# The frequency process in a non-neutral two-type continuous-state branching
process with competition and its genealogy

Imanol Nuñez∗ and  José-Luis Pérez†

###### Abstract.

We consider a population growth model given by a two-type continuous-state
branching process with immigration and competition, introduced by Ma in [17].
We study the relative frequency of one of the types in the population when the
total mass is forced to be constant at a dense set of times. The resulting
process is described as the solution to an SDE, which we call the culled
frequency process, generalizing the $\Lambda$-asymmetric frequency process
introduced by Caballero et al. in [2]. We obtain conditions for the culled
frequency process to have a moment dual and show that it is given by a
branching-coalescing continuous-time Markov chain that describes the genealogy
of the two-type CBI with competition. Finally, we obtain a large population
limit of the culled frequency process, resulting in a deterministic ordinary
differential equation (ODE). Two particular cases of the limiting ODE are
studied to determine if general two-type branching mechanisms and general
Malthusians can lead to the coexistence of the two types in the population.

$*$ Department of Probability and Statistics, Centro de Investigación en
Matemáticas A.C. Calle Jalisco s/n. C.P. 36240, Guanajuato, Mexico. Email:
<EMAIL_ADDRESS>
$\dagger$ Department of Probability and Statistics, Centro de Investigación en
Matemáticas A.C. Calle Jalisco s/n. C.P. 36240, Guanajuato, Mexico. Email:
<EMAIL_ADDRESS>
##  1\. Introduction

The main result by Birkner et al. in [1] shows that the frequency process
describing the proportion of the mass descended from an ancestral individual
associated with a measure-valued branching process can be transformed into a
$\Lambda$-Fleming–Viot process using a suitable time change involving a
functional of the total mass if and only if the measure-valued process is
$\alpha$-stable. In particular, this implies that if $X^{(1)}$ and $X^{(2)}$
are independent and identically distributed $\alpha$-stable continuous-state
branching processes (CSBPs), then the ratio process
$R=X^{(1)}/(X^{(1)}+X^{(2)})$ can be time-changed to become Markovian. This
allows describing the time-changed genealogy of an $\alpha$-stable CSBP
through the block-counting process of a
$\mathrm{Beta}(2-\alpha,\alpha)$-coalescent that is in moment duality to the
aforementioned Markovian version of the frequency process. Foucart and Hénard,
in [4], extended this result by adding an immigration term to the CSBP, making
it a continuous-state branching process with immigration (CBI). This
modification is reflected in the genealogy, which is now described by the
block-counting process of a $\mathrm{Beta}(2-\alpha,\alpha-1)$-coalescent,
where the difference with the result of Birkner et al. is due to a mutation
mechanism in the genealogy that arises because of immigration in the CBI.
Nonetheless, their work remains in the independent and identically distributed
$\alpha$-stable case as they use the time change methodology of Birkner et al.

In [2], Caballero et al. studied the case where $X^{(1)}$ and $X^{(2)}$ are
modeled by CBIs with different reproduction mechanisms that are not
necessarily $\alpha$-stable, enabling the study of a population with two
species reproducing by different mechanisms. In their work, the processes are
still assumed to be independent. The time change perspective does not apply
here, see Lemma 3.5 of [1]. Instead, inspired by the works [6] and [7] of
Gillespie, Caballero et al. used a novel method called culling (see section
4.2 [2]) that allows them to control the total population size, see Section
3.3. By using this technique, they were able to extend the result proved by
Birkner et al. in [1] to non-$\alpha$-stable cases with different reproduction
mechanisms. The resulting frequency process was called the _$\Lambda$
-asymmetric frequency process_.

In certain cases, the $\Lambda$-asymmetric frequency process has a moment dual
that is the block-counting process of a branching-coalescing process, which is
reminiscent of the ancestral selection graph of Krone and Neuhauser [14]. The
resulting genealogy includes mutation and pairwise branching, reflecting,
respectively, immigration in the associated two-type CBI and efficiency in
resource consumption (as in [11]).

Gónzalez-Casanova et al. in [8] dropped the independent and identically
distributed assumptions made by Birkner et al. on $X^{(1)}$ and $X^{(2)}$ by
considering a two-type CBI $X=(X^{(1)},X^{(2)})$. Gónzalez-Casanova et al.
showed that the frequency process $R$ can still be time-changed to be
Markovian by keeping only the $\alpha$-stability assumption on the associated
two-type CBI. In some cases, the time-changed frequency process coincides with
the $\beta$-Fleming–Viot process defined by Griffiths in [12], which is the
weighted moment dual to the block-counting process of a two-type branching-
coalescing process. This connection is enlightening as it implies that the
time-changed genealogy of the two-type $\alpha$-stable CBI $X$ is described by
a branching-coalescing continuous-time Markov chain.

This paper aims to generalize these results to a population with two types of
individuals ($1$ and $2$), where: i) individuals of one type might produce
individuals of the other type (cross-branching); ii) the reproduction
mechanisms of each one of the two types might be different; iii) there might
be limited resources for which both types compete, leading to different
carrying capacities according to the efficiency of the types in resource
usage. The two types of individuals will be modeled by $X^{(1)}$ and
$X^{(2)}$, the components of a two-type population growth model $X$ that
extends a CBI using general Malthusians (inspired by Lambert’s branching
process with logistic growth [15]), introduced by Ma in [17], which we call a
two-type CBI with competition. Hence, in this paper, we will not assume that
$X^{(1)}$ and $X^{(2)}$ are independent, identically distributed nor
$\alpha$-stable. Our approach consists of applying the culling technique
developed in [2] to obtain the frequency process of individuals of type $1$ in
the population, under the assumption that the total population size remains
constant in a dense set of times. We call the resulting process the culled
frequency process. This will allow us to study the dynamics of the genetic
profile of populations with two types of individuals where the two types are
closely related by the reproduction mechanisms and the competition for limited
resources.

The model introduced in this paper can be valuable for studying populations
with two types of individuals that interact and compete for limited resources
(see [19] for an example in ecology). By considering competition and resource
constraints, our model provides insights into how genetic diversity is
maintained or altered, and how certain traits prevail or coexist under
selective pressures. This framework is a powerful tool for exploring genetic
variation, reproduction, mutation, and competition in evolutionary biology,
ecology, and conservation biology.

The outline of the paper is the following. In section 2, we establish the
basic notation used throughout the paper and also introduce the population
growth model $X$ studied in this work. In section 3 we define the culled
frequency process of individuals of type 1, characterizing its dynamics
through a stochastic differential equation (SDE), see (3.12), that is seen to
be the limit of the culling procedure, given in section 3.3. The main result
in section 3 is Proposition 3.2, which ensures the existence of a unique
strong solution to the SDE (3.12). The proof relies on a simple but
enlightening extension of Theorem 3.2 in [5]. With the culled frequency
process defined, in section 4, we establish a result stating that whenever it
has a moment dual, it is a branching-coalescing continuous-time Markov chain,
describing the genealogy of the two-type CBI with competition. Finally, in
section 5, the large population limit of the frequency process is considered,
resulting in a deterministic ordinary differential equation (ODE), see (5.2).
Two particular cases of the limiting ODE are studied to determine if general
two-type branching mechanisms and general Malthusians can lead to the
coexistence of the two types in the population.

##  2\. Notation and preliminaries

In this section, we introduce the notation we will use in the rest of the
paper and the two-type CBI with competition that we will consider for our
results. In what follows we take $c\in\mathbb{R}_{+}^{2}$ and
$\eta\in\mathbb{R}_{+}^{2}$ to be vectors of non-negative constants. Until
further notice, for $i,j\in\\{1,2\\}$ let $b_{ij}:\mathbb{R}\to\mathbb{R}$ be
a locally Lipschitz function such that $b_{ij}(x)=0$ for $x\leq 0$ and
$b_{ij}(x)\geq 0$ for $i\neq j$. Moreover, we consider Borel measures
$\mu_{1},\mu_{2}$ and $\nu$ over $U_{2}:=\mathbb{R}_{+}^{2}\setminus\\{0\\}$
that fulfill the integrability conditions

$\displaystyle\int_{U_{2}}(1\wedge\lVert w\rVert)\nu(dw)<\infty,$
$\displaystyle\int_{U_{2}}\bigl{(}\lVert w\rVert\wedge\lVert
w\rVert^{2}+w_{j}\bigr{)}\mu_{i}(dw)<\infty,\quad i\neq j,$

where $\lVert w\rVert=(w_{1}^{2}+w_{2}^{2})^{1/2}$ for $w\in U_{2}$.

Extending one of the special cases in [16], see also [17], we consider
$X=(X^{(1)},X^{(2)})$ as the solution of the stochastic differential equation
defined by

(2.1)
$\begin{split}X_{t}^{(i)}=&x^{(i)}+\int_{0}^{t}\bigl{(}\eta_{i}+b_{ii}(X_{s}^{(i)})+b_{ij}(X_{s}^{(j)})\bigr{)}ds+\int_{0}^{t}\sqrt{2c^{i}X_{s-}^{(i)}}dB_{s}^{(i)}\\\
&+\int_{0}^{t}\int_{U_{2}}\int_{0}^{X_{s-}^{(i)}}w_{i}\widetilde{N}_{R}^{i}(ds,dw,du)+\int_{0}^{t}\int_{U_{2}}\int_{0}^{X_{s-}^{(j)}}w_{i}N_{R}^{j}(ds,dw,du)\\\
&+\int_{0}^{t}\int_{U_{2}}w_{i}N_{I}(ds,dw),\end{split}$

for $i,j\in\\{1,2\\}$, $i\neq j$, and all $t\geq 0$ such that the SDE makes
sense (before the time of explosion). In (2.1) we consider:

  * •

$x^{(i)}\geq 0$, for $i\in\\{1,2\\}$, is deterministic;

  * •

$B=(B^{1},B^{2})$ is a two-dimensional Brownian Motion;

  * •

$N_{I}(ds,dw)$ is a Poisson random measure over $\mathbb{R}_{+}\times U_{2}$
with intensity measure $ds\nu(dw)$;

  * •

for $i\in\\{1,2\\}$, $N_{R}^{i}(ds,dw,du)$ is a Poisson random measure over
$\mathbb{R}_{+}\times U_{2}\times\mathbb{R}_{+}$ with intensity measure
$ds\mu_{i}(dw)du$, and $\widetilde{N}_{R}^{(i)}$ denotes the associated
compensated random measure.

All of the previous elements are assumed to be defined in the same complete
probability space and independent. Note that the integrability conditions for
$\mu_{1}$ and $\mu_{2}$ tell us that these two measures have finite means.
This allows us to define $X$ through the SDE (2.1).

The fact that there exists a unique strong solution up to the time of
explosion, in case it is finite, to (2.1) is due to Theorems 3.1 and 3.2 in
[17]. Indeed, the locally Lipschitz condition allows us to obtain pathwise
uniqueness up to the time of explosion by virtually the same argument that
proves Theorem 3.1 in [17]. Meanwhile, the existence of a unique positive
strong solution follows by a slight modification of the proof of Theorem 3.2
in [17].

For the rest of the paper, we will set $z=x^{(1)}+x^{(2)}$ and $r=x^{(1)}/z$.
Moreover, given $z>0$ we define the function
$T_{z,2}:\mathbb{R}_{+}^{2}\to[0,1]^{2}$ by

$T_{z,2}(w)=\biggl{(}\frac{w_{1}}{z+w_{1}+w_{2}},\frac{w_{2}}{z+w_{1}+w_{2}}\biggr{)},$

and the operator
$\mathbf{T}_{2}^{(z)}:\mathcal{M}(\mathbb{R}_{+}^{2})\to\mathcal{M}([0,1]^{2})$
that maps every measure $\mu$ over $\mathbb{R}_{+}^{2}$ to the measure
$\mu\circ T_{z,2}^{-1}$ over $[0,1]^{2}$; i.e.

(2.2) $\mathbf{T}_{2}^{(z)}\mu(A)\coloneq\mu\circ
T_{z,2}^{-1}(A)\quad\text{for all }A\in\mathcal{B}([0,1]^{2}).$

##  3\. The culled frequency process

Our motivation is to study the dynamics of the genetic profile of a population
consisting of two types of individuals who might reproduce using different
mechanisms that in general are not independent, subject to competition and
immigration. To this end, we start this section by introducing two processes
that characterize the relative frequency of individuals of type $1$ and the
total size of the population.

###  3.1. Total population and frequency process

We start by defining the total population process $Z$ by

$Z_{t}=X_{t}^{(1)}+X_{t}^{(2)},\quad t\geq 0,\ Z_{0}=z,$

where we assume that $z>0$, recalling that $z=x^{(1)}+x^{(2)}$. Additionally,
we introduce the frequency process of type $1$ individuals given by

$R_{t}=\frac{X_{t}^{(1)}}{X_{t}^{(1)}+X_{t}^{(2)}}1_{\\{t<\zeta\\}}+\dagger
1_{\\{t\geq\zeta\\}},\quad t\geq 0,\ R_{0}=r,$

where $\dagger$ is a cemetery state, $\zeta:=\inf\\{t\geq
0:Z_{t}\in\\{0,\infty\\}\\}$ and $r=x^{(1)}/(x^{(1)}+x^{(2)})$. We introduce
the cemetery state in order to define the process $R$ when the process $Z$
explodes or reaches zero in finite time. We note that the frequency process
$R$ is not Markovian. To address this issue, throughout this section, we will
use the culling method developed in [2] to construct a Markov process that
encodes the dynamics of the frequency process $R$.

It is readily seen that $(R,Z)$ is a Markov process with values in
$([0,1]\cup\\{\dagger\\})\times[0,+\infty]$. To avoid using the cemetery
state, given $0<\varepsilon<z<L$, we will consider the stopping time
$\tau:=\tau_{\varepsilon}^{-}\wedge\tau_{L}^{+}$, where

$\tau_{\varepsilon}^{-}:=\inf\\{t\geq
0:Z_{t}=\varepsilon\\}\quad\text{and}\quad\tau_{L}^{+}:=\inf\\{t\geq
0:Z_{t}>L\\}.$

Then the process $(R,Z)$ stopped at time $\tau$ will encode the dynamics of
the populations described by $X$ before the total population $Z$ becomes
relatively small or explodes. This is formalized by a local martingale problem
that will be stated in the next result, which is a simple consequence of the
SDE (2.1) that defines $X$ and Itô’s formula.

######  Proposition 3.1.

For every $f\in C^{2}([0,1]\times\mathbb{R}_{+})$, the process

$M_{t}:=f(R_{t\wedge\tau},Z_{t\wedge\tau})-f(r,z)-\int_{0}^{t\wedge\tau}\mathcal{L}f(R_{s},Z_{s})ds$

is a local martingale for

(3.1) $\displaystyle\mathcal{L}f(r,z)={}$
$\displaystyle\bigl{[}\eta_{1}+b_{11}(rz)+b_{12}((1-r)z)\bigr{]}\biggl{[}\partial_{1}f(r,z)\frac{1-r}{z}+\partial_{2}f(r,z)\biggr{]}$
(3.2)
$\displaystyle+\bigl{[}\eta_{2}+b_{21}(rz)+b_{22}((1-r)z)\bigr{]}\biggl{[}-\partial_{1}f(r,z)\frac{r}{z}+\partial_{2}f(r,z)\biggr{]}$
(3.3)
$\displaystyle+\frac{c_{2}-c_{1}}{z}\partial_{1}f(r,z)2r(1-r)+\frac{r(1-r)}{z}\bigl{[}(1-r)c_{1}+rc_{2}\bigr{]}\partial_{11}^{2}f(r,z)$
(3.4)
$\displaystyle+c_{1}\bigl{[}r(1-r)\partial_{12}^{2}f(r,z)+r(1-r)\partial_{21}^{2}f(r,z)+rz\partial_{22}^{2}f(r,z)\bigr{]}$
(3.5)
$\displaystyle+c_{2}\bigl{[}-r(1-r)\partial_{12}^{2}f(r,z)-r(1-r)\partial_{21}^{2}f(r,z)+(1-r)z\partial_{22}^{2}f(r,z)\bigr{]}$
(3.6)
$\displaystyle+rz\int_{U_{2}}\biggl{[}f\biggl{(}r+(1-r)\frac{w_{1}}{z+w_{1}+w_{2}}-r\frac{w_{2}}{z+w_{1}+w_{2}},z+w_{1}+w_{2}\biggr{)}-f(r,z)$
(3.7) $\displaystyle\hskip 199.16928pt-
w_{1}\biggl{(}\frac{1-r}{z}\partial_{1}f(r,z)+\partial_{2}f(r,z)\biggr{)}\biggr{]}\mu_{1}(dw)$
(3.8)
$\displaystyle+(1-r)z\int_{U_{2}}\biggl{[}f\biggl{(}r+(1-r)\frac{w_{1}}{z+w_{1}+w_{2}}-r\frac{w_{2}}{z+w_{1}+w_{2}},z+w_{1}+w_{2}\biggr{)}-f(r,z)$
(3.9) $\displaystyle\hskip 199.16928pt-
w_{2}\biggl{(}-\frac{r}{z}\partial_{1}f(r,z)+\partial_{2}f(r,z)\biggr{)}\biggr{]}\mu_{2}(dw)$
(3.10)
$\displaystyle+\int_{U_{2}}\biggl{[}f\biggl{(}r+(1-r)\frac{w_{1}}{z+w_{1}+w_{2}}-{}r\frac{w_{2}}{z+w_{1}+w_{2}},z+w_{1}+w_{2}\biggr{)}-f(r,z)\biggr{]}\nu(dw)$

It is important to remark that even if the process $(R,Z)$ is Markovian, in
general $R$ will not be an autonomous process (i.e. it will depend on $Z$),
hence it is not Markovian.

###  3.2. Culled frequency process

Heuristically, if we want a Markov process that captures the dynamics of the
frequency process $R$, it should have an infinitesimal generator given by
(3.2) with a caveat: the function $f$ should only depend on $r$. Thus, the
desired process should not depend on $Z$, which would correspond to $z$ being
constant in the generator. The culling procedure introduced in [2] allows to
achieve this as a limiting procedure by keeping $Z$ equal to a constant $z>0$
in a dense set of times over $\mathbb{R}_{+}$. We now introduce the limiting
process of the culling procedure, $R^{(z,r)}$, with constant population size
$z>0$ and started at $r$, which we simply call _the culled frequency process_
, which is a generalization of the $\Lambda$-asymmetric frequency process
introduced by Caballero et al. in [2].

For $z>0$ and $r\in[0,1]$, let $R^{(z,r)}$ be the solution to the stochastic
differential equation

(3.11) $\displaystyle R_{t}^{(z,r)}={}$ $\displaystyle
r+\int_{0}^{t}\bigl{[}\widetilde{D}(R_{s-}^{(z,r)})+S(R_{s-}^{(z,r)})+S_{c}(R_{s-}^{(z,r)})+m(R_{s-}^{(z,r)})+m_{c}^{1}(R_{s-}^{(z,r)})+m_{c}^{2}(R_{s-}^{(z,r)})\bigr{]}ds$
(3.12)
$\displaystyle+\int_{0}^{t}\sigma(R_{s-}^{(z,r)})dB_{s}+\int_{0}^{t}\int_{U_{2}}\bigl{[}\widetilde{g}^{(z)}(R_{s-}^{(z,r)},w)+\widetilde{h}^{(z)}(R_{s-}^{(z,r)},w)\bigr{]}N_{3}(ds,dw)$
(3.13)
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\bigl{[}g_{1}^{(z)}(R_{s-}^{(z,r)},w,v)+h_{1}^{(z)}(R_{s-}^{(z,r)},w,v)\bigr{]}\widetilde{N}_{1}(ds,dw,dv)$
(3.14)
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\bigl{[}g_{2}^{(z)}(R_{s-}^{(z,r)},w,v)+h_{2}^{(z)}(R_{s-}^{(z,r)},w,v)\bigr{]}\widetilde{N}_{2}(ds,dw,dv)$

where $r>0$,

(3.15)
$\begin{split}\widetilde{D}(r)={}&\biggl{[}b_{11}(zr)\frac{1-r}{z}-b_{22}(z(1-r))\frac{r}{z}+b_{12}(z(1-r))\frac{1-r}{z}-b_{21}(zr)\frac{r}{z}\biggr{]}1_{\\{r\in[0,1]\\}},\\\
S(r)={}&\frac{2}{z}(c_{2}-c_{1})r(1-r)1_{\\{r\in[0,1]\\}},\\\
S_{c}(r)={}&\biggl{[}\sum_{i=1}^{2}(-1)^{i}\int_{U_{2}}\frac{w_{i}(w_{1}+w_{2})}{z+w_{1}+w_{2}}\mu_{i}(dw)\biggr{]}r(1-r)1_{\\{r\in[0,1]\\}},\\\
m(r)={}&\frac{1}{z}\bigl{[}\eta_{1}(1-r)-\eta_{2}r\bigr{]},\\\
m_{c}^{1}(r)={}&-r^{2}\int_{U_{2}}\frac{zw_{2}}{z+w_{1}+w_{2}}\mu_{1}(dw)1_{\\{r\in[0,1]\\}},\\\
m_{c}^{2}(r)={}&(1-r)^{2}\int_{U_{2}}\frac{zw_{1}}{z+w_{1}+w_{2}}\mu_{2}(dw)1_{\\{r\in[0,1]\\}},\\\
\sigma(r)={}&\sqrt{\frac{2}{z}r(1-r)\bigl{[}c_{1}(1-r)+c_{2}r\bigr{]}}1_{\\{r\in[0,1]\\}},\end{split}$

and

  * •

$B$ is a Brownian Motion;

  * •

for $i\in\\{1,2\\}$, $N_{i}(ds,dw,dv)$, is a Poisson random measure over
$\mathbb{R}_{+}\times U_{2}\times\mathbb{R}_{+}$ with intensity measure
$ds\mu_{i}(dw)dv$, and $\widetilde{N}_{i}$ denotes the associated compensated
random measure;

  * •

$N_{3}(ds,dw)$ is a Poisson random measure over $\mathbb{R}_{+}\times U_{2}$
with intensity measure $ds\nu(dw)$;

  * •

for $q,w\in\mathbb{R}_{+}\times U_{2}$ we set

$\widetilde{g}^{(z)}(q,w):=(1-q)\frac{w_{1}}{z+w_{1}+w_{2}}1_{\\{q\in[0,1]\\}}\quad\text{and}\quad\widetilde{h}^{(z)}(q,w):=-q\frac{w_{2}}{z+w_{1}+w_{2}}1_{\\{q\in[0,1]\\}};$

  * •

for $q,w,v\in\mathbb{R}_{+}\times U_{2}\times\mathbb{R}_{+}$ we define

$\displaystyle g_{1}^{(z)}(q,w,v)$
$\displaystyle:=\widetilde{g}^{(z)}(q,w)1_{\\{v\leq qz\\}},$ $\displaystyle
g_{2}^{(z)}(q,w,v)$
$\displaystyle:=\widetilde{g}^{(z)}(q,w)1_{\\{v\leq(1-q)z\\}}$ $\displaystyle
h_{1}^{(z)}(q,w,v)$ $\displaystyle:=\widetilde{h}^{(z)}(q,w)1_{\\{v\leq
qz\\}},$ $\displaystyle h_{2}^{(z)}(q,w,v)$
$\displaystyle:=\widetilde{h}^{(z)}(q,w)1_{\\{v\leq(1-q)z\\}}.$

The stochastic processes ($B,N_{1},N_{2}$ and $N_{3}$) used in the definition
of the SDE (3.12) are assumed to be independent and defined in the same
complete probability space. We now state a result concerning the solution of
(3.12).

######  Proposition 3.2.

There exists a unique strong solution to (3.12) with values in $[0,1]$.
Moreover, for every $t>0$ there exists $K(t)>0$ such that for any
$r,s\in[0,1]$ we get

(3.16) $\mathbb{E}\bigl{[}\lvert R_{t}^{(z,r)}-R_{t}^{(z,s)}\rvert\bigr{]}\leq
K(t)\lvert r-s\rvert.$

Before presenting the proof, let us make a couple of remarks. A key element in
the proof of the preceding proposition will be a slight extension of Theorem
3.2 in [16], taking advantage that the branching mechanism $\mu_{i}$ is of
bounded variation for cross-branching, meaning that $\int_{U_{2}}(1\wedge
w_{j})\mu_{i}(dw)<\infty$ whenever $i\neq j$. In addition, formally, we should
also prove that the solution takes values in $[0,1]$, but this is omitted as
it follows along the same lines that the first step in the proof of
Proposition 4.1 in [2]. Let us now prove Proposition 3.2.

###### Proof.

Pathwise uniqueness. We start by computing some bounds that will be needed in
the proof. First note that the local Lipschitz assumption on $b_{ij}$ for
$i,j\in\\{1,2\\}$, together with the definition of
$\widetilde{D},S,S_{c},m,m_{c}^{1}$ and $m_{c}^{2}$ in (3.15) respectively,
implies the existence of a constant $K_{d}>0$ such that for any $x,y\in[0,1]$,

(3.17) $\displaystyle\lvert\widetilde{D}(x)-\widetilde{D}(y)\rvert+\lvert
S(x)-S(y)\rvert+\lvert m(x)-m(y)\rvert$ (3.18) $\displaystyle+\lvert
S_{c}(x)-S_{c}(y)\rvert+\lvert m_{c}^{1}(x)-m_{c}^{1}(y)\rvert+\lvert
m_{c}^{2}(x)-m_{c}^{2}(y)\rvert\leq K_{d}\lvert x-y\rvert.$

Moreover, note that for $x,y\in[0,1]$ we get directly

(3.19)
$\int_{U_{2}}\lvert\widetilde{g}^{(z)}(x,w)+\widetilde{h}^{(z)}(x,w)-\widetilde{g}^{(z)}(y,w)-\widetilde{h}^{(z)}(y,w)\rvert\nu(dw)\leq\lvert
x-y\rvert\int_{U_{2}}\frac{w_{1}+w_{2}}{z+w_{1}+w_{2}}\nu(dw),$

while the bounds

$\displaystyle\int_{0}^{\infty}\int_{U_{2}}\lvert
h_{1}^{(z)}(x,w,v)-h_{1}^{(z)}(y,w,v)\rvert\mu_{1}(dw)dv$ $\displaystyle\leq
2\lvert x-y\rvert\int_{U_{2}}w_{2}\mu_{1}(dw),$
$\displaystyle\int_{0}^{\infty}\int_{U_{2}}\lvert
g_{2}^{(z)}(x,w,v)-g_{2}^{(z)}(y,w,v)\rvert\mu_{2}(dw)dv$ $\displaystyle\leq
2\lvert x-y\rvert\int_{U_{2}}w_{1}\mu_{2}(dw),$

follow respectively from the inequalities

$\displaystyle\lvert h_{1}^{(z)}(x,w,v)-h_{1}^{(z)}(y,w,v)\rvert$
$\displaystyle\leq\frac{w_{2}}{z+w_{1}+w_{2}}(\lvert x-y\rvert
1_{\\{v\leq(x\wedge y)z\\}}+1_{\\{(x\wedge y)z<v\leq(x\vee y)z\\}}),$
$\displaystyle\lvert g_{2}^{(z)}(x,w,v)-g_{2}^{(z)}(y,w,v)\rvert$
$\displaystyle\leq\frac{w_{1}}{z+w_{1}+w_{2}}(\lvert x-y\rvert
1_{\\{v\leq(1-x\vee y)z\\}}+1_{\\{(1-x\vee y)z<v\leq(1-x\wedge y)z\\}}),$

valid for any $(w,v)\in U_{2}\times\mathbb{R}_{+}$. On the other hand, it is
readily seen that for $x,y\in[0,1]$ and $\sigma$ defined in (3.15),

(3.20) $\lvert\sigma(x)-\sigma(y)\rvert^{2}\leq\frac{6(c_{1}+c_{2})}{z}\lvert
x-y\rvert.$

In addition, similar bounds to those obtained for $h_{1}^{(z)}$ and
$g_{2}^{(z)}$ allow us to obtain the inequalities

(3.21) $\displaystyle\int_{U_{2}}\int_{0}^{\infty}\lvert
g_{1}^{(z)}(x,w,v)-g_{1}^{(z)}(y,w,v)\rvert^{2}dv\mu_{1}(dw)$
$\displaystyle\leq 2z\lvert
x-y\rvert\int_{U_{2}}\biggl{(}\frac{w_{1}}{z+w_{1}+w_{2}}\biggr{)}^{2}\mu_{1}(dw),$

and

(3.22) $\displaystyle\int_{U_{2}}\int_{0}^{\infty}\lvert
h_{2}^{(z)}(x,w,v)-h_{2}^{(z)}(y,w,v)\rvert^{2}dv\mu_{1}(dw)$
$\displaystyle\leq 2z\lvert
x-y\rvert\int_{U_{2}}\biggl{(}\frac{w_{2}}{z+w_{1}+w_{2}}\biggr{)}^{2}\mu_{2}(dw),$

which are valid for any $x,y\in[0,1]$. Finally notice that for any $(w,v)\in
U_{2}\times\mathbb{R}_{2}$, the mappings

(3.23) $\displaystyle x\mapsto
x+g_{1}^{(z)}(x,w,v)=x+(1-x)\frac{w_{1}}{z+w_{1}+w_{2}}1_{\\{v\leq xz\\}}$ and
(3.24) $\displaystyle x\mapsto
x+h_{2}^{(z)}(x,w,v)=x\biggl{(}1-\frac{w_{2}}{z+w_{1}+w_{2}}1_{\\{v\leq(1-x)z\\}}\biggr{)}$

are non-decreasing for $x\in[0,1]$.

Now let $\xi^{1}$ and $\xi^{2}$ be two solutions to (3.12) and set
$\zeta=\xi^{1}-\xi^{2}$. For notational convenience, for
$\psi:\mathbb{R}_{+}\times U_{2}\to\mathbb{R}$ and $\rho:\mathbb{R}_{+}\times
U_{2}\times\mathbb{R}_{+}\to\mathbb{R}$, we define

$\Delta_{\psi}(x,y,w):=\psi(x,w)-\psi(y,w)\quad\text{and}\quad\Delta_{\rho}(x,y,w,v):=\rho(x,w,v)-\rho(y,w,v)$

for $w\in U_{2}$, $x,y\in\mathbb{R}_{+}$, and $v\in\mathbb{R}_{+}$. Let us
consider the sequence of functions $\\{\phi_{k}:k\in\mathbb{N}\\}$ as in [16],
recalling that they fulfill that $\phi_{k}(z)\to\lvert z\rvert$ non-
decreasingly as $k\to\infty$, $0\leq\operatorname{sgn}(z)\phi^{\prime}(z)\leq
1$, and $0\leq\lvert z\rvert\phi_{k}^{\prime\prime}(z)\leq 2/k$ for every
$k\in\mathbb{N}$. Then applying Itô’s formula yields, for
$t\in\mathbb{R}_{+}$,

$\displaystyle\phi_{k}(\zeta_{t})={}$ $\displaystyle
M_{t}^{k}+\int_{0}^{t}\phi_{k}^{\prime}(\zeta_{s})\bigl{[}\widetilde{D}(\xi_{s}^{1})-\widetilde{D}(\xi_{s}^{2})\bigr{]}ds+\int_{0}^{t}\phi_{k}^{\prime}(\zeta_{s})\bigl{[}S(\xi_{s}^{1})-S(\xi_{s}^{2})\bigr{]}ds$
$\displaystyle+\int_{0}^{t}\phi_{k}^{\prime}(\zeta_{s})\bigl{[}S_{c}(\xi_{s}^{1})-S_{c}(\xi_{s}^{2})\bigr{]}ds+\int_{0}^{t}\phi_{k}^{\prime}(\zeta_{s})\bigl{[}m(\xi_{s}^{1})-m(\xi_{s}^{2})\bigr{]}ds$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\phi_{k}^{\prime\prime}(\zeta_{s})\bigl{[}\sigma(\xi_{s}^{1})-\sigma(\xi_{s}^{2})\bigr{]}^{2}ds$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{\widetilde{g}^{(z)}+\widetilde{h}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w)\bigr{)}-\phi_{k}(\zeta_{s})\Bigr{]}\nu(dw)ds$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{g_{1}^{(z)}+h_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}-\phi_{k}\bigl{(}\zeta_{s}+\Delta_{g_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}\Bigr{]}dv\mu_{1}(dw)ds$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{g_{2}^{(z)}+h_{2}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}-\phi_{k}\bigl{(}\zeta_{s}+\Delta_{h_{2}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}\Bigr{]}dv\mu_{2}(dw)ds$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{g_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}-\phi_{k}(\zeta_{s})-\Delta_{g_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\phi_{k}^{\prime}(\zeta_{s})\Bigr{]}dv\mu_{1}(dw)ds$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{h_{2}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}-\phi_{k}(\zeta_{s})-\Delta_{h_{2}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\phi_{k}^{\prime}(\zeta_{s})\Bigr{]}dv\mu_{2}(dw)ds,$

where

$\displaystyle M_{t}^{k}={}$
$\displaystyle\phi_{k}(\zeta_{0})+\int_{0}^{t}\phi_{k}^{\prime}(\zeta_{s})\bigl{[}\sigma(\xi_{s}^{1})-\sigma(\xi_{s}^{2})\bigr{]}dB_{s}$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s-}+\Delta_{\widetilde{g}^{(z)}+\widetilde{h}^{(z)}}(\xi_{s-}^{1},\xi_{s-}^{2},w)\bigr{)}-\phi_{k}(\zeta_{s-})\Bigr{]}\widetilde{N}_{3}(ds,dw)$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s-}+\Delta_{g_{1}^{(z)}+h_{1}^{(z)}}(\xi_{s-}^{1},\xi_{s-}^{2},w,v)\bigr{)}-\phi_{k}(\zeta_{s-})\Bigr{]}\widetilde{N}_{1}(ds,dw,dv)$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s-}+\Delta_{g_{2}^{(z)}+h_{2}^{(z)}}(\xi_{s-}^{1},\xi_{s-}^{2},w,v)\bigr{)}-\phi_{k}(\zeta_{s-})\Bigr{]}\widetilde{N}_{2}(ds,dw,dv),\qquad
t\in\mathbb{R}_{+},$

is a martingale due to the fact that $\zeta_{s}\in[0,1]$ for all
$s\in\mathbb{R}_{+}$ almost surely. One might wonder why do the terms
involving $\Delta_{h_{1}^{(z)}}$ and $\Delta_{g_{2}^{(z)}}$ do not appear as
compensators in the expansion of $\phi_{k}(\zeta_{t})$, and why neither the
terms involving $m_{c}^{1}$ and $m_{c}^{2}$ appear. This is answered by noting
that for every $s\in[0,t]$ we have the equalities

$\displaystyle\int_{U_{2}}\int_{0}^{\infty}\Delta_{h_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)dv\mu_{1}(du)$
$\displaystyle=m_{c}^{1}(\xi_{s}^{1})-m_{c}^{1}(\xi_{s}^{2}),$
$\displaystyle\int_{U_{2}}\int_{0}^{\infty}\Delta_{g_{2}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)dv\mu_{2}(du)$
$\displaystyle=m_{c}^{2}(\xi_{s}^{1})-m_{c}^{2}(\xi_{s}^{2}),$

so the terms of the form
$(m_{c}^{i}(\xi_{s}^{1})-m_{c}^{i}(\xi_{s}^{2}))\phi_{k}^{\prime}(\zeta_{s})ds$
end up cancelling with the $\Delta$ terms. By the previous remarks, we can
deduce bounds to obtain the pathwise uniqueness of (3.12) in a similar fashion
to Theorem 3.2 in [16]. Indeed, we first note that by using (3.18) we obtain,
for $t\in\mathbb{R}_{+}$,

$\displaystyle\int_{0}^{t}\phi_{k}^{\prime}(\zeta_{s})\Bigl{[}\bigl{(}\widetilde{D}(\xi_{s}^{1})-\widetilde{D}(\xi_{s}^{2})\bigr{)}+\bigl{(}S(\xi_{s}^{1})-S(\xi_{s}^{2})\bigr{)}+\bigl{(}S_{c}(\xi_{s}^{1})-S_{c}(\xi_{s}^{2})\bigr{)}+\bigl{(}m(\xi_{s}^{1})-m(\xi_{s}^{2})\bigr{)}\Bigr{]}ds$
$\displaystyle\leq
K_{d}\int_{0}^{t}\lvert\zeta_{s}\rvert\,\lvert\phi_{k}^{\prime}(\zeta_{s})\rvert
ds\leq K_{d}\int_{0}^{t}\lvert\zeta_{s}\rvert ds,$

where the second inequality follows from the fact that
$\lvert\phi_{k}^{\prime}(z)\rvert\leq 1$. For the term involving $\sigma$ note
that by (3.19),

(3.25)
$\displaystyle\int_{0}^{t}\phi_{k}^{\prime\prime}(\zeta_{s})\bigl{[}\sigma(\xi_{s}^{1})-\sigma(\xi_{s}^{2})\bigr{]}^{2}ds$
$\displaystyle\leq\frac{6(c_{1}+c_{2})}{z}\int_{0}^{t}\lvert\zeta_{s}\rvert\phi_{k}^{\prime\prime}(\zeta_{s})ds\leq\frac{12(c_{1}+c_{2})}{zk}t,\qquad
t\in\mathbb{R}_{+},$

where for the last inequality we have used that $0\leq\lvert
z\rvert\phi_{k}^{\prime\prime}(z)\leq 2/k$. To control the term with the
measure $\nu$ we note that, using (3.20),

$\displaystyle\int_{0}^{t}\int_{U_{2}}\bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{\widetilde{g}^{(z)}+\widetilde{h}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w)\bigr{)}-\phi_{k}(\zeta_{s})\bigr{]}\nu(dw)ds$
$\displaystyle=\int_{0}^{t}\int_{U_{2}}\int_{\zeta_{s}}^{\zeta_{s}+\Delta_{\widetilde{g}^{(z)}+\widetilde{h}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w)}\phi_{k}^{\prime}(y)dy\nu(dw)ds$
$\displaystyle\leq\int_{0}^{t}\int_{U_{2}}\lvert\widetilde{g}^{(z)}(\xi_{s}^{1},w)+\widetilde{h}^{(z)}(\xi_{s}^{1},w)-\widetilde{g}^{(z)}(\xi_{s}^{2},w)-\widetilde{h}^{(z)}(\xi_{s}^{2},w)\rvert\nu(dw)ds$
$\displaystyle\leq\biggl{(}\int_{U_{2}}\frac{w_{1}+w_{2}}{z+w_{1}+w_{2}}\nu(dw)\biggr{)}\int_{0}^{t}\lvert\zeta_{s}\rvert
ds,\qquad t\in\mathbb{R}_{+}.$

Similarly, we obtain both of the following bounds

$\displaystyle\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{g_{1}^{(z)}+h_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)-\phi_{k}\bigl{(}\zeta_{s}+\Delta_{g_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}\Bigr{]}dv\mu_{1}(dw)ds$
$\displaystyle\leq\Bigl{(}2\int_{U_{2}}w_{2}\mu_{1}(dw)\Bigr{)}\int_{0}^{t}\lvert\zeta_{s}\rvert
ds,$
$\displaystyle\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{g_{2}^{(z)}+h2^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)-\phi_{k}\bigl{(}\zeta_{s}+\Delta_{h_{2}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}\Bigr{]}dv\mu_{2}(dw)ds$
$\displaystyle\leq\Bigl{(}2\int_{U_{2}}w_{1}\mu_{2}(dw)\Bigr{)}\int_{0}^{t}\lvert\zeta_{s}\rvert
ds,\qquad t\in\mathbb{R}_{+}.$

Finally, by using Lemma 3.1 in [16], together with (3.21) and (3.22), we get
the first inequality in

$\displaystyle\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{g_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}-\phi_{k}(\zeta_{s})-\Delta_{g_{1}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\phi_{k}^{\prime}(\zeta_{s})\Bigr{]}dv\mu_{1}(dw)ds$
$\displaystyle\leq\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\frac{2\bigl{(}g_{1}^{(z)}(\xi_{s}^{1},w,v)-g_{1}^{(z)}(\xi_{s}^{2},w,v)\bigr{)}^{2}}{k\lvert\zeta_{s}\rvert}dv\mu_{1}(dw)ds$
$\displaystyle\leq\biggl{(}\frac{2}{k}\int_{U_{2}}\biggl{(}\frac{w_{1}}{z+w_{1}+w_{2}}\biggr{)}^{2}\mu_{1}(dw)\biggr{)}t,\qquad
t\in\mathbb{R}_{+}.$

Similarly,

$\displaystyle\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\Bigl{[}\phi_{k}\bigl{(}\zeta_{s}+\Delta_{h_{2}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\bigr{)}-\phi_{k}(\zeta_{s})-\Delta_{h_{2}^{(z)}}(\xi_{s}^{1},\xi_{s}^{2},w,v)\phi_{k}^{\prime}(\zeta_{s})\Bigr{]}dv\mu_{2}(dw)ds$
$\displaystyle\leq\biggl{(}\frac{2}{k}\int_{U_{2}}\biggl{(}\frac{w_{2}}{z+w_{1}+w_{2}}\biggr{)}^{2}\mu_{2}(dw)\biggr{)},\qquad
t\in\mathbb{R}_{+}.$

The bounds established give us the inequality, valid for $t\geq 0$,

$\phi_{k}(\zeta_{t})\leq M_{t}^{k}+K^{\prime}\int_{0}^{t}\lvert\zeta_{s}\rvert
ds+\frac{t\tilde{K}}{k},$

where

$\displaystyle K^{\prime}$
$\displaystyle=K_{d}+2\int_{U_{2}}w_{2}\mu_{1}(dw)+2\int_{U_{2}}w_{1}\mu_{2}(dw)+\int_{U_{2}}\frac{w_{1}+w_{2}}{z+w_{1}+w_{2}}\nu(dw),$
$\displaystyle\tilde{K}$
$\displaystyle=\frac{6(c_{1}+c_{2})}{z}+2\int_{U_{2}}\biggl{(}\frac{w_{1}}{z+w_{1}+w_{2}}\biggr{)}^{2}\mu_{1}(dw)+2\int_{U_{2}}\biggl{(}\frac{w_{2}}{z+w_{1}+w_{2}}\biggr{)}^{2}\mu_{2}(dw).$

By taking expectations, recalling that $M^{k}$ is a true martingale,

$\mathbb{E}[\phi_{k}(\zeta_{t})]\leq\mathbb{E}[\phi_{k}(\zeta_{0})]+K^{\prime}\int_{0}^{t}\mathbb{E}[\lvert\zeta_{s}\rvert]ds+\frac{t\tilde{K}}{k}$

for any $t>0$. Then, letting $k$ tend to infinity and using that
$\phi_{k}(z)\to\lvert z\rvert$ non-decreasingly as $k\to\infty$ we deduce, by
the Monotone Convergence Theorem, that for any fixed $t>0$,

(3.26)
$\mathbb{E}[\lvert\zeta_{t}\rvert]\leq\mathbb{E}[\lvert\zeta_{0}\rvert]+K^{\prime}\int_{0}^{t}\mathbb{E}[\lvert\zeta_{s}\rvert]ds.$

If $\xi_{0}^{1}=r=\xi_{0}^{2}$, then we obtain, by Gronwall’s inequality, that
$\xi_{t}^{1}=\xi_{t}^{2}$ almost surely for every $t\geq 0$, from where
$\xi^{1}=\xi^{2}$ by the right-continuity of the trajectories. Moreover, using
again Gronwall’s inequality, we deduce (3.16) from (3.26) by considering
$\xi_{0}^{1}=s$ and $\xi_{0}^{2}=r$.

Existence of a strong solution. It suffices to note that for $x\in[0,1]$ we
get the bounds

(3.27)
$\displaystyle\int_{U_{2}}\lvert\widetilde{g}^{(z)}(x,w)+\widetilde{h}^{(z)}(x,w)\rvert\nu(dw)$
$\displaystyle\leq\int_{U_{2}}\frac{w_{1}+w_{2}}{z+w_{1}+w_{2}}\nu(dw)$ (3.28)
$\displaystyle\int_{U_{2}}\lvert\widetilde{g}^{(z)}(x,w)+\widetilde{h}^{(z)}(x,w)\rvert^{2}\nu(dw)$
$\displaystyle\leq 2\int_{U_{2}}\frac{w_{1}+w_{2}}{z+w_{1}+w_{2}}\nu(dw),$
(3.29) $\displaystyle\sum_{i=1}^{2}\int_{0}^{\infty}\int_{U_{2}}\lvert
g_{i}^{(z)}(x,w,v)+h_{i}^{(z)}(x,w,v)\rvert^{2}\mu_{i}(dw)dv$
$\displaystyle\leq
4\int_{U_{2}}\frac{(w_{1}+w_{2})^{2}}{z+w_{1}+w_{2}}(\mu_{1}+\mu_{2})(dw).$

Indeed, these bounds along with the definition of $\widetilde{D}$, $S$,
$S_{c}$, $m$, $m_{c}^{1}$, $m_{c}^{2}$ and $\sigma$ imply the existence of a
constant $K_{l}>0$ such that for all $x\in\mathbb{R}$ we have

$\displaystyle\sigma(x)^{2}+(\widetilde{D}(x)+S(x)+S_{c}(x)+m(x)+m_{c}^{1}(x)+m_{c}^{2}(x))^{2}$
$\displaystyle+\sum_{i=1}^{2}\int_{0}^{\infty}\int_{U_{2}}\lvert
g_{i}^{(z)}(x,w,v)+h_{i}^{(z)}(x,w,v)\rvert^{2}\mu_{i}(dw)dv+\int_{U_{2}}\lvert\widetilde{g}^{(z)}(x,w)+\widetilde{h}^{(z)}(x,w)\rvert^{2}\nu(dw)$
$\displaystyle+\biggl{(}\int_{U_{2}}\lvert\widetilde{g}^{(z)}(x,w)+\widetilde{h}^{(z)}(x,w)\rvert\nu(dw)\biggr{)}^{2}\leq
K_{l}(1+x^{2}).$

Therefore, the existence of a strong solution, which is _a fortiori_ unique,
is given by similar arguments to those in the proof of Theorem 5.1 in [16].
This finishes the proof of Proposition 3.2. ∎

To use the culling approach we will follow closely the methodology described
in [2], where the convergence to the limiting process $R^{(z,r)}$ is proved
using the convergence of infinitesimal generators. Hence, we will need
$R^{(z,r)}$ to be Feller. This is the content of the next result.

######  Proposition 3.3.

For any given $z>0$, the process $R^{(z,r)}$ is Feller and its infinitesimal
generator $\mathcal{L}^{(z)}$ acts on $f\in C^{2}([0,1])$ as follows:

(3.30) $\displaystyle\mathcal{L}^{(z)}f(r)={}$
$\displaystyle\bigl{[}\widetilde{D}(r)+S(r)+m(r)\bigr{]}f^{\prime}(r)+\frac{1}{2}\sigma(r)^{2}f^{\prime\prime}(r)$
(3.31)
$\displaystyle+rz\int_{[0,1]^{2}}\biggl{(}f\bigl{(}r+(1-r)u_{1}-ru_{2}\bigr{)}-f(r)-\frac{(1-r)u_{1}}{1-u_{1}-u_{2}}f^{\prime}(r)\biggr{)}\mathbf{T}_{2}^{(z)}\mu_{1}(du)$
(3.32)
$\displaystyle+(1-r)z\int_{[0,1]^{2}}\biggl{(}f\bigl{(}r+(1-r)u_{1}-ru_{2}\bigr{)}-f(r)+\frac{ru_{2}}{1-u_{1}-u_{2}}f^{\prime}(r)\biggr{)}\mathbf{T}_{2}^{(z)}\mu_{2}(du)$
(3.33)
$\displaystyle+\int_{[0,1]^{2}}\Bigl{(}f\bigl{(}r+(1-r)u_{1}-ru_{2}\bigr{)}-f(r)\Bigr{)}\mathbf{T}_{2}^{(z)}\nu(du),$

where $\mathbf{T}_{2}^{(z)}(\cdot)$ is defined in (2.2).

###### Proof.

We define $T_{t}f(r):=\mathbb{E}[f(R_{t}^{(z,r)})]$. The fact that the family
$\\{T_{t}:t\geq 0\\}$ forms a semigroup such that $T_{t}(C[0,1])\subset
C([0,1])$ for all $t\geq 0$, follows the same lines as the first part of the
proof of Proposition 4.1 in [2].

By Itô’s formula we deduce, see Proposition 3.1, that for any $f\in
C^{2}([0,1])$,

$f(R_{t}^{(z,r)})=f(r)+\int_{0}^{t}\mathcal{L}^{(z)}(R_{s}^{(z,r)})ds+M_{t}^{f},$

where $M^{f}$ is a local martingale. By using the fact that $f\in
C^{2}([0,1])$, it is readily seen that

$\sup_{r\in[0,1]}\biggl{\lvert}\bigl{[}\widetilde{D}(r)+S(r)+m(r)\bigr{]}f^{\prime}(r)+\frac{1}{2}\sigma(r)^{2}f^{\prime\prime}(r)\biggr{\rvert}<\infty.$

On the other hand, for any $r\in[0,1]$ and $u_{1},u_{2}\in[0,1]$ such that
$u_{1}+u_{2}<1$, standard arguments reveal that

$\displaystyle\Bigl{\lvert}f\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}-f(r)\Bigr{\rvert}$
$\displaystyle\leq(u_{1}+u_{2})\lVert f^{\prime}\rVert_{\infty},$
$\displaystyle\biggl{\lvert}f\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}-f(r)-\frac{(1-r)u_{1}}{1-u_{1}-u_{2}}f^{\prime}(r)\biggr{\rvert}$
$\displaystyle\leq\lVert u\rVert^{2}\lVert
f^{\prime\prime}\rVert_{\infty}+\frac{u_{1}^{2}+u_{1}u_{2}}{1-u_{1}-u_{2}}\lVert
f^{\prime}\rVert_{\infty}+u_{2}\lVert f^{\prime}\rVert_{\infty},$
$\displaystyle\biggl{\lvert}f\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}-f(r)+\frac{ru_{2}}{1-u_{1}-u_{2}}f^{\prime}(r)\biggr{\rvert}$
$\displaystyle\leq\lVert u\rVert^{2}\lVert
f^{\prime\prime}\rVert_{\infty}+\frac{u_{1}u_{2}+u_{2}^{2}}{1-u_{1}-u_{2}}\lVert
f^{\prime}\rVert_{\infty}+u_{1}\lVert f^{\prime}\rVert_{\infty},$

from where we deduce the existence of a constant $K>0$ such that

$\sup_{r\in[0,1]}\lvert\mathcal{L}^{(z)}f(r)\rvert\leq K.$

Now the rest of the proof follows as in Proposition 4.1 in [2]. ∎

Just before describing the culling procedure, let us remark that the generator
given in Proposition 3.3 coincides with the form of the generator in (3.2)
when $f$ only depends on $r$ and $z$ is held constant. Hence, the process
$R^{(z,r)}$ is seen to describe the dynamics of the ratio process $R$ under
the assumption that the total population stays constant through time.

###  3.3. Culling

We now turn to show that we can recover the dynamics of $R$ with a Markovian
process by applying the culling method, which we recall is a sampling
technique, inspired by the works of Gillespie [6, 7], that takes advantage of
the Markovianity of $(R,Z)$ and restarts the process with total population
$z>0$ at every sample time.

Before going into the technical aspect of the process, let us describe the
underlying idea behind it. For a fixed $n\in\mathbb{N}$ we let the process
$\\{(R_{t\wedge\tau},Z_{t\wedge\tau}):t\geq 0\\}$ evolve starting from the
point $(r,z)$, where $\tau$ is defined in Section 3.1. At time $t=1/n$ we
sample the first coordinate $R_{n^{-1}\wedge\tau}$, which serves as the value
to which a continuous-time Markov chain $\overline{R}^{(z,n)}$ jumps to. Then,
by using that $(R,Z)$ is a homogeneous Markov process, we restart the process
$\\{(R_{t\wedge\tau},Z_{t\wedge,\tau})\\}$ from the point
$(R_{n^{-1}\wedge\tau},z)$ by culling (or growing) the population accordingly.
The procedure just described is then followed inductively. Hence, we construct
a collection of continuous-time pure jump processes
$\\{\overline{R}^{(z,n)}:n\in\mathbb{N}\\}$ such that for a fixed
$n\in\mathbb{N}$, the trajectory $t\mapsto\overline{R}_{t}^{(z,n)}$ follows
the first coordinate of the bivariate process $(R,Z)$, which is observed
discretely, and with the property that as $n$ tends to infinity, the
fluctuations of $Z$ around $z$ become zero in a dense set of times over
$\mathbb{R}_{+}$.

Formally, for every $n\in\mathbb{N}$ we define a pure-jump Markov process
$\overline{R}^{(z,n)}$ with jump times $\\{T_{m}^{(n)}:m\geq 1\\}$ given by
independent exponentially distributed random variables with common rate $n$.
We denote the law of the process started at $r\in[0,1]$ by $P_{r}$. The
transition kernel $\rho^{(z,n)}:[0,1]\times\mathcal{B}([0,1])\to[0,1]$ of the
process will be given by

$\rho^{(z,n)}(r,A)\equiv P_{r}\bigl{(}\overline{R}_{T_{1}^{(n)}}^{(z,n)}\in
A\bigr{)}:=\mathbb{P}_{(r,z)}\bigl{(}R_{n^{-1}\wedge\tau}\in
A,Z_{n^{-1}\wedge\tau}\in\mathbb{R}_{+}\bigr{)},$

where $\mathbb{P}_{(r,z)}$ stands for the law of $(R,Z)$ started in $(r,z)$.
With these specifications the infinitesimal generator $\mathcal{A}^{(z,n)}$ of
$\overline{R}^{(z,n)}$ is bounded, see Proposition 17.2 in [13], and for $f\in
C([0,1])$ we get

(3.34)
$\mathcal{A}^{(z,n)}f(r)=n\int_{[0,1]}\bigl{(}f(y)-f(r)\bigr{)}\rho^{(z,n)}(r,dy).$

Using this process we can obtain an analogous result to Theorem 4.1 of [2],
which states that the sequence of processes $R^{(z,n)}$ converges weakly to
the unique solution of (3.12).

######  Theorem 3.1.

For any fixed $z>0$ and $T>0$, $\overline{R}^{(z,n)}\Rightarrow R^{(z,r)}$ as
$n\to\infty$ whenever $\overline{R}_{0}^{(z,n)}\Rightarrow R_{0}^{(z,r)}=r$ as
$n\to\infty$ in $D([0,T],[0,1])$ with the $J_{1}$ Skorokhod topology.

The proof of the previous result follows by a minor modification of the proof
of Theorem 4.1 in [2]. We include the proof for completeness.

###### Proof.

We start by noting that $[0,1]$ is compact and, by Proposition 3.3,
$R^{(z,r)}$ is Feller. Therefore, due to Theorem 17.28 in [13], it is enough
to prove that, for any $f\in C^{2}([0,1])$,

$\mathcal{A}^{(z,n)}f(r)\to\mathcal{L}^{(z)}f(r)\quad\text{uniformly for
$r\in[0,1]$ as $n\to\infty$,}$

where $\mathcal{L}^{(z)}$ is the generator defined in (3.30). By definition of
$\rho^{(z,n)}$ and (3.34) we obtain

(3.35)
$\mathcal{A}^{(z,n)}f(r)=n\Bigl{(}\mathbb{E}_{(r,z)}\bigl{[}f(R_{n^{-1}\wedge\tau})\bigr{]}-f(r)\Bigr{)}.$

Due to Proposition 3.1 we know that, for any $f\in C^{2}([0,1])$, we have the
equality

(3.36)
$f(R_{n^{-1}\wedge\tau})-f(r)=\int_{0}^{n^{-1}\wedge\tau}\mathcal{G}f(R_{s},Z_{s})ds+M_{n^{-1}\wedge\tau}^{f},$

where $M^{f}$ is a local martingale and

$\displaystyle\mathcal{G}f(r,z)={}$
$\displaystyle\bigl{[}\widetilde{D}(r)+S(r)+m(r)\bigr{]}f^{\prime}(r)+\frac{1}{2}\sigma(r)^{2}f^{\prime\prime}(r)$
$\displaystyle+rz\int_{U_{2}}\biggl{(}f\biggl{(}r+(1-r)\frac{w_{1}}{z+w_{1}+w_{2}}-r\frac{w_{2}}{z+w_{1}+w_{2}}\biggr{)}-f(r)-(1-r)\frac{w_{1}}{z}f^{\prime}(r)\biggr{)}\mu_{1}(dw)$
$\displaystyle+(1-r)z\int_{U_{2}}\biggl{(}f\biggl{(}r+(1-r)\frac{w_{1}}{z+w_{1}+w_{2}}-r\frac{w_{2}}{z+w_{1}+w_{2}}\biggr{)}-f(r)+r\frac{w_{2}}{z}f^{\prime}(r)\biggr{)}\mu_{2}(dw)$
$\displaystyle+\int_{U_{2}}\biggl{(}f\biggl{(}r+(1-r)\frac{w_{1}}{z+w_{1}+w_{2}}-r\frac{w_{2}}{z+w_{1}+w_{2}}\biggr{)}-f(r)\biggr{)}\nu(dw).$

Therefore, as $f\in C^{2}([0,1])$ and the fact that, by the definition of
$\tau$, $(R_{s},Z_{s})\in[0,1]\times[\varepsilon,L]$ for all
$s\in[0,n^{-1}\wedge\tau)$, by similar arguments to those given in the proof
of Proposition 3.3, we deduce the existence of a constant $K_{f}>0$ such that

(3.37) $\lvert\mathcal{G}f(R_{s},Z_{s})\rvert\leq K_{f}\quad\text{for all
$s\in[0,n^{-1}\wedge\tau)$}\quad\text{$\mathbb{P}$-a.s.}$

This entails that $\\{M_{t\wedge\tau}^{f}:t\in\mathbb{R}_{+}\\}$ is bounded,
and thus a martingale that starts in $0$.

Hence, taking expectation with respect to $\mathbb{P}_{(r,z)}$ in (3.36) and
using (3.35),

$\mathcal{A}^{(z,n)}f(r)=\mathbb{E}_{r,z}\Bigl{[}n\int_{0}^{n^{-1}\wedge\tau}\mathcal{G}f(R_{s},Z_{s})ds\Bigr{]}.$

Now, by the bound (3.37),

$\Bigl{\lvert}n\int_{0}^{n^{-1}\wedge\tau}\mathcal{G}f(R_{s},Z_{s})ds\Bigr{\rvert}\leq
K_{f}\quad\text{with whole probability.}$

Then, we may apply the Dominated Convergence Theorem to deduce

$\lim_{n\to\infty}\mathcal{A}^{(z,n)}f(r)=\lim_{n\to\infty}\mathbb{E}_{(r,z)}\Bigl{[}n\int_{0}^{n^{-1}\wedge\tau}\mathcal{G}f(R_{s},Z_{s})ds\Bigr{]}=\mathcal{G}f(r,z)=\mathcal{L}^{(z)}f(r).$

The uniform convergence follows by Lemma 31.7 of [18], concluding the proof. ∎

##  4\. Moment duality

The objective of the section is to present a process that will serve as a
moment dual to the culled frequency process defined in section 3. This will be
a block-counting process where blocks are allowed to branch and coalesce,
where the branching in the dual process is due to selection and coalescence
that arises as a result of the asymmetry in the reproductive mechanisms that
we consider in the two-type CBI with competition.

###  4.1. Block-counting process

To define the transition rates of the block counting process we will consider,
let us establish further notation for convenience. For $n\in\mathbb{N}$,
$k\in\\{1,2,3,\ldots,n\\}$ and $i\in\\{1,2\\}$ set

$\displaystyle\lambda_{n,k}^{i}:={}$
$\displaystyle\int_{[0,1]^{2}}u_{1}^{k}(1-u_{1}-u_{2})^{n-k}\mathbf{T}_{2}^{(z)}\mu_{i}(du),$
$\displaystyle\lambda_{n,k}^{I}:={}$
$\displaystyle\int_{[0,1]^{2}}u_{1}^{k}(1-u_{1}-u_{2})^{n-k}\mathbf{T}_{2}^{(z)}\nu(du).$

In addition, let us define

$\displaystyle\gamma_{n}^{1}:={}$
$\displaystyle\int_{[0,1]^{2}}\biggl{(}1-(1-u_{1}-u_{2})^{n}-n\frac{u_{1}}{1-u_{1}-u_{2}}\biggr{)}\mathbf{T}_{2}^{(z)}\mu_{1}(du),$
$\displaystyle\gamma_{n}^{2}:={}$
$\displaystyle\int_{[0,1]^{2}}\biggl{(}1-(1-u_{1}-u_{2})^{n}-n\frac{u_{2}}{1-u_{1}-u_{2}}\biggr{)}\mathbf{T}_{2}^{(z)}\mu_{2}(du),$
$\displaystyle\vartheta_{n,\mu}:={}$
$\displaystyle\int_{[0,1]^{2}}\bigl{(}1-(1-u_{2})^{n}\bigr{)}\mathbf{T}_{2}^{(z)}\mu_{1}(du),$
$\displaystyle\vartheta_{n,\nu}:={}$
$\displaystyle\int_{[0,1]^{2}}\bigl{(}1-(1-u_{2})^{n}\bigr{)}\mathbf{T}_{2}^{(z)}\nu(du).$

We aim to deduce moment duality, which is why we will consider the generator
$\mathcal{L}^{(z)}$ in Proposition 3.3 applied to monomials of the form
$f(r)=r^{n}$. Therefore we will need to assume that the functions $b_{ij}$ are
polynomials, this is

(AP) $b_{ij}(x)=\sum_{k=0}^{m_{ij}}a_{ij}^{(k)}x^{k},\quad i,j\in\\{1,2\\}.$

Suppose now that $n\in\mathbb{N}$, $r\in[0,1]$, and $f(r)=r^{n}$. Then we get

$f^{\prime}(r)=nr^{n-1}\quad\text{and}\quad
f^{\prime\prime}(r)=2\binom{n}{2}r^{n-2},$

where we consider $\binom{1}{2}=0$. Thus, by using (3.15) we deduce

$\frac{1}{2}\sigma(r)^{2}f^{\prime\prime}(r)=\binom{n}{2}\frac{2}{z}\bigl{[}c_{1}(r^{n-1}-r^{n})+(c_{1}-c_{2})(r^{n+1}-r^{n})\bigr{]}.$

Using $S$ as in (3.15) we get

$S(r)f^{\prime}(r)=n\frac{2}{z}(c_{1}-c_{2})(r^{n+1}-r^{n}),$

so

(4.1)
$\displaystyle\frac{1}{2}\sigma(r)^{2}f^{\prime\prime}(r)+S(r)f^{\prime}(r)$
(4.2)
$\displaystyle=\binom{n}{2}\frac{2}{z}c_{1}(r^{n-1}-r^{n})+\binom{n+1}{2}\frac{2}{z}(c_{1}-c_{2})(r^{n+1}-r^{n}).$

On the other hand, considering $m$ as in (3.15),

(4.3)
$m(r)f^{\prime}(r)=n\frac{\eta_{1}}{z}(r^{n-1}-r^{n})-n\frac{\eta_{2}}{z}r^{n}.$

Let us now consider the term $\widetilde{D}(r)f^{\prime}(r)$, with
$\widetilde{D}$ defined in (3.15). We start by noting that

$\displaystyle\frac{1-r}{z}b_{11}(zr)nr^{n-1}$
$\displaystyle=n\sum_{k=0}^{m_{11}}a_{11}^{(k)}z^{k-1}(r^{n+k-1}-r^{n+k})$
$\displaystyle=n\frac{a_{11}^{(0)}}{z}(r^{n-1}-r^{n})+n\sum_{k=2}^{m_{11}}a_{11}^{(k)}z^{k-1}(r^{n+k-1}-r^{n})-n\sum_{k=1}^{m_{11}}a_{11}^{(k)}z^{k-1}(r^{n+k}-r^{n})$
$\displaystyle=n\frac{a_{11}^{(0)}}{z}(r^{n-1}-r^{n})+n\sum_{k=1}^{m_{11}-1}\bigl{(}a_{11}^{(k+1)}z^{k}-a_{11}^{(k)}z^{k-1}\bigr{)}(r^{n+k}-r^{n})-na_{11}^{(m_{11})}z^{m_{11}-1}(r^{n+m_{11}}-r^{n}).$

Now we note that

$\displaystyle-\frac{r}{z}b_{22}\bigl{(}z(1-r)\bigr{)}nr^{n-1}$
$\displaystyle=n\sum_{l=0}^{m_{22}}(-a_{22}^{(l)}z^{l-1})\sum_{k=0}^{l}\binom{l}{k}(-1)^{k}r^{n+k}=n\sum_{k=0}^{m_{22}}(-1)^{k}\bigl{[}\sum_{l=k}^{m_{22}}\binom{l}{k}(-a_{22}^{(l)}z^{l-1})\bigr{]}r^{n+k}$
$\displaystyle=n\sum_{k=0}^{m_{22}}(-1)^{k}\bigl{[}\sum_{l=k}^{m_{22}}\binom{l}{k}(-a_{22}^{(l)}z^{l-1})\bigr{]}(r^{n+k}-r^{n})+n\sum_{k=0}^{m_{22}}(-1)^{k}\bigl{[}\sum_{l=k}^{m_{22}}\binom{l}{k}(-a_{22}^{(l)}z^{l-1})\bigr{]}r^{n}$
$\displaystyle=n\sum_{k=1}^{m_{22}}(-1)^{k+1}\bigl{[}\sum_{l=k}^{m_{22}}\binom{l}{k}a_{22}^{(l)}z^{l-1}\bigr{]}(r^{n+k}-r^{n})-nr^{n}\sum_{l=0}^{m_{22}}a_{22}^{(l)}z^{l-1}\sum_{k=0}^{l}\binom{l}{k}(-1)^{k}$
$\displaystyle=n\sum_{k=1}^{m_{22}}(-1)^{k+1}\bigl{[}\sum_{l=k}^{m_{22}}\binom{l}{k}a_{22}^{(l)}z^{l-1}\bigr{]}(r^{n+k}-r^{n})-n\frac{a_{22}^{(0)}}{z}r^{n}.$

Similarly, we obtain

$\displaystyle-\frac{r}{z}b_{21}(rz)nr^{n-1}$
$\displaystyle=\sum_{k=1}^{m_{21}}n(-a_{21}^{(k)})z^{k-1}(r^{n+k}-r^{n})-n\frac{b_{21}(z)}{z}r^{n},$

and

$\displaystyle\frac{1-r}{z}b_{12}\bigl{(}z(1-r)\bigr{)}nr^{n-1}={}$
$\displaystyle n\frac{b_{12}(z)}{z}(r^{n-1}-r^{n})$
$\displaystyle+n\sum_{k=1}^{m_{12}-1}(-1)^{k+1}\bigl{[}a_{12}^{(k)}z^{k-1}+\sum_{l=k+1}^{m_{12}}\binom{l+1}{k+1}a_{12}^{(l)}z^{l-1}\bigr{]}(r^{n+k}-r^{n})$
$\displaystyle+n(-1)^{m_{12}+1}a_{12}^{(m_{12})}z^{m_{12}-1}(r^{n+m_{12}}-r^{n}).$

Thus, by defining for $k\in\mathbb{N}$,

$\displaystyle\theta_{k}:={}$ $\displaystyle
a_{11}^{(k+1)}z^{k}1_{\\{k<m_{11}\\}}-a_{11}^{(k)}z^{k-1}1_{\\{k\leq
m_{11}\\}}-a_{21}^{(k)}z^{k-1}1_{\\{k\leq
m_{21}\\}}+\biggl{[}(-1)^{k+1}\sum_{l=k}^{m_{22}}\binom{l}{k}a_{22}^{(l)}z^{l-1}\biggr{]}1_{\\{k\leq
m_{22}\\}}$
$\displaystyle+(-1)^{k+1}\biggl{[}\sum_{l=k+1}^{m_{12}}\binom{l+1}{k+1}a_{12}^{(l)}z^{l-1}\biggr{]}1_{\\{k<m_{12}\\}}+(-1)^{k+1}a_{12}^{(k)}z^{k-1}1_{\\{k\leq
m_{12}\\}},$

and employing the definition of $\widetilde{D}$ in (3.15) we deduce

(4.4)
$\widetilde{D}(r)f^{\prime}(r)=n\frac{a_{11}^{(0)}+b_{12}(z)}{z}(r^{n-1}-r^{n})+\sum_{k=1}^{\tilde{m}}n\theta_{k}(r^{n+k}-r^{n})-n\frac{a_{22}^{(0)}+b_{21}(z)}{z}r^{n},$

where $\tilde{m}=m_{11}\vee m_{12}\vee m_{21}\vee m_{22}$.

We now turn to the terms corresponding to jumps in the culled frequency
process. By using the equalities

$\displaystyle
zr\biggl{[}\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}^{n}-r^{n}-\frac{(1-r)u_{1}}{1-u_{1}-u_{2}}nr^{n-1}\biggr{]}$
$\displaystyle=z\sum_{k=1}^{n}\binom{n}{k}u_{1}^{k}(1-u_{1}-u_{2})^{n-k}(r^{n-k+1}-r^{n})-z\bigl{(}1-(1-u_{2})^{n}\bigr{)}r^{n}$
$\displaystyle\quad-z\biggl{[}1-(1-u_{1}-u_{2})^{n}-\frac{nu_{1}}{1-u_{1}-u_{2}}\biggr{]}(r^{n+1}-r^{n}),$
$\displaystyle
z(1-r)\biggl{[}\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}^{n}-r^{n}+\frac{ru_{2}}{1-u_{1}-u_{2}}nr^{n-1}\biggr{]}$
$\displaystyle=z\sum_{k=2}^{n+1}\binom{n}{k-1}u_{1}^{k-1}(1-u_{1}-u_{2})^{n-k+1}(r^{n-k+1}-r^{n})$
$\displaystyle\quad-z\sum_{k=2}^{n}\binom{n}{k}u_{1}^{k}(1-u_{1}-u_{2})^{n-k}(r^{n-k+1}-r^{n})$
$\displaystyle\quad+z\biggl{[}1-(1-u_{1}-u_{2})^{n}-\frac{nu_{2}}{1-u_{1}-u_{2}}\biggr{]}(r^{n+1}-r^{n}),$

and

$\displaystyle\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}^{n}-r^{n}$
$\displaystyle=\sum_{k=1}^{n}\binom{n}{k}u_{1}^{k}(1-u_{1}-u_{2})^{n-k}(r^{n-k}-r^{n})-\bigl{(}1-(1-u_{2})^{n}\bigr{)}r^{n},$

we deduce

(4.5) $\displaystyle
zr\int_{[0,1]^{2}}\biggl{[}f\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}-f(r)-\frac{(1-r)u_{1}}{1-u_{1}-u_{2}}f^{\prime}(r)\biggr{]}\mathbf{T}_{2}^{(z)}\mu_{1}(du)$
(4.6)
$\displaystyle\quad+z(1-r)\int_{[0,1]^{2}}\biggl{[}f\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}-f(r)+\frac{ru_{2}}{1-u_{1}-u_{2}}f^{\prime}(r)\biggr{]}\mathbf{T}_{2}^{(z)}\mu_{2}(du)$
(4.7)
$\displaystyle\quad+\int_{[0,1]^{2}}\Bigl{[}f\bigl{(}r+u_{1}(1-r)-u_{2}r\bigr{)}-f(r)\Bigr{]}\mathbf{T}_{2}^{(z)}\nu(du)$
(4.8)
$\displaystyle=\sum_{k=2}^{n}\binom{n}{k}z\lambda_{n,k}^{1}(r^{n-k+1}-r^{n})-z\vartheta_{n,\mu}r^{n}-z\gamma_{n}^{1}(r^{n+1}-r^{n})$
(4.9)
$\displaystyle\quad+z\lambda_{n,n}^{2}(1-r^{n})+\sum_{k=2}^{n}\biggl{[}\binom{n}{k-1}z\lambda_{n,k-1}^{2}-\binom{n}{k}z\lambda_{n,k}^{2}\biggr{]}(r^{n-k+1}-r^{n})+z\gamma_{n}^{2}(r^{n+1}-r^{n})$
(4.10)
$\displaystyle\quad+\sum_{k=1}^{n}\binom{n}{k}\lambda_{n,k}^{I}(r^{n-k}-r^{n})-\vartheta_{n,\nu}r^{n}.$

Finally, let us define the collection
$\\{q_{nm}:n,m\in\mathbb{N}_{0}\cup\\{\dagger\\}\\}$ by setting

(4.11) $q_{nm}=\begin{dcases*}\begin{aligned}
&1_{\\{m=n-1\\}}\biggl{(}\binom{n}{2}\frac{2}{z}c_{1}+n\frac{a_{11}^{0}+b_{12}(z)+\eta_{1}}{z}\biggr{)}\\\
&\quad+1_{\\{m>0\\}}\binom{n}{n-m+1}z(\lambda_{n,n-m+1}^{1}-\lambda_{n,n-m+1}^{2})\\\
&\quad+\binom{n}{n-m}z\lambda_{n,n-m}^{2}+\binom{n}{n-m}\lambda_{n,n-m}^{I}\end{aligned}&if
$n\in\mathbb{N}$ and $0\leq m<n$,\\\
1_{\\{m=n+1\\}}\biggl{(}\binom{n+1}{2}\frac{2}{z}(c_{1}-c_{2})-z(\gamma_{n}^{1}-\gamma_{n}^{2})\biggr{)}+n\theta_{m-n}&if
$n\in\mathbb{N}$ and $n<m\leq n+\tilde{m}$,\\\
z\vartheta_{n,\mu}+\vartheta_{n,\nu}+n\frac{a_{22}^{(0)}+b_{21}(z)+\eta_{2}}{z}&if
$n\in\mathbb{N}$ and $j=\dagger$,\\\ 0&otherwise.\end{dcases*}$

Note that if $q_{nm}\geq 0$ for all $n\neq m$, then this collection defines
the transition rates of a continuous-time Markov Chain with state space
$\mathbb{N}_{0}\cup\\{\dagger\\}$. Now we consider the function
$H(r,n):[0,1]\times(\mathbb{N}_{0}\cup\\{\dagger\\})\to[0,1]$ defined by
$H(r,n)=r^{n}$ for $n\in\mathbb{N}_{0}$ and $H(r,\dagger)=0$. Under the
assumption of $q_{nm}\geq 0$ for all $n\neq m$, using the equalities (4.1),
(4.3), (4.4) and (4.5) we deduce that for any $r\in[0,1]$ and any
$n\in\mathbb{N}_{0}$,

$\displaystyle\mathcal{L}^{(z)}H(r,n)$
$\displaystyle=\sum_{m=0}^{n-1}q_{nm}(r^{m}-r^{n})+\sum_{m=n+1}^{n+\tilde{m}}q_{nm}(r^{m}-r^{n})-q_{n\dagger}r^{n}=\mathcal{Q}^{(z)}H(r,n),$

where $\mathcal{Q}^{(z)}$ is the generator defined by the collection
$\\{q_{nm}:n,m\in\mathbb{N}_{0}\cup\\{\dagger\\}\\}$. This reasoning proves
the following theorem.

######  Theorem 4.1.

Assume that (AP) holds. If $q_{nm}\geq 0$ for all $n\neq m$, then for any
$r\in[0,1]$ and $n\in\mathbb{N}_{0}\cup\\{\dagger\\}$ and any $t>0$ we have

$\mathbb{E}_{r}\bigl{[}(R_{t}^{(z,r)})^{n}\bigr{]}=\mathbb{E}_{n}\bigl{[}r^{N_{t}}\bigr{]},$

where $N$ is a continuous-time Markov Chain over
$\mathbb{N}_{0}\cup\\{\dagger\\}$ whose transition rates are defined by
(4.11).

######  Remark 4.1.

As pointed out in section 6 of [2], the assumption $q_{nm}\geq 0$ for all
$n\neq m$ restricts the cases we may study by duality to those where there
exists a selective advantage of one type (in this case type 1) over the other.
Moreover, it imposes restrictions, apart from $b_{12},b_{21}\geq 0$ on the
polynomials we can take into account for the model.

###  4.2. Biological interpretation of duality

Whenever the dual process of the culled frequency process exists, the rates
given in (4.11) allow us to give a characterization of the evolutionary forces
that arise due to the different reproduction mechanisms of the two species
that constitute the population. Most of the mechanisms we will mention also
appear in Section 9 of [2], but there are new terms that appear in the moment
dual because, unlike the population model in [2], we are not assuming
independence between the processes $X^{(1)}$ and $X^{(2)}$. These effects are
the following:

Selection:

Considering the branching rate in (4.11), we find two types of selection:

  * (i):

Classical selection: This type can be found in the following two terms: The
term $\theta_{1}$, which is the difference between the drift terms; and
$2(c_{1}-c_{2})/z$, observed previously by Gillespie [6, 7], corresponding to
the difference between diffusion terms.

  * (ii):

Frequency dependent selection: This kind of selection arises in two forms. The
first one is present in the transition rates (4.11) through the terms
$\theta_{m}$ for $m\geq 2$. This type of selection was observed in [9], and we
want to emphasize that this form of selection can lead to coexistence in the
culled frequency process. The second one is due to the difference in the jump
measures of the two-type CBI with competition in the terms
$-z(\gamma_{n}^{1}-\gamma_{n}^{2})$. This kind of selection arises from the
asymmetry in large reproduction events between the types in the population and
was recently studied in the context of the ancestral selection graph of a
$\Lambda$-asymmetric Moran model in [10].

Frequency dependent variance:

As noted in [2], from the form of the culled frequency process (given as the
solution to (3.12)), one observes that the asymmetry between the diffusion
terms $c_{1}$ and $c_{2}$ modifies the variance. In [11], this term was
obtained in the context of populations that require different amounts of
resources to reproduce and is referred to as _efficiency_.

Coalescence:

Both the terms $2c_{1}/z$ and $z(\lambda_{n,n-m+1}^{1}-\lambda_{n,n-m+1}^{2})$
are associated with coalescence; the former being the rate of pairwise
mergers, while the latter corresponds to mergers of multiple ancestral lines.

Mutation:

Mutation is obtained through the inclusion of immigration and cross-branching
into the model, and is found in the terms $(a_{11}^{0}+b_{12}(z)+\eta_{1})/z$,
$\lambda_{n,n-m}^{2}$, $\lambda_{n,n-m}^{I}$,
$(a_{22}^{0}+b_{21}(z)+\eta_{2})/z$, $z\vartheta_{n,\mu}$ and
$\vartheta_{n,\nu}$. Apart from the terms included in [2] we obtain new terms
thanks to the cross-branching; namely, $\lambda_{n,n-m}^{2}$, $b_{12}(z)/z$,
$b_{21}(z)/z$ and $z\vartheta_{n,\mu}$.

##  5\. Large population limit

In previous sections, we developed the culled frequency process, $R^{(z,r)}$,
specifically designed to describe the dynamics of the first coordinate of
$(R,Z)$, while keeping the total population size constant at $z>0$. In this
section, we explore the large population limit of the culled frequency
process, proving that it exists under certain hypotheses, and present two
examples that exhibit coexistence. The latter term means that, in these
examples, neither population becomes extinct; equivalently, the frequency
process never reaches $0$ or $1$.

###  5.1. The large population limit

We start by noting that if we see the culled frequency process as the solution
of the SDE (3.12), we can actually consider that the functions $b_{ij}$ depend
on $z$ (which is considered fixed for (3.12)). We stress this fact by writing
$b_{ij}^{(z)}$ for the rest of the section. This dependence will allow us to
introduce suitable rescalings such that the limit

(5.1) $\lim_{z\to\infty}\frac{b_{ij}^{(z)}(zr)}{z}$

is well defined and finite for all $r\in[0,1]$. To see that this is necessary,
notice that if $b_{11}^{(z)}(x)=x(1-x)$ for all $z>0$, then the limit in (5.1)
exists but is equal to $-\infty$, while the limit does not exist if
$b_{11}^{(z)}=x\sin(x)$ for all $z>0$.

Let us assume then that the limit in (5.1) exists uniformly for $r\in[0,1]$
and denote said limit by $\beta_{ij}$. Let us further assume that
$\beta_{ij}(r)=\lim_{z\to\infty}b_{ij}^{(z)}(rz)/z$ is Lipschitz in $[0,1]$.
Having stated the assumptions on $\beta_{ij}$, consider the ordinary
differential equation

(5.2) $\displaystyle dR_{t}^{(\infty,r)}={}$
$\displaystyle\beta_{11}(R_{t}^{(\infty,r)})(1-R_{t}^{(\infty,r)})dt-\beta_{22}(1-R_{t}^{(\infty,r)})R_{t}^{(\infty,r)}dt$
(5.3)
$\displaystyle+\beta_{12}(1-R_{t}^{(\infty,r)})(1-R_{t}^{(\infty,r)})dt-\beta_{21}(R_{t}^{(\infty,r)})R_{t}^{(\infty,r)}dt$
(5.4)
$\displaystyle+(1-R_{t}^{(\infty,r)})^{2}\int_{U_{2}}w_{1}\mu_{2}(dw)-(R_{t}^{(\infty,r)})^{2}\int_{U_{2}}w_{2}\mu_{1}(dw),$

with initial condition $R_{0}^{(\infty,r)}=r$. By comparing (3.12) with (5.2)
one might guess that the solution of (5.2) is the large population limit of
the culled frequency process. Indeed, this is what the following result
states.

######  Theorem 5.1.

For any $T>0$ we get

$\lim_{z\to\infty}\mathbb{E}\biggl{[}\sup_{t\leq T}\lvert
R_{t}^{(z,r)}-R_{t}^{(\infty,r)}\rvert^{2}\biggr{]}=0;$

this is, $R^{(z,r)}\to R^{(\infty,r)}$ uniformly in compacts in $L^{2}$ as
$z\to\infty$.

###### Proof.

Considering the SDE (3.12) and the ODE (5.2) we get

$R_{t}^{(z,r)}-R_{t}^{(\infty,r)}=A_{t}^{(z,1)}+A_{t}^{(z,2)}+A_{t}^{(z,3)},$

where

$\displaystyle A_{t}^{(z,1)}={}$
$\displaystyle\int_{0}^{t}\sigma(R_{s-}^{(z,r)})dB_{s}+\int_{0}^{t}\int_{U_{2}}\bigl{[}\widetilde{g}^{(z)}(R_{s-}^{(z,r)},w)+\widetilde{h}^{(z)}(R_{s-}^{(z,r)},w)\bigr{]}\widetilde{N}_{3}(ds,dw)$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\bigl{[}g_{1}^{(z)}(R_{s-}^{(z,r)},w,v)+h_{1}^{(z)}(R_{s-}^{(z,r)},w,v)\bigr{]}\widetilde{N}_{1}(ds,dw,dv)$
$\displaystyle+\int_{0}^{t}\int_{U_{2}}\int_{0}^{\infty}\bigl{[}g_{2}^{(z)}(R_{s-}^{(z,r)},w,v)+h_{2}^{(z)}(R_{s-}^{(z,r)},w,v)\bigr{]}\widetilde{N}_{2}(ds,dw,dv),$
$\displaystyle A_{t}^{(z,2)}={}$
$\displaystyle\int_{0}^{t}\bigl{[}S(R_{s-}^{(z,r)})+S_{c}(R_{s-}^{(z,r)})+m(R_{s-}^{(z,r)})\bigr{]}ds+\int_{0}^{t}\int_{U_{2}}\bigl{[}\widetilde{g}^{(z)}(R_{s-}^{(z,r)},w)+\widetilde{h}^{(z)}(R_{s-}^{(z,r)},w)\bigr{]}\nu(dw)ds,$
and $\displaystyle A_{t}^{(z,3)}={}$
$\displaystyle\int_{0}^{t}\Bigl{[}\widetilde{D}(R_{s-}^{(z,r)})-\begin{aligned}
\bigl{(}&\beta_{11}(R_{s-}^{(\infty,r)})(1-R_{s-}^{(\infty,r)})-\beta_{22}(1-R_{s-}^{(\infty,r)})R_{s-}^{(\infty,r)}\\\
&+\beta_{12}(1-R_{s-}^{(\infty,r)})(1-R_{s-}^{(\infty,r)})-\beta_{21}(R_{s-}^{(\infty,r)})R_{s-}^{(\infty,r)}\bigr{)}\Bigr{]}ds\end{aligned}$
$\displaystyle+\int_{0}^{t}\Bigl{[}m_{c}^{1}(R_{s-}^{(z,r)})+(R_{s-}^{(\infty,r)})^{2}\int_{U_{2}}w_{2}\mu_{1}(dw)\Bigr{]}ds$
$\displaystyle+\int_{0}^{t}\Bigl{[}m_{c}^{2}(R_{s-}^{(z,r)})-(1-R_{s-}^{(\infty,r)})^{2}\int_{U_{2}}w_{1}\mu_{2}(dw)\Bigr{]}ds.$

We first consider the term $A_{t}^{(z,1)}$. Note that by the definition of
$\sigma$ given in (3.15) together with Doob’s $L^{2}$ inequality and Itô’s
isometry we obtain, for any $t\in[0,T]$,

$\displaystyle\mathbb{E}\biggl{[}\sup_{u\leq
t}\biggl{(}\int_{0}^{u}\sigma(R_{s-}^{(z,r)})dB_{s}\biggr{)}^{2}\biggr{]}$
$\displaystyle\leq
4\mathbb{E}\biggl{[}\int_{0}^{T}\sigma(R_{s-}^{(z,r)})^{2}ds\biggr{]}\leq\frac{8}{z}(c_{1}+c_{2})T.$

Applying again Doob’s $L^{2}$ inequality we deduce

$\displaystyle\mathbb{E}\biggl{[}\sup_{u\leq
t}\biggl{(}\int_{0}^{u}\int_{U_{2}}\bigl{[}\widetilde{g}^{(z)}(R_{s-}^{(z,r)},w)+\widetilde{h}^{(z)}(R_{s-}^{(z,r)},w)\bigr{]}\widetilde{N}_{3}(ds,dw)\biggr{)}^{2}\biggr{]}$
$\displaystyle\leq
4\mathbb{E}\biggl{[}\int_{0}^{T}\int_{U_{2}}\bigl{[}\widetilde{g}^{(z)}(R_{s-}^{(z,r)},w)+\widetilde{h}^{(z)}(R_{s-}^{(z,r)},w)\bigr{]}^{2}\nu(dw)ds\biggr{]}$
$\displaystyle\leq
4T\int_{U_{2}}\frac{(w_{1}+w_{2})^{2}}{(z+w_{1}+w_{2})}\nu(dw).$

Analogous computations yield

$\displaystyle\sum_{i=1}^{2}\mathbb{E}\biggl{[}\sup_{u\leq
t}\biggl{(}\int_{0}^{u}\int_{U_{2}}\int_{0}^{\infty}\bigl{[}g_{i}^{(z)}(R_{s-}^{(z,r)},w,v)+h_{i}^{(z)}(R_{s-}^{(z,r)},w,v)\bigr{]}\widetilde{N}_{i}(ds,dw,dv)\biggr{)}^{2}\biggr{]}$
$\displaystyle\leq
4T\int_{U_{2}}\frac{z(w_{1}+w_{2})^{2}}{(z+w_{1}+w_{2})^{2}}(\mu_{1}+\mu_{2})(dw).$

By the Dominated Convergence Theorem we note that

$\int_{U_{2}}\frac{(w_{1}+w_{2})^{2}}{(z+w_{1}+w_{2})}\nu(dw)+\int_{U_{2}}\frac{z(w_{1}+w_{2})^{2}}{(z+w_{1}+w_{2})^{2}}(\mu_{1}+\mu_{2})(dw)\to
0\quad\text{as $z\to\infty$.}$

Hence, we deduce the existence of a collection of constants
$\\{K_{1}(T,z):z>0\\}$ that fulfill $K_{1}(T,z)\to 0$ as $z\to\infty$ and

$\mathbb{E}\Bigl{[}\sup_{u\leq t}(A_{u}^{(z,1)})^{2}\Bigr{]}\leq
K_{1}(T,z)\quad\text{for all $t\in[0,T]$ and $z>0$.}$

Meanwhile, to control the term $A_{t}^{(z,2)}$ we simply use the triangle
inequality to deduce, for any $t\in[0,T]$, that

$\displaystyle\mathbb{E}\Bigl{[}\sup_{u\leq
t}(A_{u}^{(z,2)})^{2}\Bigr{]}\leq{}$ $\displaystyle
4\biggl{(}\int_{0}^{T}\biggl{(}\frac{\lvert
c_{2}-c_{1}\rvert}{2z}+\frac{\eta_{1}+\eta_{2}}{z}\biggr{)}ds\biggr{)}^{2}+4\biggl{(}\int_{0}^{T}\int_{U_{2}}\frac{w_{1}(w_{1}+w_{2})}{z+w_{1}+w_{2}}\mu_{1}(dw)ds\biggr{)}^{2}$
$\displaystyle+4\biggl{(}\int_{0}^{T}\int_{U_{2}}\frac{w_{2}(w_{1}+w_{2})}{z+w_{1}+w_{2}}\mu_{2}(dw)ds\biggr{)}^{2}+4\biggl{(}\int_{0}^{T}\int_{U_{2}}\frac{w_{1}+w_{2}}{z+w_{1}+w_{2}}\nu(dw)ds\biggr{)}^{2}$
$\displaystyle={}$ $\displaystyle 4T^{2}\begin{aligned}
\biggl{(}&\frac{1}{4z^{2}}\bigl{(}\lvert
c_{2}-c_{1}\rvert+2\eta_{1}+2\eta_{2}\bigr{)}^{2}+\biggl{(}\int_{U_{2}}\frac{w_{1}(w_{1}+w_{2})}{z+w_{1}+w_{2}}\mu_{1}(dw)\biggr{)}^{2}\\\
&+\biggl{(}\int_{U_{2}}\frac{w_{2}(w_{1}+w_{2})}{z+w_{1}+w_{2}}\mu_{2}(dw)\biggr{)}^{2}+\biggl{(}\int_{U_{2}}\frac{w_{1}+w_{2}}{z+w_{1}+w_{2}}\nu(dw)\biggr{)}^{2}\biggr{)}.\end{aligned}$

Thus, by noting that

$\sum_{i=1}^{2}\int_{U_{2}}\frac{w_{i}(w_{1}+w_{2})}{z+w_{1}+w_{2}}\mu_{i}(dw)+\int_{U_{2}}\frac{w_{1}+w_{2}}{z+w_{1}+w_{2}}\nu(dw)\to
0\quad\text{as }z\to\infty$

by the Dominated Convergence Theorem, we obtain a collection of constants
$\\{K_{2}(T,z):z>0\\}$ that satisfy both $K_{2}(T,z)\to 0$ as $z\to\infty$ and

$\mathbb{E}\Bigl{[}\sup_{u\leq t}(A_{u}^{(z,2)})^{2}\Bigr{]}\leq
K_{2}(T,z)\quad\text{for all $t\in[0,T]$ and $z>0$.}$

Let us now tackle the term $A_{t}^{(z,3)}$. Recall, from (3.15), that for
$r\in[0,1]$,

$\widetilde{D}(r)=\frac{b_{11}^{(z)}(zr)}{z}(1-r)-\frac{b_{22}^{(z)}(z(1-r))}{z}r+\frac{b_{12}{(z)}(z(1-r))}{z}(1-r)-\frac{b_{21}^{(z)}(zr)}{z}r.$

Now note that, for $t\in[0,T]$,

$\displaystyle\mathbb{E}\biggl{[}\sup_{u\leq
t}\biggl{(}\int_{0}^{u}\biggl{(}\frac{b_{11}^{(z)}(R_{s-}^{(z,r)})}{z}(1-R_{s-}^{(z,r)})-\beta_{11}(R_{s-}^{(\infty,r)})(1-R_{s-}^{(\infty,r)})\biggr{)}ds\biggr{)}^{2}\biggr{]}$
$\displaystyle\leq{}$ $\displaystyle 3\mathbb{E}\biggl{[}\sup_{u\leq
t}\biggl{(}\int_{0}^{u}\biggl{[}\frac{b_{11}^{(z)}(zR_{s-}^{(z,r)})}{z}-\beta_{11}(R_{s-}^{(z,r)})\biggr{]}(1-R_{s-}^{z,r})ds\biggr{)}^{2}\biggr{]}$
$\displaystyle+3\mathbb{E}\biggl{[}\sup_{u\leq
t}\Bigl{(}\int_{0}^{u}\bigl{[}\beta_{11}(R_{s-}^{(z,r)})-\beta_{11}(R_{s-}^{(\infty,r)})\bigr{]}(1-R_{s-}^{(z,r)})ds\Bigr{)}^{2}\biggr{]}$
$\displaystyle+3\mathbb{E}\biggl{[}\sup_{u\leq
t}\Bigl{(}\int_{0}^{u}\beta_{11}(R_{s-}^{(\infty,r)})(R_{s-}^{(\infty,r)}-R_{s-}^{(z,r)})ds\Bigr{)}^{2}\biggr{]},$

By assumption $\beta_{11}$ is Lipschitz, say with constant $L_{11}>0$, so
$\lVert\beta_{11}\rVert_{\infty}:=\sup\\{\lvert\beta_{11}(r)\rvert:r\in[0,1]\\}<\infty$.
Moreover, $b_{11}^{(z)}(zr)/z\to\beta_{11}(r)$ uniformly on $r\in[0,1]$, so
there exists a collection of constants $\\{C_{z}^{(11)}:z>0\\}$, with
$C_{z}^{11}\to 0$ as $z\to\infty$, for which

$\sup_{r\in[0,1]}\biggl{\lvert}\frac{b_{11}^{(z)}(zr)}{z}-\beta_{11}(r)\biggr{\rvert}\leq
C_{z}^{(11)}\quad\text{for $z>0$.}$

By these remarks, we deduce

$\displaystyle\mathbb{E}\biggl{[}\sup_{u\leq
t}\biggl{(}\int_{0}^{u}\biggl{(}\frac{b_{11}^{(z)}(R_{s-}^{(z,r)})}{z}(1-R_{s-}^{(z,r)})-\beta_{11}(R_{s-}^{(\infty,r)})(1-R_{s-}^{(\infty,r)})\biggr{)}ds\biggr{)}^{2}\biggr{]}$
$\displaystyle\leq
3T^{2}C_{z}^{(11)}+3(L_{11}^{2}+\lVert\beta_{11}\rVert_{\infty}^{2})\mathbb{E}\biggl{[}\sup_{u\leq
t}\Bigl{(}\int_{0}^{u}\lvert R_{s-}^{(z,r)}-R_{s-}^{(\infty,r)}\rvert
ds\Bigr{)}^{2}\biggr{]}$ $\displaystyle\leq
3T^{2}C_{z}^{(11)}+3(L_{11}^{2}+\lVert\beta_{11}\rVert_{\infty}^{2})T\mathbb{E}\Bigl{[}\int_{0}^{t}\sup_{u\leq
s}\lvert R_{u}^{(z,r)}-R_{u}^{(\infty,r)}\rvert^{2}ds\Bigr{]}.$

By proceeding in the same way for the rest of the terms in $\widetilde{D}$, it
is seen that there is a collection of constants $\\{C_{z}^{(D)}:z>0\\}$, with
$C_{z}^{(D)}\to 0$ as $z\to\infty$, and a constant $C_{D}>0$ such that

$\displaystyle\mathbb{E}\biggl{[}\sup_{u\leq
t}\Bigl{(}\int_{0}^{u}(\widetilde{D}(R_{s-}^{(z,r)})-\widetilde{D}_{\infty}(R_{s-}^{(\infty,r)}))ds\Bigr{)}^{2}\biggr{]}$
$\displaystyle\leq
T^{2}C_{z}^{(D)}+TC_{D}\mathbb{E}\Bigl{[}\int_{0}^{t}\sup_{u\leq s}\lvert
R_{u}^{(z,r)}-R_{u}^{(\infty,r)}\rvert^{2}ds\Bigr{]},$

where

$\widetilde{D}_{\infty}(r)\coloneq\beta_{11}(r)(1-r)-\beta_{22}(1-r)r+\beta_{12}(1-r)(1-r)-\beta_{21}(r)r.$

On the other hand, we see that

$\displaystyle\mathbb{E}\biggl{[}\sup_{u\leq
t}\biggl{(}\int_{0}^{u}\Bigl{(}m_{c}^{1}(R_{s-}^{(z,r)})+(R_{s-}^{(\infty,r)})^{2}\int_{U_{2}}w_{2}\mu_{1}(dw)\Bigr{)}ds\biggr{)}^{2}\biggr{]}$
$\displaystyle\leq{}$ $\displaystyle 2\mathbb{E}\biggl{[}\sup_{u\leq
t}\biggl{(}\int_{0}^{u}\int_{U_{2}}\frac{w_{2}(w_{1}+w_{2})}{z+w_{1}+w_{2}}\mu_{1}(dw)(R_{s-}^{(z,r)})^{2}ds\biggr{)}^{2}\biggr{]}$
$\displaystyle+2\mathbb{E}\biggl{[}\sup_{u\leq
t}\Bigl{(}\int_{0}^{u}\int_{U_{2}}w_{2}\mu_{1}(dw)\bigl{[}(R_{s-}^{(\infty,r)})^{2}-(R_{s-}^{(z,r)})^{2}\bigr{]}ds\Bigr{)}^{2}\biggr{]}$
$\displaystyle\leq{}$ $\displaystyle
2\Bigl{(}\int_{U_{2}}\frac{w_{2}(w_{1}+w_{2})}{z+w_{1}+w_{2}}\mu_{1}(dw)\Bigr{)}^{2}T^{2}+8\Bigl{(}\int_{U_{2}}w_{2}\mu_{1}(dw)\Bigr{)}^{2}T\mathbb{E}\Bigl{[}\int_{0}^{t}\sup_{u\leq
s}\lvert R_{u}^{(z,r)}-R_{u}^{(\infty,r)}\rvert^{2}ds\Bigr{]}.$

By doing similar computations to the term involving $m_{c}^{2}$, from these
bounds we deduce the existence of $\\{K_{3}(T,z):z>0\\}$ and $K_{4}(T)$ such
that $K_{3}(T,z)\to 0$ as $z\to\infty$ and

$\mathbb{E}\Bigl{[}\sup_{u\leq t}(A_{u}^{(z,3)})^{2}\Bigr{]}\leq
K_{3}(T,z)+K_{4}(T)\mathbb{E}\Bigl{[}\int_{0}^{t}\sup_{u\leq s}\lvert
R_{u}^{(z,r)}-R_{u}^{(\infty,r)}\rvert^{2}ds\Bigr{]}\quad\text{for all
$t\in[0,T]$ and $z>0$.}$

As a consequence, for all $t\in[0,T]$ and $z>0$, we get the inequality

$\mathbb{E}\Bigl{[}\sup_{u\leq t}\lvert
R_{u}^{(z,r)}-R_{u}^{(\infty,r)}\rvert^{2}\Bigr{]}\leq
3\bigl{(}K_{1}(T,z)+K_{2}(T,z)+K_{3}(T,z)\bigr{)}+3K_{4}(T)\int_{0}^{t}\mathbb{E}\Bigl{[}\sup_{u\leq
s}\lvert R_{u}^{(z,r)}-R_{u}^{(\infty,r)}\rvert^{2}\Bigr{]}ds.$

Now, Gronwall’s lemma entