}=\
n_{{}_{\mathrm{W}}}\ \Big{[}\dfrac{r\ (1-n)+1}{n}\Big{[}\
n_{{}_{\mathrm{W}}}+(1-sh)\ (1-c)\
n_{{}_{\mathrm{D}}}\Big{]}-1\Big{]}=F^{g}_{W}(n_{{}_{\mathrm{D}}},n_{{}_{\mathrm{W}}}).\end{array}\right.$
(27)

The density $n_{{}_{\mathrm{W}}}$ (resp. $n_{{}_{\mathrm{D}}}$) corresponds to
one half of the wild-type (resp. drive) allele density at the time of zygote
formation. When conversion happens in the germline, heterozygous individuals
undergo a conversion of their wild-type alleles with probability $c$, and
produce a fraction $(1+c)/2$ of drive-carrying gametes.

Preliminary statements on the model

As before, we detail the the minimal speed of the problem linearized at low
densities, for both drive and wild-type alleles.

In case of drive invasion, the minimal speed of the problem linearized at low
drive density, i.e. the speed of any pulled monostable wave with positive
speed is given by:

$2\sqrt{\partial_{n_{{}_{\mathrm{D}}}}F^{g}_{D}(0,1)}=2\sqrt{(1-sh)(1+c)-1}.$
(28)

Note that
$F^{g}_{W}(n_{{}_{\mathrm{D}}},n_{{}_{\mathrm{W}}})=F^{z}_{W}(n_{{}_{\mathrm{D}}},n_{{}_{\mathrm{W}}})$:
in case of a wild-type invasion, the minimal speeds are already given by (19)
and (20) (Section 3.2.1).

For our analysis, it will be convenient to rewrite model (27) so that it
follows the frequency of the drive
$p_{{}_{\mathrm{D}}}=\frac{n_{{}_{\mathrm{D}}}}{n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}}$
and the total population density
$n=n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DW}}}+n_{{}_{\mathrm{DD}}}=n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}$
(details in Appendix B.2.2):

$\left\\{\begin{array}[]{ll}\partial_{t}n-\partial_{xx}^{2}n&=\big{(}r\
(1-n)+1\big{)}\ \Big{(}(1-s)\ p_{{}_{\mathrm{D}}}^{2}+2\ (1-sh)\
p_{{}_{\mathrm{D}}}\
(1-p_{{}_{\mathrm{D}}})+(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n,\\\ \\\
\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}&=2\
\partial_{x}\log(n)\ \partial_{x}p_{{}_{\mathrm{D}}}+\big{(}r\
(1-n)+1\big{)}\Big{(}(2h-1)\ s\ p_{{}_{\mathrm{D}}}+(1-sh)(1+c)-1\Big{)}\
p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}}).\end{array}\right.$ (29)

$r=+\infty$

Similarly as in Section 3.1.2, we can compute formally the limiting equation
on $p_{{}_{\mathrm{D}}}$ when $r=+\infty$:

$\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
=\dfrac{\Big{(}-(1-2h)\ s\ p_{{}_{\mathrm{D}}}+[(1-sh)(1+c)-1]\Big{)}\
p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}})}{(1-s)\
p_{{}_{\mathrm{D}}}^{2}+2\ (1-sh)\ p_{{}_{\mathrm{D}}}\
(1-p_{{}_{\mathrm{D}}})+(1-p_{{}_{\mathrm{D}}})^{2}}.$ (30)

Note that as in section 3.1, this equation does not depend on $n$. We
introduce:

$\mathscr{A}_{g}:=s\ (1-2h),\quad\quad s_{1}:=\dfrac{c}{1-h(1-c)},\quad\quad
s_{2,g}:=\dfrac{c}{2ch+h(1-c)}=\dfrac{c}{h(1+c)}.$ (31)

where $g$ stands for germline. Note that $\mathscr{A}_{g}>0\iff
s_{1}<s_{2,g}$. We define a set $\mathscr{S}_{g}$ of $s$ values:

$\mathscr{S}_{g}:=\Big{\\{}s\in(0,1)|(1-2sh)\big{(}c-sh(c+1)\big{)}+s(1-2h)>0\Big{\\}}.$
(32)

Results are exactly the same as in Section 3.2.1, substituting
$\mathscr{A}_{z}$ by $\mathscr{A}_{g}$, $s_{2,z}$ by $s_{2,g}$,
$\mathscr{S}_{z}$ by $\mathscr{S}_{g}$, and the minimal speed of the problem
linearized at low drive density (18) by (28) (see Appendix E.2).

$r=0$

Using the relation $n=n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}$, system (27)
can be rewritten as follows when $r=0$:

$\left\\{\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{D}}}-\partial_{xx}^{2}n_{{}_{\mathrm{D}}}\
=\ \Big{(}c\ (1-sh)+s\ (1-h)\Big{)}\ \dfrac{n_{{}_{\mathrm{D}}}\
n_{{}_{\mathrm{W}}}}{n_{{}_{\mathrm{D}}}+n_{{}_{\mathrm{W}}}}-s\
n_{{}_{\mathrm{D}}},\\\ \\\
\partial_{t}n_{{}_{\mathrm{W}}}-\partial_{xx}^{2}n_{{}_{\mathrm{W}}}=\
-\Big{(}1-(1-sh)\ (1-c)\Big{)}\ \dfrac{n_{{}_{\mathrm{D}}}\
n_{{}_{\mathrm{W}}}}{n_{{}_{\mathrm{D}}}+n_{{}_{\mathrm{W}}}}.\end{array}\right.$
(33)

We apply the results of Appendix D with $\beta_{1}=1-(1-sh)\ (1-c)$ and
$\beta_{2}=c\ (1-sh)+s\ (1-h)$. There exists a monostable and pulled drive
invasion wave if:

$\beta_{2}>\gamma\ \ \iff\ \
s<s_{2,g}=\dfrac{c}{h(1+c)}=\dfrac{c}{2ch+h(1-c)}.$ (34)

On the other hand when $\beta_{2}<\gamma$, the reaction term of $n_{D}$ in
(33) is strictly negative. As before, the density $n_{{}_{\mathrm{D}}}$
converges to zero uniformly in space at rate at least $\beta_{2}-\gamma$ (gene
drive clearance) and the final density of wild-type is not clearly determined:
the problem boils down to diffusion only in the large time asymptotics
(details in Appendix C.2.1).

Note that the same intuitions as in section 6 hold: when $\mathscr{A}_{g}>0$,
a condition for having a pulled wave with positive speed for both $r=0$ and
$r=+\infty$ is $s<s_{2,g}$ ; when $\mathscr{A}_{g}<0$, a condition for having
a pulled wave with positive speed for both $r=0$ and $r=+\infty$ is
$s\in\mathscr{S}_{g}\subseteq[0,s_{2,g}]$. This suggests that, under those
conditions, whatever the value of the demographic parameter $r$ is, the drive
invasion wave is always pulled and consequently, travels at a speed which does
not depend on $r$ either (speed given by (28)). As before, we verify this
intuition numerically (vertical level lines) in the following section.

Numerical illustrations

(a) $c=0.25$, $h=0.3$

(b) $c=0.25,h=0.75$

Figure 6: Heatmap representing the speed of the wave when conversion occurs in
the germline. When the drive invades the population, the speed is positive (in
yellow-orange-red). On the contrary, when the wild-type invades the
population, the speed is negative (in blue). Below the pure drive persistence
line (light green), a well-mixed population containing only drive homozygous
individuals will necessarily get extinct. Below the composite persistence line
(dark green), it is the whole population that gets extinct (calculations for
both lines available in Appendix F.2). The gray zone corresponds to the gene
drive clearance area. In (a) we have $\mathscr{A}_{g}>0$, therefore the system
is always monostable for $r=+\infty$: when $s<s_{1}$ the drive always invades;
when $s_{1}<s<s_{2,g}$ the final state is a coexistence state; when
$s>s_{2,g}$ the wild-type invades or there is gene drive clearance. When both
drive and wild types invade (coexistence), only the speed of the drive is
shown in the heatmap, resulting in an apparent discontinuity at $s=s_{2,z}$.
In (b) we have $\mathscr{A}_{z}<0$, therefore when $r=+\infty$: when
$s<s_{2,g}$ the system is monostable and the drive always invades; when
$s_{2,g}<s<s_{1}$ the system is bistable and the final state depends on the
initial condition; when $s>s_{1}$ the system is monostable and the wild-type
invades or there is gene drive clearance.

For a better understanding of Figures 6(a) and 6(b), we detail the effect of
fitness disadvantage $s$ and dominance coefficient $h$ on drive dynamics for
$r=+\infty$ and $c=0.25$, without spatial structure, in Appendix F.1 (in
Figure 14(a)).

####  3.2.3 Conclusion

When conversion occurs in the zygote (resp. in the germline) for
$\mathscr{A}_{z}<0$ (resp. $\mathscr{A}_{g}<0$), demographics influence the
speed of the drive propagation at least for $s\in(s_{2,z},s_{1})$ (resp.
$s\in(s_{2,g},s_{1})$). More precisely, the sign of the speed can switch,
changing the type of the invasion (drive or wild-type). When
$\mathscr{A}_{z}>0$ (resp. $\mathscr{A}_{g}>0$) however, a model following
only frequencies will always predict the correct speed of expansion.However, a
model following only frequencies will not provide information on population
size, and in particular whether the population is suppressed or eradicated,
while this point is of great biological relevance.

For both zygote and germline conversion timings, the critical values of
$\mathscr{A}_{z}$ and $\mathscr{A}_{g}$ can be interpreted as the values at
which the fitness of adults who were born heterozygous ($f_{H}^{\prime}$) is
the arithmetic mean of the fitness of adults born homozygote
($(f_{D}+f_{W})/2$). The fitness of adults who were born heterozygous depends
on the timing of gene conversion. For germline conversion,
$f_{H}^{\prime}=f_{H}$, and $\mathscr{A}_{g}=0$ when $h=1/2$, i.e. when there
is co-dominance between the drive and wild-type alleles, i.e. when
$f_{H}^{\prime}=f_{H}=(f_{D}+f_{W})/2$. For zygote conversion, the fitness of
adults born heterozygous depends on whether gene conversion has taken place or
not ($f_{H}^{\prime}=(1-c)(1-hs)+c(1-s)$). The condition $\mathscr{A}_{z}=0$
is equivalent to $(1-c)(1-h)=1/2$, which happens for
$f_{H}^{\prime}=(f_{D}+f_{W})/2$.

##  4 Discussion

Following [20], we quantified the impact of demography in the case of the
propagation of a super-Mendelian drive. We extended the analysis of reference
[20] to the case of partial conversion ($0<c<1$), implying the presence of
heterozygotes.

### On the final state of the population

The final size of the population naturally varies. In case where no wild type
can survive, the final size is the same regardless of the details of gene
conversion(timing nor probability):
$n_{{}_{\mathrm{D}}}^{*}=\min(0,1-\frac{s}{r(1-s)})$. In case of coexistence
between wild-type and drive alleles, the final size depends on all parameters
(see Appendix F.2.2). Interestingly, in the case of coexistence, the drive
allele can persist in the population even if a pure drive population would not
($n_{{}_{\mathrm{D}}}^{*}=0$), see Figure 4 (note the area between the
composite persistence line and the pure drive persistence line). In contrast
with standard Mendelian genetics (corresponding to $c=0$), coexistence can
occur even if the dominance parameter is such that $h\in(0,1)$ [15, 35]. More
precisely, when conversion is partial and, either $h<1-\frac{1}{2(1-c)}$
(zygote conversion), or $h<\frac{1}{2}$ (germline conversion), there exists a
stable coexistence state if $s$ takes intermediate values
$s\in(s_{1},s_{2,z})$ (zygote conversion), or $s\in(s_{1},s_{2,g})$ (germline
conversion), where $s_{1},s_{2,z},s_{2,g}$ depend on $(c,h)$ but do not depend
on the demographic parameter $r$ (see details in Section 3.2.1 and 3.2.2).
While the final size of the population naturally depends on $r$.

### On the transient regime (propagation of waves)

In order to evaluate the impact of demography on the dynamics of drive
expansion, we compared the extreme cases $r\to\infty$ and $r\to 0$ (resp. low
demographic variations versus large demographic variations).

For $r=0$, we found that, when the drive propagates, it does so through a
monostable and pulled wave. This happens when the drive is not too costly. In
contrary, the drive gets uniformly extinct if it is too costly. The threshold
on the fitness cost $s_{2,z}$ (zygote conversion), or $s_{2,g}$ (germline
conversion), depends on $(c,h)$. The situation is analogous to the spatial
spreading of an epidemic following a SI type model.

The case $r=0$ gives the possibility to measure the importance of the
demographic advection term $2\partial_{x}\log(n)\partial_{x}p$ when the
problem is formulated in frequency, see equations (11), (21), (29). In fact,
we show that ignoring this term can lead to an overestimation of the wave
speed. This happens, for instance, in case of perfect conversion in the
zygote, when $s\in(\frac{2}{5},\frac{1}{2})$, then the equation (11) without
$2\partial_{x}\log(n)\partial_{x}p$ would lead to a pushed front with velocity
$\frac{2-3s}{\sqrt{2s}}$ [22]. However, we proved that the front is actually
pulled with velocity $2\sqrt{1-2s}<\frac{2-3s}{\sqrt{2s}}$. Intuitively,
advection due to demographic variations slows down the expansion of the bulk.
Noticeably, the effect is so strong that it prevents the front from being
pushed.

In contrast, for $r=\infty$, the analysis formally boils down to a single
equation on the drive allele frequency [37, 20]. According to [35], where the
case of germline conversion was investigated, there is a large panel of
frequency-dependence relationships, including monostable fixation of one
allele, bi-stability, and stable coexistence between the two alleles, even if
$h\in(0,1)$. This leads to a variety of propagation phenomena, either pulled
or pushed, as described in Section 3.2.2. The same panel of relationships
arises in the case of zygote conversion, with qualitative similarities but
quantitative differences in the thresholds and in the wave speeds, compare
Section 3.2.2 with Section 3.2.1.

To connect $r=0$ and $r=\infty$, we conjecture that, if the wave of the drive
is pulled at $r=\infty$, then it is pulled for any value of $r>0$, and the
wave speed is independent of $r$. In particular, this occurs when the
frequency-dependence term induces monostable dynamics and $s$ is small enough,
or when there is stable co-existence. This conjecture is supported by
numerical investigations (Figures 4, 5, 6(a) and 6(b)). Still, the final size
of population naturally depends on $r$.

### Perspectives

We have focused on the classical dichotomy between pulled and pushed waves,
even if the transition between the two is subject to current research both in
theoretical studies [2, 6, 9], and in experimental works [14].

Pulled and pushed waves are associated with different outcomes on the
maintenance of neutral diversity (which was not considered in our study). The
genetic diversity of a population expanding by a pulled wave is very limited
(with possible accumulation of deleterious mutations [34]), while more
diversity is maintained under a pushed wave [36]. It could be interesting to
investigate how gene conversion influences the maintenance of neutral
diversity along an expanding wave. More generally, the bottleneck following
spread of a suppression drive will affect neutral diversity, which may have
long-lasting consequences even if wild-type individuals later recolonize the
area.

It would be highly relevant to explore stochastic dynamics beyond our
deterministic approach. When population sizes get to be small, as in the drive
eradication case, large fluctuations and even chasing events are expected, as
described in [12]. It would also be extremely interesting to extend the scope
of the model, including by distinguishing between males and females which may
have different fitnesses (especially in transgenic mosquitoes [8, 32, 26,
23]). Plural life stages or haploid phases might also influence modeling
conclusions [28, 27].

##  5 Acknowledgements

This work is funded by ANR-19-CE45-0009-01 "TheoGeneDrive".

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programm
(grant agreement No 865711).

We are grateful to the INRAE MIGALE bioinformatics facility (MIGALE, INRAE,
2020. Migale bioinformatics Facility, doi: 10.15454/1.5572390655343293E12) for
providing computing resources.

Appendix

##  Appendix A Model with partial conversion: growth term details

To obtain the global growth term for each genotypes in models (4) and (5), we
calculate type proportions among the offspring for each possible couple, and
then we sum the corresponding terms. The calculations follow standard lines of
population genetics, differing only by the timing of gene conversion. When
conversion occurs in the zygote, the parameter $c$ appears in between the
gametes and the offspring production, whereas when conversion occurs in the
germline, it appears before gametes production. These equations in densities
are consistent with the one obtained in frequency in the literature (when
conversion occurs in the zygote [15, 40], or in the germline [15, 35]).

###  A.1 Conversion occurring in the zygote

Parents |  Egg Adult |  Fitness |  Growth term
---|---|---|---
WW + WW |  WWWW$1$$1$ |  1 |  $\dfrac{n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{WW}}}}{n}$
WW + WD |  WDDDWDWWWW$\frac{1}{2}$$c$$1-c$$\frac{1}{2}$$1$ |  $1-s$$1-sh$$1$ |  $\frac{1}{2}\ c\ \ (1-s)\ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}}{n}$$\frac{1}{2}\ (1-c)\ \ (1-sh)\ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}}{n}$$\frac{1}{2}\ \ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}}{n}$
WW + DD |  WDDDWD$1$$c$$1-c$ |  $1-s$$1-sh$ |  $c\ \ (1-s)\ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n}$$(1-c)\ \ (1-sh)\ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n}$
WD + WD |  WWWWWDDDWDDDDD$\frac{1}{4}$$1$$\frac{1}{2}$$c$$1-c$$\frac{1}{4}$$1$ |  $1$$1-s$$1-sh$$1-s$ |  $\frac{1}{4}\ \ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$$\frac{1}{2}\ c\ \ (1-s)\ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$$\frac{1}{2}\ (1-c)\ \ (1-sh)\ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$$\frac{1}{4}\ \ (1-s)\ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$
WD + DD |  WDDDWDDDDD$\frac{1}{2}$$c$$1-c$$\frac{1}{2}$$1$ |  $1-s$$1-sh$$1-s$ |  $\frac{1}{2}\ c\ \ (1-s)\ \dfrac{2n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}}{n}$$\frac{1}{2}\ (1-c)\ \ (1-sh)\ \dfrac{2n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}}{n}$$\frac{1}{2}\ \ (1-s)\ \dfrac{2n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}}{n}$
DD + DD |  DDDD$1$$1$ |  $1-s$ |  $(1-s)\ \dfrac{n_{{}_{\mathrm{DD}}}n_{{}_{\mathrm{DD}}}}{n}$
Table 8: Growth term details when conversion occurs in the zygote.

###  A.2 Conversion occurring in the germline

Parents |  Gametes Adult |  Fitness |  Growth term
---|---|---|---
WW + WW |  W,W + W,WWW$1$$1$ |  $1$ |  $\dfrac{n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{WW}}}}{n}$
WW + WD |  W,W + D,DWDWWW,W + W,DWD$c$$1$$\frac{1}{2}$$1-c$$\frac{1}{2}$ |  $1-sh$$1$$1-sh$ |  $c\ \ (1-sh)\ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}}{n}$$(1-c)\ \frac{1}{2}\ \ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}}{n}$$(1-c)\ \frac{1}{2}\ \ (1-sh)\ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DW}}}}{n}$
WW + DD |  W,W + D,DWD$1$$1$ |  $1-sh$ |  $(1-sh)\ \dfrac{2n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n}$
WD + WD |  D,D + D,DDDD,D + W,DDDWDW,D + W,DDDWDWW$c^{2}$$1$$2c(1-c)$$\frac{1}{2}$$\frac{1}{2}$$(1-c)^{2}$$\frac{1}{4}$$\frac{1}{2}$$\frac{1}{4}$ |  $1-s$$1-s$$1-sh$$1-s$$1-sh$$1$ |  $c^{2}\ \ (1-s)\ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$$c\ (1-c)\ \ (1-s)\ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$$c\ (1-c)\ \ (1-sh)\ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$$(1-c)^{2}\ \frac{1}{4}\ \ (1-s)\ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$$(1-c)^{2}\ \frac{1}{2}\ \ (1-sh)\ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$$(1-c)^{2}\ \frac{1}{4}\ \ \dfrac{n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DW}}}}{n}$
WD + DD |  D,D + D,DDDWDW,D + D,DDD$c$$1$$\frac{1}{2}$$1-c$$\frac{1}{2}$ |  $1-s$$1-sh$$1-s$ |  $c\ \ (1-s)\ \dfrac{2n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}}{n}$$(1-c)\ \frac{1}{2}\ \ (1-sh)\ \dfrac{2n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}}{n}$$(1-c)\ \frac{1}{2}\ \ (1-s)\ \dfrac{2n_{{}_{\mathrm{DW}}}n_{{}_{\mathrm{DD}}}}{n}$
DD + DD |  D,D + D,DDD$1$$1$ |  $1-s$ |  $(1-s)\ \dfrac{n_{{}_{\mathrm{DD}}}n_{{}_{\mathrm{DD}}}}{n}$
Table 9: Growth term details when conversion occurs in the germline.

##  Appendix B System rewritten with variables $(n,p_{{}_{\mathrm{D}}})$

Below, we present the details of the reformulation from models (6), (17) and
(27) in terms of total population density $n$ and drive allele frequency
$p_{{}_{\mathrm{D}}}$.

###  B.1 Model with perfect conversion

We rewrite model (6) with variables:

$n=n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}},\quad\quad\quad\quad
p_{{}_{\mathrm{D}}}=\dfrac{n_{{}_{\mathrm{D}}}}{n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}}=\dfrac{n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}.$
(35)

where $n$ is the total population density and $p_{{}_{\mathrm{D}}}$ is the
drive allele frequency, or equivalently in this model, the frequency of drive
homogeneous individuals.

Equation on n:

$\displaystyle\partial_{t}n-\partial_{xx}^{2}n$ $\displaystyle=\Big{(}r\
(1-n)+1\Big{)}\ \Big{(}(1-s)\ \dfrac{n_{{}_{\mathrm{DD}}}^{2}+2\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}+\
\dfrac{n_{{}_{\mathrm{WW}}}^{2}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}\Big{)}-n,$
(36) $\displaystyle=\Big{(}r\ (1-n)+1\Big{)}\ \Big{(}(1-s)\
p_{{}_{\mathrm{D}}}^{2}+(1-s)\ 2\
p_{{}_{\mathrm{D}}}(1-p_{{}_{\mathrm{D}}})+(1-s)(1-p_{{}_{\mathrm{D}}})^{2}+s(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n,$
$\displaystyle=\Big{(}r\ (1-n)+1\Big{)}\
\Big{(}(1-s)(p_{{}_{\mathrm{D}}}+1-p_{{}_{\mathrm{D}}})^{2}+s(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n,$
$\displaystyle=\Big{(}r\ (1-n)+1\Big{)}\
\Big{(}(1-s)+s(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n,$ $\displaystyle=\Big{(}r\
(1-n)+1\Big{)}\ \Big{(}(1-s)\ p_{{}_{\mathrm{D}}}\
(2-p_{{}_{\mathrm{D}}})+(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n.$

Equation on $\mathbf{n_{{}_{\mathrm{DD}}}}$:

$\displaystyle\partial_{t}n_{{}_{\mathrm{DD}}}$
$\displaystyle=\partial_{xx}^{2}n_{{}_{\mathrm{DD}}}\ +\ (1-s)\ \Big{(}r\
(1-n_{{}_{\mathrm{DD}}}-n_{{}_{\mathrm{WW}}})+1\Big{)}\
\dfrac{n_{{}_{\mathrm{DD}}}^{2}+2\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}-n_{{}_{\mathrm{DD}}},$
(37) $\displaystyle\stackrel{{\scriptstyle}}{{=}}p_{{}_{\mathrm{D}}}\
\partial_{xx}^{2}n+2\ \partial_{x}n\ \partial_{x}p_{{}_{\mathrm{D}}}+n\
\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\ +\ (1-s)\ \Big{(}r\ (1-n)+1\Big{)}\
(2-p_{{}_{\mathrm{D}}})\ n\ p_{{}_{\mathrm{D}}}-n\ p_{{}_{\mathrm{D}}}.$

Equation on $\mathbf{p_{{}_{\mathrm{D}}}=\dfrac{n_{{}_{\mathrm{DD}}}}{n}}$:

$\displaystyle\partial_{t}p_{{}_{\mathrm{D}}}$ $\displaystyle=\dfrac{n\
(\partial_{t}n_{{}_{\mathrm{DD}}})-n_{{}_{\mathrm{DD}}}\
(\partial_{t}n)}{n^{2}}=\dfrac{n\ (\partial_{t}n_{{}_{\mathrm{DD}}})-n\
p_{{}_{\mathrm{D}}}\ (\partial_{t}n)}{n^{2}}=\dfrac{1}{n}\big{(}\
\partial_{t}n_{{}_{\mathrm{DD}}}-p_{{}_{\mathrm{D}}}\ (\partial_{t}n)\big{)},$
(38) $\displaystyle=\dfrac{1}{n}\Big{[}\ p_{{}_{\mathrm{D}}}\
\partial_{xx}^{2}n+2\ \partial_{x}n\ \partial_{x}p_{{}_{\mathrm{D}}}+n\
\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\ +\ \Big{(}r\ (1-n)+1\Big{)}\ (1-s)\
(2-p_{{}_{\mathrm{D}}})\ n\ p_{{}_{\mathrm{D}}}-n\ p_{{}_{\mathrm{D}}}\
\Big{]}$ $\displaystyle\quad\quad-\dfrac{1}{n}\big{[}\
p_{{}_{\mathrm{D}}}\partial_{xx}^{2}n\ +\ \Big{(}r\ (1-n)+1\Big{)}\
\Big{(}(1-s)\ p_{{}_{\mathrm{D}}}\
(2-p_{{}_{\mathrm{D}}})+(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n\
p_{{}_{\mathrm{D}}}-n\ p_{{}_{\mathrm{D}}}\ \Big{]},$
$\displaystyle=\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\ +\ 2\
\partial_{x}\log(n)\ \partial_{x}p_{{}_{\mathrm{D}}}\ +\Big{(}r\
(1-n)+1\Big{)}\ p_{{}_{\mathrm{D}}}\
\Big{(}(1-s)(2-p_{{}_{\mathrm{D}}})-(1-s)(2-p_{{}_{\mathrm{D}}})p_{{}_{\mathrm{D}}}-(1-p_{{}_{\mathrm{D}}})^{2}\Big{)},$
$\displaystyle=\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\ +\ 2\
\partial_{x}\log(n)\ \partial_{x}p_{{}_{\mathrm{D}}}\ +\Big{(}r\
(1-n)+1\Big{)}\ s\ p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}})\
(p_{{}_{\mathrm{D}}}-\frac{2s-1}{s}).$

Combining equations on $n$ and $p_{{}_{\mathrm{D}}}$, we obtain model (11).

###  B.2 Model with partial conversion

####  B.2.1 Conversion in the zygote

We rewrite model (17) with variables:

$n=n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}},\quad\quad\quad\quad
p_{{}_{\mathrm{D}}}=\dfrac{n_{{}_{\mathrm{D}}}}{n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}}.$
(39)

where $n$ is the total population density and $p_{{}_{\mathrm{D}}}$ is the
drive allele frequency.

Equation on $\mathbf{n}$:

$\displaystyle\partial_{t}n-\partial_{xx}^{2}n$ $\displaystyle=\dfrac{r\
(1-n)+1}{n}\ \Big{(}(1-s)\ n_{{}_{\mathrm{D}}}^{2}+[2\ c\ (1-s)+2\ (1-sh)\
(1-c)]\
n_{{}_{\mathrm{D}}}n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{W}}}^{2}\Big{)}-n,$ (40)
$\displaystyle=\dfrac{r\ (1-n)+1}{n}\ \Big{(}(1-s)\
(np_{{}_{\mathrm{D}}})^{2}+[2\ c\ (1-s)+2\ (1-sh)\ (1-c)]\
p_{{}_{\mathrm{D}}}(1-p_{{}_{\mathrm{D}}})n^{2}+(1-p_{{}_{\mathrm{D}}})^{2}n^{2}\Big{)}-n,$
$\displaystyle=\big{(}r\ (1-n)+1\big{)}\ \Big{(}(1-s)\
p_{{}_{\mathrm{D}}}^{2}+[2\ c\ (1-s)+2\ (1-sh)\ (1-c)]\ p_{{}_{\mathrm{D}}}\
(1-p_{{}_{\mathrm{D}}})+(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n,$
$\displaystyle=\big{(}r\ (1-n)+1\big{)}\ \Big{(}(1-s)\
p_{{}_{\mathrm{D}}}^{2}+2\ p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}})\ [c\
(1-s)+(1-c)\ (1-sh)]+(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n.$

Equation on $\mathbf{n_{{}_{\mathrm{D}}}}$:

$\displaystyle\partial_{t}n_{{}_{\mathrm{D}}}$
$\displaystyle=\partial_{xx}^{2}n_{{}_{\mathrm{D}}}\ +\dfrac{r\
(1-n)+1}{n}\Big{[}\ (1-s)(n_{{}_{\mathrm{D}}}+2\ c\
n_{{}_{\mathrm{W}}})+(1-sh)\ (1-c)\
n_{{}_{\mathrm{W}}}\Big{]}n_{{}_{\mathrm{D}}}-n_{{}_{\mathrm{D}}},$ (41)
$\displaystyle=\partial_{xx}^{2}n_{{}_{\mathrm{D}}}\ +\dfrac{r\
(1-n)+1}{n}\Big{[}\ (1-s)(n\ p_{{}_{\mathrm{D}}}+2\ c\ n\
(1-p_{{}_{\mathrm{D}}}))+(1-sh)\ (1-c)\ n\ (1-p_{{}_{\mathrm{D}}})\Big{]}\ n\
p_{{}_{\mathrm{D}}}-n\ p_{{}_{\mathrm{D}}},$
$\displaystyle\stackrel{{\scriptstyle}}{{=}}p_{{}_{\mathrm{D}}}\
\partial_{xx}^{2}n+2\ \partial_{x}n\ \partial_{x}p_{{}_{\mathrm{D}}}+n\
\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\ $ $\displaystyle\quad\quad+\big{(}r\
(1-n)+1\big{)}\Big{[}\ (1-s)(p_{{}_{\mathrm{D}}}+2\ c\
(1-p_{{}_{\mathrm{D}}}))+(1-sh)\ (1-c)\ (1-p_{{}_{\mathrm{D}}})\Big{]}\ n\
p_{{}_{\mathrm{D}}}-n\ p_{{}_{\mathrm{D}}}.$

Equation on $\mathbf{p_{{}_{\mathrm{D}}}=\dfrac{n_{{}_{\mathrm{D}}}}{n}}$:

$\displaystyle\partial_{t}p_{{}_{\mathrm{D}}}$ $\displaystyle=\dfrac{n\
(\partial_{t}n_{{}_{\mathrm{D}}})-n_{{}_{\mathrm{D}}}\
(\partial_{t}n)}{n^{2}}=\dfrac{n\ (\partial_{t}n_{{}_{\mathrm{D}}})-n\
p_{{}_{\mathrm{D}}}\ (\partial_{t}n)}{n^{2}}=\dfrac{1}{n}\big{(}\
\partial_{t}n_{{}_{\mathrm{D}}}-p_{{}_{\mathrm{D}}}\ (\partial_{t}n)\big{)},$
(42) $\displaystyle=\dfrac{1}{n}\Big{[}2\ \partial_{x}n\
\partial_{x}p_{{}_{\mathrm{D}}}+n\ \partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
+\big{(}r\ (1-n)+1\big{)}\Big{(}\ (1-s)(p_{{}_{\mathrm{D}}}+2\ c\
(1-p_{{}_{\mathrm{D}}}))+(1-sh)\ (1-c)\ (1-p_{{}_{\mathrm{D}}})$
$\displaystyle\quad\quad-(1-s)\ p_{{}_{\mathrm{D}}}^{2}-2\
p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}})\ [c\ (1-s)+(1-c)\
(1-sh)]-(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}\ p_{{}_{\mathrm{D}}}\ n\Big{]},$
$\displaystyle=2\ \partial_{x}\log(n)\
\partial_{x}p_{{}_{\mathrm{D}}}+\partial_{xx}^{2}p_{{}_{\mathrm{D}}}+\big{(}r\
(1-n)+1\big{)}\Big{(}\ p_{{}_{\mathrm{D}}}\ (1-s)\ (1-p_{{}_{\mathrm{D}}})+2\
(1-s)\ c\ (1-p_{{}_{\mathrm{D}}})+(1-sh)\ (1-c)\ (1-p_{{}_{\mathrm{D}}})$
$\displaystyle\quad\quad-2\ p_{{}_{\mathrm{D}}}\ c\ (1-s)\
(1-p_{{}_{\mathrm{D}}})-2\ p_{{}_{\mathrm{D}}}\ (1-c)\ (1-sh)\
(1-p_{{}_{\mathrm{D}}})-(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}\
p_{{}_{\mathrm{D}}}\Big{]},$ $\displaystyle=2\ \partial_{x}\log(n)\
\partial_{x}p_{{}_{\mathrm{D}}}+\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
+\big{(}r\ (1-n)+1\big{)}\Big{(}p_{{}_{\mathrm{D}}}\big{[}(1-s)\ (1-2c)-2\
(1-sh)\ (1-c)+1\big{]}+2\ (1-s)\ c$ $\displaystyle\quad\quad+(1-sh)\
(1-c)-1\Big{)}\ (1-p_{{}_{\mathrm{D}}})\ p_{{}_{\mathrm{D}}},$
$\displaystyle=2\ \partial_{x}\log(n)\
\partial_{x}p_{{}_{\mathrm{D}}}+\partial_{xx}^{2}p_{{}_{\mathrm{D}}}$
$\displaystyle\quad\quad+\big{(}r\
(1-n)+1\big{)}\Big{(}[1-2(1-c)(1-h)]sp_{{}_{\mathrm{D}}}-s[1-(1-c)(1-h)]+c(1-s)\Big{)}\
(1-p_{{}_{\mathrm{D}}})\ p_{{}_{\mathrm{D}}}.$

Combining equations on $n$ and $p_{{}_{\mathrm{D}}}$, we obtain model (21).

####  B.2.2 Conversion in the germline

We rewrite model (27) with variables:

$n=n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}},\quad\quad\quad\quad
p_{{}_{\mathrm{D}}}=\dfrac{n_{{}_{\mathrm{D}}}}{n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{D}}}}.$
(43)

where $n$ is the total population density and $p_{{}_{\mathrm{D}}}$ is the
drive allele frequency.

Equation on n:

$\displaystyle\partial_{t}n-\partial_{xx}^{2}n$ $\displaystyle=\dfrac{r\
(1-n)+1}{n}\ \Big{(}(1-s)\ n_{{}_{\mathrm{D}}}^{2}+2\ (1-sh)\
n_{{}_{\mathrm{D}}}n_{{}_{\mathrm{W}}}+n_{{}_{\mathrm{W}}}^{2}\Big{)}-n,$ (44)
$\displaystyle=\dfrac{r\ (1-n)+1}{n}\ \Big{(}(1-s)\
(np_{{}_{\mathrm{D}}})^{2}+2\ (1-sh)\
p_{{}_{\mathrm{D}}}(1-p_{{}_{\mathrm{D}}})n^{2}+((1-p_{{}_{\mathrm{D}}})n)^{2}\Big{)}-n,$
$\displaystyle=\big{(}r\ (1-n)+1\big{)}\ \Big{(}(1-s)\
p_{{}_{\mathrm{D}}}^{2}+2\ (1-sh)\ p_{{}_{\mathrm{D}}}\
(1-p_{{}_{\mathrm{D}}})+(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}n-n.$

Equation on $\mathbf{n_{{}_{\mathrm{D}}}}$:

$\displaystyle\partial_{t}n_{{}_{\mathrm{D}}}$
$\displaystyle=\partial_{xx}^{2}n_{{}_{\mathrm{D}}}\ +\dfrac{r\
(1-n)+1}{n}\Big{[}\ (1-s)n_{{}_{\mathrm{D}}}+(1-sh)\ (1+c)\
n_{{}_{\mathrm{W}}}\Big{]}n_{{}_{\mathrm{D}}}-n_{{}_{\mathrm{D}}},$ (45)
$\displaystyle=p_{{}_{\mathrm{D}}}\ \partial_{xx}^{2}n+2\ \partial_{x}n\
\partial_{x}p_{{}_{\mathrm{D}}}+n\ \partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
+\dfrac{r\ (1-n)+1}{n}\Big{[}\ (1-s)np_{{}_{\mathrm{D}}}+(1-sh)\ (1+c)\ n\
(1-p_{{}_{\mathrm{D}}})\Big{]}\ n\ p_{{}_{\mathrm{D}}}-n\
p_{{}_{\mathrm{D}}},$
$\displaystyle\stackrel{{\scriptstyle}}{{=}}p_{{}_{\mathrm{D}}}\
\partial_{xx}^{2}n+2\ \partial_{x}n\ \partial_{x}p_{{}_{\mathrm{D}}}+n\
\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\ +\Big{(}r\ (1-n)+1\Big{)}\Big{[}\
(1-s)p_{{}_{\mathrm{D}}}+(1-sh)\ (1+c)\ (1-p_{{}_{\mathrm{D}}})\Big{]}\ n\
p_{{}_{\mathrm{D}}}-n\ p_{{}_{\mathrm{D}}}.$

Equation on $\mathbf{p_{{}_{\mathrm{D}}}=\dfrac{n_{{}_{\mathrm{D}}}}{n}}$:

$\displaystyle\partial_{t}p_{{}_{\mathrm{D}}}$ $\displaystyle=\dfrac{n\
(\partial_{t}n_{{}_{\mathrm{D}}})-n_{{}_{\mathrm{D}}}\
(\partial_{t}n)}{n^{2}}=\dfrac{n\ (\partial_{t}n_{{}_{\mathrm{D}}})-n\
p_{{}_{\mathrm{D}}}\ (\partial_{t}n)}{n^{2}}=\dfrac{1}{n}\big{(}\
\partial_{t}n_{{}_{\mathrm{D}}}-p_{{}_{\mathrm{D}}}\ (\partial_{t}n)\big{)},$
(46) $\displaystyle=\dfrac{1}{n}\Big{[}2\ \partial_{x}n\
\partial_{x}p_{{}_{\mathrm{D}}}+n\ \partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
+\big{(}r\ (1-n)+1\big{)}\Big{(}\ (1-s)\ p_{{}_{\mathrm{D}}}+(1-sh)\ (1+c)\
(1-p_{{}_{\mathrm{D}}})$ $\displaystyle\quad\quad-(1-s)\
p_{{}_{\mathrm{D}}}^{2}-2\ (1-sh)\ p_{{}_{\mathrm{D}}}\
(1-p_{{}_{\mathrm{D}}})-(1-p_{{}_{\mathrm{D}}})^{2}\Big{)}\
p_{{}_{\mathrm{D}}}\ n\ \Big{]},$ $\displaystyle=2\ \partial_{x}\log(n)\
\partial_{x}p_{{}_{\mathrm{D}}}+\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
+\big{(}r\ (1-n)+1\big{)}\Big{(}(1-s)\ p_{{}_{\mathrm{D}}}+(1-sh)(1+c)-2\
(1-sh)\ p_{{}_{\mathrm{D}}}-(1-p_{{}_{\mathrm{D}}})\Big{)}\
p_{{}_{\mathrm{D}}}\ (1-p_{{}_{\mathrm{D}}}),$ $\displaystyle=2\
\partial_{x}\log(n)\
\partial_{x}p_{{}_{\mathrm{D}}}+\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
+\big{(}r\ (1-n)+1\big{)}\Big{(}(2h-1)\ s\
p_{{}_{\mathrm{D}}}+(1-sh)(1+c)-1\Big{)}\ p_{{}_{\mathrm{D}}}\
(1-p_{{}_{\mathrm{D}}}).$

Combining equations on $n$ and $p_{{}_{\mathrm{D}}}$, we obtain model (29).

##  Appendix C Proofs for model (11) with perfect conversion in the zygote

###  C.1 Numerical evidence for the continuity when $r\rightarrow 0$

Below, we plot the speed of the traveling wave solutions of model (11) for a
range of $r$ and $s$ values, and for $r=0$. A positive speed correspond to
drive invasion.

Figure 7: Wave speed values in model with perfect conversion in the zygote
(11), regarding parameters $r$ the intrinsic growth rate (log scale in between
$0.01$ and $10$, plus the exact value $r=0$ in the bottom color line) and $s$
the fitness disadvantage for drive (normal scale). Below the pure drive
persistence line (light green), a well-mixed population containing only drive
homozygous individuals will necessarily get extinct.

We observe continuity in the speed value when $r\rightarrow 0$ away from
$s=\frac{1}{2}$, meaning that the case $r=0$ is relevant to approximate very
small intrinsic growth rates.

###  C.2 Proof of the statements in Tables 3 and 4 when perfect conversion
occurs in the zygote

In this section we prove the statements of Tables 3 and 4 on the two models of
interest:

$\mathbf{r=\infty}$

$\partial_{t}p-\partial_{xx}^{2}p=\dfrac{s\ p\ (1-p)\
\big{(}p-\dfrac{2s-1}{s}\big{)}}{1-s+s(1-p)^{2}}=f^{\infty}(p).$ (47)

$\mathbf{r=0}$

$\left\\{\begin{array}[]{ll}\partial_{t}n_{{}_{\mathrm{DD}}}-\partial_{xx}^{2}n_{{}_{\mathrm{DD}}}&=\
(1-s)\
\dfrac{n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}-s\
n_{{}_{\mathrm{DD}}}=f^{0}(n_{{}_{\mathrm{DD}}},n_{{}_{\mathrm{WW}}})\\\ \\\
\partial_{t}n_{{}_{\mathrm{WW}}}-\partial_{xx}^{2}n_{{}_{\mathrm{WW}}}&=\
\dfrac{-\
n_{{}_{\mathrm{WW}}}n_{{}_{\mathrm{DD}}}}{n_{{}_{\mathrm{WW}}}+n_{{}_{\mathrm{DD}}}}.\end{array}\right.$
(48)

Monostable / Bistable

  * $\mathbf{r=\infty}$

$0<s<0.5$

The equation admits two admissible steady states $0$ and $1$.

As $(f^{\infty})^{\prime}(0)>0$ and $(f^{\infty})^{\prime}(1)<0$, the only
stable state is $p=1$.

$0.5<s<1$

The equation admits three admissible steady states $0$, $\dfrac{2s-1}{s}$ and
$1$.

As $(f^{\infty})^{\prime}(0)<0$, $(f^{\infty})^{\prime}(\frac{2s-1}{s})>0$ and
$(f^{\infty})^{\prime}(1)<0$, both $p=0$ and $p=1$ are stable states.

  * $\mathbf{r=0}$

$0<s<0.5$

The system admits $(n_{{}_{\mathrm{DD}}}=0,n_{{}_{\mathrm{WW}}}\in[0,1])$ as
admissible steady states. The Jacobian matrix, when switching to $n$ and
$p_{{}_{\mathrm{D}}}$ variables, indicates that the only stable state is
$(n=0,p_{{}_{\mathrm{D}}}=1)$, i.e.
$(n_{{}_{\mathrm{DD}}}=0,n_{{}_{\mathrm{WW}}}=0)$.

$0.5<s<1$

The system admits $(n_{{}_{\mathrm{DD}}}=0,n_{{}_{\mathrm{WW}}}\in[0,1])$ as
admissible steady states. The Jacobian matrix, when switching to $n$ and
$p_{{}_{\mathrm{D}}}$ variables, indicates that the stable states are
$(n=0,p_{{}_{\mathrm{D}}}=1)$ and $(n\in[0,1],p_{{}_{\mathrm{D}}}=0)$, i.e.
$(n_{{}_{\mathrm{DD}}}=0,n_{{}_{\mathrm{WW}}}\in[0,1])$.

Existence of critical traveling waves

  * $\mathbf{r=\infty}$

    * The existence of traveling waves for the scalar equation (13) in both monostable and bistable cases is a classical result in the theory of reaction-diffusion equations, see for instance the seminal works in [4, 5].

  * $\mathbf{r=0}$

$0<s<0.5$

We apply the results of Appendix D with $\beta_{1}=1$ and $\beta_{2}=1-s$.
Therefore system (48) admits a traveling wave when $0<s<0.5$.

$0.5<s<1$

There is no drive propagation due to the gene drive clearance: the drive
allele density decreases uniformly in space (details in section C.2.1).
Regarding the wild-type alleles, their dynamic is given by the heat equation
implying only diffusion and no growth. It cannot admit traveling wave
solutions.

Pulled/pushed waves and speed values

For both models, the speed of the linearized problem around zero density of
drive allele is given by
$2\sqrt{1-2s}=2\sqrt{(f^{\infty})^{\prime}(0)}=2\sqrt{\partial_{p}f^{0}(0,1)}$.

  * $\mathbf{r=\infty}$

$0<s\lesssim 0.35$

Numerically, we observe that the speed of the wave is equal to the minimal
speed of the linearized problem: the wave is pulled (detail in section C.2.2)

$0.35\lesssim s<0.5$

Numerically, we observe that the speed of the wave is strictly above the
minimal speed of the linearized problem: the wave is pushed (detail in section
C.2.2).

$0.5<s<1$

As the system is bistable, the wave is necessarily pushed. The numerical
approximation $s\approx 0.70$ indicating whether the drive of the wild-type
population will invade the environment was already determined in the work of
Tanaka et al [38].

  * $\mathbf{r=0}$

$0<s<0.5$

We apply the results of Appendix D with $\beta_{1}=1$ and $\beta_{2}=1-s$.
Therefore system (48) admits a traveling wave with speed $v=2\sqrt{1-2s}$ when
$0<s<0.5$. This value corresponds to the KPP speed, by definition the wave is
pulled.

$0.5<s<1$

No wave (see above, in Existence of critical traveling waves).

####  C.2.1 Gene drive clearance for $s\in(0.5,1)$ when $r=0$

Consider the model with perfect conversion in the zygote (6). The densities
$n_{{}_{\mathrm{WW}}}$ and $n_{{}_{\mathrm{DD}}}$ dynamics are qualitatively
given in Figure 8 for $r=0$.

(a) Spreading eradication drive when $0<s<0.5$.

(b) Gene drive clearance when $0.5<s<1$

Figure 8: Qualitative dynamics of the drive homozygotes density
$n_{{}_{\mathrm{DD}}}$ (red line) and the wild-type homozygotes density
$n_{{}_{\mathrm{WW}}}$ (blue line) in space.

When $s>0.5$ and $r=0$, we observe gene drive clearance (in Figure 8(b)). More
precisely, we have the following estimate, deduced from (15):

$\partial_{t}n_{{}_{\mathrm{DD}}}-\partial_{xx}^{2}n_{{}_{\mathrm{DD}}}\leq\
(1-2s)\ n_{{}_{\mathrm{DD}}},$ (49)

Therefore, $n_{{}_{\mathrm{DD}}}$ is exponentially decaying in time, uniformly
in space. The dynamics of the wild type then boil down to the standard heat
equation, there cannot exist a traveling wave.

####  C.2.2 Numerical approximation of $s$ threshold value for the
pulled/pushed wave when $r=+\infty$

In order to determine an approximation of the threshold value at which the
wave switches from a pulled wave to a pushed wave, we used the recent
continuation procedure published in [6]. Figure 9 presents the value of the
wave speed obtained via the latter continuation numerical scheme [24], for a
wide range of $s$ values. Notice the transition between pulled fronts (plain
red) and pushed fronts (plain green). For the sake of clarity, the value of
the minimal speed of the linearized problem $v=2\sqrt{1-2s}$ is shown in red
for $s\in(0,\frac{1}{2})$. Notice that the speed of the pushed front changes
sign approximately at $s\approx 0.70$, in agreement with the theoretical
criterion (9).

Figure 9: Value of the wave speed when $r=+\infty$ obtained via the numerical
scheme in [24], for a wide range of $s$ values. The transition between pulled
fronts (plain red) and pushed fronts (plain green) is approximately $0.35$.
For the sake of clarity, the value of the minimal speed of the linearized
problem $v=2\sqrt{1-2s}$ is shown in red for $s\in(0,\frac{1}{2})$.

##  Appendix D Critical traveling wave for an SI similar model.

Consider the following epidemiological model,

$\left\\{\begin{array}[]{ll}\partial_{t}S-\partial_{xx}^{2}S&=\ -\beta\
\dfrac{\ S\ I}{S+I},\\\ \\\ \partial_{t}I-\partial_{xx}^{2}I&=\ \beta\
\dfrac{S\ I}{S+I}-\gamma I.\\\ \end{array}\right.$ (50)

where $S$ is the density of susceptible individuals, $I$ is the density of
infected individuals, $\gamma$ is the mortality of infected individuals and
$\beta$ is the transmission coefficient. This model has already been studied
in the literature, see [41] and references therein. In particular, the
existence of a minimal traveling wave has been established in the latter
reference.

Models (15), (25) and (33) are very similar to the above SI model (50), except
that the coefficient $\beta$ is different in the first and the second equation
of the system. We write this new system with two coefficients
$\beta_{1},\beta_{2}$:

$\left\\{\begin{array}[]{ll}\partial_{t}S-\partial_{xx}^{2}S&=\ -\beta_{1}\
\dfrac{\ S\ I}{S+I},\\\ \\\ \partial_{t}I-\partial_{xx}^{2}I&=\ \beta_{2}\
\dfrac{S\ I}{S+I}-\gamma I.\end{array}\right.$ (51)

###  D.1 Existence of critical traveling wave solutions

We are able to establish the following Theorem by adapting the proof in [41].

######  Theorem.

Suppose that $\beta_{1}>0$, and $\beta_{2}>\gamma$, then system (51) admits a
positive and bounded traveling wave solution with profile $(S^{*},I^{*})$, and
speed $v=2\sqrt{\beta_{2}-\gamma}$. Furthermore, both $S^{*}$ and $I^{*}$ are
positive, and bounded by $1$ and $\frac{\beta_{2}-\gamma}{\gamma}$
respectively.

By adapting further the elements of [41], it would be possible to prove that
the profile $S^{*}$ is increasing, whereas the profil $I^{*}$ is unimodal, as
shown in Figure 10.

Figure 10: Qualitative shape of the solution ($I^{*}$ in red, $S^{*}$ in
blue).

###  D.2 Proof of the theorem

We proceed as follows:

  1. 1.

Although the system does not satisfy the comparison principle due to a lack of
monotonicity, the construction of traveling waves is performed through a
construction of sub-solutions
(${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}$)
and super-solutions
(${\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}$,${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>for the system.

  2. 2.

Using Schauder’s fixed point theorem, we prove the existence of a critical
traveling wave solution $(S^{*},I^{*})$ with speed $v$ such that
${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}(z)\leq
S^{*}(z)\leq{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}(z)$
and
${\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}(z)\leq
I(z)\leq{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}(z)$
for all $z$ in $\mathbb{R}$.

  3. 3.

Finally, we conclude with the positivity of the critical traveling wave
solution thanks to the strong maximum principle.

####  D.2.1 Construction of sub- and super-solutions

We are seeking sub- and super-solutions, respectively
(${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}$),
(${\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}$,${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}$).
Because of the non-monotonic coupling in the system, the following set of
cross-relationships must be satisfied:

  1. 1.

$-v\
{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}}^{\prime}-{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>\dfrac{{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>I}{{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}+I}\quad\forall{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}\leq
I\leq{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>
  2. 2.

$-v\
{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}^{\prime}-{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}^{\prime\prime}\leq-\beta_{1}\dfrac{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>I}{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}+I}\quad\forall{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}\leq
I\leq{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>
  3. 3.

$-v\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}^{\prime}-{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>\dfrac{S\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}}{S+{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}\quad\forall{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>S\leq{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>
  4. 4.

$-v\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime}-{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime\prime}\leq\beta_{2}\
\dfrac{S\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}{S+{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}-\gamma\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}\quad\forall{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>S\leq{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>
To define our sub and super-solutions, it is useful to introduce the the
following family of functions $\mathcal{I}(z)=e^{-\lambda^{*}z}$, where
$\lambda^{*}$ is solution of the following dispersion equation:

$(\lambda^{*})^{2}-v\lambda^{*}+(\beta_{2}-\gamma)=0.$ (52)

They are solutions of the linearized problem

$v\mathcal{I}^{\prime}+\mathcal{I}^{\prime\prime}+(\beta_{2}-\gamma)\
\mathcal{I}=0.$ (53)

For the critical speed $v=2\sqrt{(\beta_{2}-\gamma)}$, the corresponding
double root is $\lambda^{*}=\dfrac{v}{2}=\sqrt{(\beta_{2}-\gamma)}$.

######  Lemma.

There exist two large enough constants $L_{1}>0$ and $L_{2}>0$, such that the
functions
${\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}$
defined below satisfy the conditions 1. 2. 3. and 4.:

${\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>z\leq-L_{1}\log(\dfrac{1}{L_{1}}),\\\ \\\ 1-L_{1}e^{-\dfrac{z}{L_{1}}}&\forall
z>-L_{1}\log(\dfrac{1}{L_{1}}).\\\ \end{array}}$ (54)
${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>z\leq L_{1}\log(L_{1}),\\\ \\\ 1-L_{1}e^{-\dfrac{z}{L_{1}}}&\forall
z>L_{1}\log(L_{1}).\\\ \end{array}\right.$ (55)

${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>z\leq\dfrac{1}{\lambda^{*}},\\\ \\\ eM\lambda^{*}ze^{-\lambda^{*}z}&\forall
z>\dfrac{1}{\lambda^{*}}.\\\ \end{array}\right.$ (56)
${\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}=\left\\{\begin{array}[]{ll}0&\forall
z\leq\big{(}\dfrac{L_{2}}{eM\lambda^{*}}\big{)}^{2},\\\ \\\
\big{(}eM\lambda^{*}z-L_{2}\sqrt{z}\big{)}e^{-\lambda^{*}z}&\forall
z>\big{(}\dfrac{L_{2}}{eM\lambda^{*}}\big{)}^{2}.\end{array}\right.$ (57)

with $M=\dfrac{\beta_{2}-\gamma}{\gamma}=\dfrac{(\lambda^{*})^{2}}{\gamma}$.

###### Proof.

Before we proceed with the proof, we introduce the following set of
conditions, which are more restrictive than 1,2,3,4, but may appear more
useful at some point in the calculations. When satisfied, they clearly imply
1,2,3,4.

  1. (i)

$-v\
{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}^{\prime}-{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>0$;

  2. (ii)

$-v\
{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}^{\prime}-{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>;

  3. (iii)

$-v\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}^{\prime}-{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}^{\prime\prime}\geq(\beta_{2}-\gamma)\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>;

  4. (iv)

$-v\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime}-{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime\prime}\leq\beta_{2}\
\dfrac{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}+{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}-\gamma\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}$.

We verify each of the four conditions on
${\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>${\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}$:

Condition 1: $-v\
{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}}^{\prime}-{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>\dfrac{{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>I}{{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>
The constant function
${\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>satisfies the more restrictive condition (i) $-v\
{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}^{\prime}-{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>0$ .

Condition 2:
${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}^{\prime}-{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}^{\prime\prime}\leq-\beta_{1}\dfrac{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>I}{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>
Let us take $L_{1}$ sufficiently large such that
$\dfrac{1}{\lambda^{*}}<L_{1}\log(L_{1})$.

  * $\bullet$

For $z>L_{1}\log(L_{1})$:

Since
${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>M\ \lambda^{*}\ z\ e^{\lambda^{*}z}$ and $\ \
{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>the condition (ii) $-v\
{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}^{\prime}-{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>holds for a sufficiently large $L_{1}>0$:

$\displaystyle-v\
{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}^{\prime}-{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}^{\prime\prime}=(\dfrac{1}{L_{1}}-v)\
e^{-\dfrac{z}{L_{1}}}$ $\displaystyle\leq-\beta_{1}\ e\ M\ \lambda^{*}\ z\
e^{-\lambda^{*}z}=-\beta_{1}\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}},$
(58) $\displaystyle\iff\beta_{1}\ e\ M\ \lambda^{*}\ z\
e^{(\dfrac{1}{L_{1}}-\lambda^{*})z}$ $\displaystyle\leq(v-\dfrac{1}{L_{1}}).$
(59)

  * $\bullet$

For $z\leq L_{1}\log(L_{1})$:

The condition 2. is verified since
${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>
Condition 3: $-v\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}^{\prime}-{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>\dfrac{S\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}}{S+{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>
  * $\bullet$

For $z\leq\dfrac{1}{\lambda^{*}}$:

With
${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>
$-v\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}^{\prime}-{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>\dfrac{M}{1+M}-\gamma\ M=\beta_{2}\
\dfrac{{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}}{{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color@cmyk@fill{0.32}{0.64}{0}{0}\bar{S}}+{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>\dfrac{S\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}}{S+{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>S\leq{\color[rgb]{0.68,0.36,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0.36,1}\pgfsys@color@cmyk@stroke{0.32}{0.64}{0}{0}\pgfsys@color<EMAIL_ADDRESS>(60)

  * $\bullet$

For $z>\dfrac{1}{\lambda^{*}}$:

Since
${\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>is proportional to $ze^{-\lambda^{*}z}$, and $\lambda^{*}$ is precisely the
double root of the characteristic equation (52), we deduce that condition
(iii) $-v\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}^{\prime}-{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color@cmyk@fill{0}{0.77}{0.87}{0}\bar{I}}^{\prime\prime}-(\beta_{2}-\gamma)\
{\color[rgb]{1,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.23,0.13}\pgfsys@color@cmyk@stroke{0}{0.77}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>0$ is verified.

Condition 4: $-v\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime}-{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime\prime}\leq\beta_{2}\
\dfrac{S\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}{S+{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}-\gamma\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}$.
Let us take $L_{2}$ sufficiently large such that
$L_{2}>eM\lambda^{*}\sqrt{L_{1}\log\left(L_{1}\right)}$.

  * $\bullet$

For $z\leq\Big{(}\dfrac{L_{2}}{e\ M\ \lambda^{*}}\Big{)}^{2}$:

${\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}=0$
so condition 4. is satisfied.

  * $\bullet$

For $z>\Big{(}\dfrac{L_{2}}{e\ M\ \lambda^{*}}\Big{)}^{2}$:

The choice of $L_{2}$ implies $z>L_{1}\log(L_{1})$, which means
${\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>
We can reformulate condition (iv) as follows:

$-v\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime}-{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime\prime}\leq\beta_{2}\
\dfrac{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}+{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}-\gamma\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}\iff-v\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime}-{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime\prime}-(\beta_{2}-\gamma)\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}\leq-\beta_{2}\
\dfrac{{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{2}}{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}+{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}.$
(61)

With $L_{3}=eM\lambda^{*}$, and:

$\displaystyle{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}=[L_{3}\
z-L_{2}\ \sqrt{z}]\ e^{-\lambda^{*}z},$ (62)
$\displaystyle{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime}=[L_{3}-L_{2}\
\dfrac{1}{2\ \sqrt{z}}]\ e^{-\lambda^{*}z}-[L_{3}z-L_{2}\ \sqrt{z}]\
\lambda^{*}\ e^{-\lambda^{*}z},$ (63)
$\displaystyle{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime\prime}=[L_{2}\
\dfrac{1}{4\ z\ \sqrt{z}}]\ e^{-\lambda^{*}z}-2\ [L_{3}-L_{2}\ \dfrac{1}{2\
\sqrt{z}}]\ \lambda^{*}\ e^{-\lambda^{*}z}+[L_{3}z-L_{2}\ \sqrt{z}]\
(\lambda^{*})^{2}\ e^{-\lambda^{*}z}$ (64)

On the one hand, we obtain the following identities:

$\displaystyle-v\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime}-{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{\prime\prime}-(\beta_{2}-\gamma)\
{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}$
$\displaystyle=e^{-\lambda^{*}z}\Big{[}L_{3}\Big{(}-v+v\ \lambda^{*}\ z+2\
\lambda^{*}-(\lambda^{*})^{2}\ z-(\beta_{2}-\gamma)z\Big{)}$ (65)
$\displaystyle\quad+\ L_{2}\Big{(}v\ \dfrac{1}{2\ \sqrt{z}}-v\ \sqrt{z}\
\lambda^{*}-\dfrac{1}{4\ z\ \sqrt{z}}-\lambda^{*}\
\dfrac{1}{\sqrt{z}}+(\lambda^{*})^{2}\ \sqrt{z}+(\beta_{2}-\gamma)\
\sqrt{z}\Big{)}\Big{]},$ (66)
$\displaystyle=e^{-\lambda^{*}z}\Big{[}L_{3}\Big{(}(2\lambda^{*}-v)-z\
\Big{(}-v\lambda^{*}+(\lambda^{*})^{2}+(\beta_{2}-\gamma)\Big{)}\Big{)}$ (67)
$\displaystyle\quad+\ L_{2}\Big{(}\sqrt{z}\ \Big{(}-v\
\lambda^{*}+(\lambda^{*})^{2}+(\beta_{2}-\gamma)\Big{)}+\dfrac{1}{2\
\sqrt{z}}(v-2\lambda^{*})-\dfrac{1}{4\ z\ \sqrt{z}}\Big{)}\Big{]},$ (68)
$\displaystyle=-L_{2}\ e^{-\lambda^{*}z}\ \dfrac{1}{4\ z\ \sqrt{z}}$ (69)

On the other hand, we have:

$-\beta_{2}\
\dfrac{{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}^{2}}{{\color[rgb]{0.0600000000000001,0.89,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.89,1}\pgfsys@color@cmyk@stroke{0.94}{0.11}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.11}{0}{0}\underline{S}}+{\color[rgb]{0.99,0.69,0.07}\definecolor[named]{pgfstrokecolor}{rgb}{0.99,0.69,0.07}\underline{I}}}=-\beta_{2}\
\dfrac{[L_{3}\ z-L_{2}\ \sqrt{z}]^{2}\
e^{-2\lambda^{*}z}}{1-L_{1}e^{-\dfrac{z}{L_{1}}}+[L_{3}\ z-L_{2}\ \sqrt{z}]\
e^{-\lambda^{*}z}}.$ (70)

We resume with the reformulation (61), which is now equivalent to the
following:

$\displaystyle-L_{2}\ e^{-\lambda^{*}z}\ (1-L_{1}e^{-\dfrac{z}{L_{1}}}+[L_{3}\
z-L_{2}\ \sqrt{z}]\ e^{-\lambda^{*}z})\leq-4\beta_{2}\ [L_{3}\ z-L_{2}\
\sqrt{z}]^{2}\ e^{-2\lambda^{*}z}\ z\ \sqrt{z}$ (71) $\displaystyle\iff$
$\displaystyle 4\beta_{2}\ [L_{3}\ z-L_{2}\ \sqrt{z}]^{2}\ e^{-\lambda^{*}z}\
z\ \sqrt{z}-L_{2}\ [L_{3}\ z-L_{2}\ \sqrt{z}]\ e^{-\lambda^{*}z}\leq\ L_{2}\
(1-L_{1}e^{-\dfrac{z}{L_{1}}}),$ (72) $\displaystyle\iff$ $\displaystyle
4\beta_{2}\ e^{-\lambda^{*}z}\ z^{3}\sqrt{z}\
(L_{3})^{2}+e^{-\lambda^{*}z}\Big{(}(1-8\beta_{2}\ z^{2})\ L_{3}\ L_{2}\
z+(L_{2})^{2}\sqrt{z}(1-4\beta_{2}\ z^{2})\Big{)}\leq\ L_{2}\
(1-L_{1}e^{-\dfrac{z}{L_{1}}}).$ (73)

We may increase $L_{2}$ such that $1-4\beta_{2}\
\Big{(}\dfrac{L_{2}}{eM\lambda^{*}}\Big{)}^{4}\leq 0$. Then, since
$z>(\dfrac{L_{2}}{eM\lambda^{*}})^{2}$:

$1-8\beta_{2}\ z^{2}\leq 1-4\beta_{2}\ z^{2}<1-4\beta_{2}\
\Big{(}\dfrac{L_{2}}{eM\lambda^{*}}\Big{)}^{4}\leq 0.$ (74)

Since $(1-8\beta_{2}\ z^{2})$ and $(1-4\beta_{2}\ z^{2})$ are negative terms,
we need to show $L_{2}\ (1-L_{1}e^{-\dfrac{z}{L_{1}}})\geq 4\beta_{2}\
e^{-\lambda^{*}z}\ z^{3}\sqrt{z}\ (L_{3})^{2}$.

Let $g(z)=\beta_{2}\ e^{-\lambda^{*}z}\ z^{3}\sqrt{z}\ (L_{3})^{2}$ be a
$\mathscr{C}^{1}([0;-\infty[)$ function. Since $\lim\limits_{z\rightarrow
0}(g(z))=0$ and $\lim\limits_{z\rightarrow+\infty}(g(z))=0$ there exists a
constant C (which is independent from $L_{2}$) such that $g(z)<C\ \ \forall
z\geq 0$. We finally increase $L_{2}$ so that condition (iv) is verified. ∎

####  D.2.2 Existence and positivity of a critical traveling wave solution

Now, exactly as in [41], we are in a position to define a set of functions

$\Gamma=\\{(S,I)\in B_{\mu}(\mathbb{R},\mathbb{R}^{2})\ |\ \underline{S}\leq
S\leq\bar{S},\ \underline{I}\leq I\leq\bar{I}\\},$

where $B_{\mu}(\mathbb{R},\mathbb{R}^{2})$ is the set of two-component
continuous functions with each component growing at infinity slower than
$e^{\mu|z|}$, as well as an operator $F:\Gamma\to
C(\mathbb{R},\mathbb{R}^{2})$ that will satisfy the assumptions of the
Schauder fixed point theorem and whose fixed point in $\Gamma$ will precisely
be the solution $(S,I)$ we seek. Note that the inequalities 1., 2., 3., 4.
(beginning of Section D.2.1) are precisely what we use to prove that
$F(\Gamma)\subset\Gamma$. Details can be found in [41].

The positivity of both $S$ and $I$ comes from the use of the strong maximum
principle, again exactly as in [41].

##  Appendix E Study of the reaction term when $r=+\infty$ in section 3.2

We are searching for conditions implying a pulled monostable wave, using
criterion (10).

###  E.1 Conversion occurring in the zygote

We rewrite limit equation (22):

$\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
=\dfrac{\Big{(}-\big{[}2(1-c)(1-h)-1\big{]}\ s\
p_{{}_{\mathrm{D}}}-s[1-(1-c)(1-h)]+c(1-s)\Big{)}\ (1-p_{{}_{\mathrm{D}}})\
p_{{}_{\mathrm{D}}}}{-s\big{[}2(1-c)(1-h)-1\big{]}p_{{}_{\mathrm{D}}}^{2}-2s\big{[}1-(1-c)(1-h)\big{]}p_{{}_{\mathrm{D}}}+1}.$
(75)

With $\mathscr{A}_{z}:=s\ \big{[}2(1-c)(1-h)-1\big{]}\in[-s,s]$ :

$\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
=\dfrac{\big{(}-\mathscr{A}_{z}\
p_{{}_{\mathrm{D}}}+\frac{1}{2}(\mathscr{A}_{z}-s)+c(1-s)\big{)}\
(1-p_{{}_{\mathrm{D}}})\
p_{{}_{\mathrm{D}}}}{-\mathscr{A}_{z}p_{{}_{\mathrm{D}}}^{2}+(\mathscr{A}_{z}-s)\
p_{{}_{\mathrm{D}}}+1}.$ (76)

Note that the mean fitness
$M_{z}(p_{{}_{\mathrm{D}}})=-\mathscr{A}_{z}p_{{}_{\mathrm{D}}}^{2}+(\mathscr{A}_{z}-s)\
p_{{}_{\mathrm{D}}}+1\in[1-s,1]$444$M_{z}^{\prime}(p_{{}_{\mathrm{D}}})=\mathscr{A}_{z}(1-2p_{{}_{\mathrm{D}}})-s\leq
0$ therefore $M_{z}(1)\leq M_{z}(p_{{}_{\mathrm{D}}})\leq M_{z}(0)$.. When
$\mathscr{A}_{z}\neq 0$, equation (76) can be rewritten:

$\partial_{t}p_{{}_{\mathrm{D}}}-\partial_{xx}^{2}p_{{}_{\mathrm{D}}}\
=\dfrac{-\mathscr{A}_{z}\ (p_{{}_{\mathrm{D}}}-{p_{{}_{\mathrm{D}}}^{*}}_{z})\
(1-p_{{}_{\mathrm{D}}})\
p_{{}_{\mathrm{D}}}}{-\mathscr{A}_{z}p_{{}_{\mathrm{D}}}^{2}+(\mathscr{A}_{z}-s)\
p_{{}_{\mathrm{D}}}+1}\quad\text{with}\quad{p_{{}_{\mathrm{D}}}^{*}}_{z}:=\frac{1}{2}+\dfrac{2c\
(1-s)-s}{2\mathscr{A}_{z}}.$ (77)

Let us introduce $s_{1}:=\dfrac{c}{1-h(1-c)}$ and
$s_{2,z}:=\dfrac{c}{2c+h(1-c)}$. Note that $\mathscr{A}_{z}>0\iff
s_{1}<s_{2,z}$. We draw the reaction term regarding the sign of
$\mathscr{A}_{z}$ and the $s$ values in Figure 11.

When $\mathscr{A}_{z}<0$ and $s\in(s_{2,z},s_{1})$, equation (22) admits two
stable steady states (bistability). The final proportion will then strongly
depend on the initial condition. On the other hand, when $\mathscr{A}_{z}>0$
and $s\in(s_{1},s_{2,z})$, the only possible equilibrium state is a
coexistence state: the final proportion $p_{{}_{\mathrm{D}}}$ will be strictly
in between $0$ and $1$.

Independently of the sign of $\mathscr{A}_{z}$, if $s<\min(s_{1},s_{2,z})$ the
only stable steady state is $p_{{}_{\mathrm{D}}}=1$ meaning that for an
initial condition outside of the steady states, we expect that the drive
always invades the population. If $s>\max(s_{1},s_{2,z})$, the only stable
steady state is $p_{{}_{\mathrm{D}}}=0$ meaning that for an initial condition
outside of the steady states, we expect that the wild-type always invades the
population.

Figure 11: Reaction term of equation (76) regarding the sign of
$\mathscr{A}_{z}=s[2(1-c)(1-h)-1]$ and the $s$ values. The $s$ threshold
values are $s_{1}=\frac{c}{1-h(1-c)}$ and $s_{2,z}=\frac{c}{2c+h(1-c)}$. The
dots on the x axis correspond to the steady states: $p_{{}_{\mathrm{D}}}=0$
and $p_{{}_{\mathrm{D}}}=1$ in black,
${p_{{}_{\mathrm{D}}}^{*}}_{z}=\frac{1}{2}+\frac{2c\
(1-s)-s}{2\mathscr{A}_{z}}$ in red when it exists.

In case of bistability, the wave is always pushed [22]; we can dismiss
condition 2 in the research of pulled monostable waves. We use criterion (10)
on monostable cases, i.e. drive invasion 1 4 6, wild type invasion 3 5 8, and
coexistence state 7 (the numbers refer to the subgraphs in Figure 11).

####  E.1.1 Monostable drive invasion

* 4 When $\mathscr{A}_{z}=0$ and $s<s_{1}=s_{2,z}=\dfrac{2c}{2c+1}\iff s<2c\ (1-s)$

From equation (76), we have for all $p_{{}_{\mathrm{D}}}\in[0,1]$:

$\sigma(0)-(1-p_{{}_{\mathrm{D}}})\
\sigma(p_{{}_{\mathrm{D}}})=\Big{(}c(1-s)-\dfrac{s}{2}\Big{)}-\dfrac{\Big{(}c(1-s)-\dfrac{s}{2}\Big{)}\
(1-p_{{}_{\mathrm{D}}})}{1-sp_{{}_{\mathrm{D}}}}=\Big{(}c(1-s)-\dfrac{s}{2}\Big{)}\
\dfrac{(1-s)\ p_{{}_{\mathrm{D}}}}{1-sp_{{}_{\mathrm{D}}}}\geq 0,$ (78)

where $\sigma$ is the selection term defined by equation (8). Criterion (10)
is verified.

* 6 When $\mathscr{A}_{z}>0$ and ${p_{{}_{\mathrm{D}}}^{*}}_{z}>1$

From equation (77), we have for all $p_{{}_{\mathrm{D}}}\in[0,1]$:

$\sigma(0)-(1-p_{{}_{\mathrm{D}}})\
\sigma(p_{{}_{\mathrm{D}}})=\mathscr{A}_{z}{p_{{}_{\mathrm{D}}}^{*}}_{z}-\dfrac{-\mathscr{A}_{z}\
(p_{{}_{\mathrm{D}}}-{p_{{}_{\mathrm{D}}}^{*}}_{z})\
(1-p_{{}_{\mathrm{D}}})}{-\mathscr{A}_{z}p_{{}_{\mathrm{D}}}^{2}+(\mathscr{A}_{z}-s)\
p_{{}_{\mathrm{D}}}+1}=\mathscr{A}_{z}p_{{}_{\mathrm{D}}}\dfrac{-(\mathscr{A}_{z}{p_{{}_{\mathrm{D}}}^{*}}_{z}+1)\
p_{{}_{\mathrm{D}}}+(\mathscr{A}_{z}+1-s)\
{p_{{}_{\mathrm{D}}}^{*}}_{z}+1}{-\mathscr{A}_{z}p_{{}_{\mathrm{D}}}^{2}+(\mathscr{A}_{z}-s)\
p_{{}_{\mathrm{D}}}+1}.$ (79)

Note that $-\mathscr{A}_{z}p_{{}_{\mathrm{D}}}^{2}+(\mathscr{A}_{z}-s)\
p_{{}_{\mathrm{D}}}+1>(1-s)>0$ and $\mathscr{A}_{z}p_{{}_{\mathrm{D}}}>0$. The
affine term $-(\mathscr{A}_{z}{p_{{}_{\mathrm{D}}}^{*}}_{z}+1)\
p_{{}_{\mathrm{D}}}+(\mathscr{A}_{z}+1-s)\ {p_{{}_{\mathrm{D}}}^{*}}_{z}+1$
decreases with $p_{{}_{\mathrm{D}}}$. In order to show that it is positive for
all $p_{{}_{\mathrm{D}}}\in[0,1]$, we just need to verify that this it is true
for $p_{{}_{\mathrm{D}}}=1$:

$-(\mathscr{A}_{z}{p_{{}_{\mathrm{D}}}^{*}}_{z}+1)+(\mathscr{A}_{z}+1-s)\
{p_{{}_{\mathrm{D}}}^{*}}_{z}+1=(1-s)\ {p_{{}_{\mathrm{D}}}^{*}}_{z}\geq
0\quad\Rightarrow\quad\sigma(0)-(1-p_{{}_{\mathrm{D}}})\
\sigma(p_{{}_{\mathrm{D}}})\geq 0\quad\forall p_{{}_{\mathrm{D}}}\in[0,1].$
(80)

Criterion (10) is verified.

* 1 When $\mathscr{A}_{z}<0$ and ${p_{{}_{\mathrm{D}}}^{*}}_{z}<0$ and $s<s_{2,z}\iff 0<c-2sc-sh+sch$

We consider equation (79) with
$-\mathscr{A}_{z}p_{{}_{\mathrm{D}}}^{2}+(\mathscr{A}_{z}-s)\
p_{{}_{\mathrm{D}}}+1>1-s>0$ and $\mathscr{A}_{z}p_{{}_{\mathrm{D}}}<0$. The
affine term $-(\mathscr{A}_{z}{p_{{}_{\mathrm{D}}}^{*}}_{z}+1)\
p_{{}_{\mathrm{D}}}+(\mathscr{A}_{z}+1-s)\ {p_{{}_{\mathrm{D}}}^{*}}_{z}+1$
decreases with $p_{{}_{\mathrm{D}}}$. In order to show that it is negative for
all $p_{{}_{\mathrm{D}}}\in[0,1]$, we introduce a condition implying the
negativity for $p_{{}_{\mathrm{D}}}=0$:

$(\mathscr{A}_{z}+1-s)\
{p_{{}_{\mathrm{D}}}^{*}}_{z}+1=\dfrac{(\mathscr{A}_{z}+1-s)\
(\mathscr{A}_{z}+2c(1-s)-s+2\mathscr{A}_{z})}{2\mathscr{A}_{z}}<0$ (81)
$\iff\Big{(}1-2s[1-(1-h)(1-c)]\Big{)}\Big{(}c-2sc-sh+sch\Big{)}+s\
\big{[}2(1-c)(1-h)-1\big{]}>0$ (82)

Criterion (10) is verified when condition (82) is true.

####  E.1.2 Monostable wild-type invasion

In case of a monostable wild-type invasion, we need to consider the wild-type
proportion $p_{{}_{\mathrm{W}}}=1-p_{{}_{\mathrm{D}}}\in[0,1]$ and rewrite the
equation (76):

$\displaystyle-\partial_{t}p_{{}_{\mathrm{W}}}+\partial_{xx}^{2}p_{{}_{\mathrm{W}}}\
$ $\displaystyle=\dfrac{\big{(}-\mathscr{A}_{z}\
(1-p_{{}_{\mathrm{W}}})+\frac{1}{2}\ (\mathscr{A}_{z}-s)+c(1-s)\big{)}\
(1-p_{{}_{\mathrm{W}}})\
p_{{}_{\mathrm{W}}}}{-\mathscr{A}_{z}(1-p_{{}_{\mathrm{W}}})^{2}+(\mathscr{A}_{z}-s)(1-p_{{}_{\mathrm{W}}})+1}$
$\displaystyle\iff\partial_{t}p_{{}_{\mathrm{W}}}-\partial_{xx}^{2}p_{{}_{\mathrm{W}}}\
$ $\displaystyle=\dfrac{\big{(}-\mathscr{A}_{z}\
p_{{}_{\mathrm{W}}}+\frac{1}{2}\ (\mathscr{A}_{z}+s)-c(1-s)\big{)}\
(1-p_{{}_{\mathrm{W}}})\
p_{{}_{\mathrm{W}}}}{-\mathscr{A}_{z}p_{{}_{\mathrm{W}}}^{2}+(\mathscr{A}_{z}+s)\
p_{{}_{\mathrm{W}}}+(1-s)}$ (83)

When $\mathscr{A}_{z}\neq 0$, equation (E.1.2) can be rewritten: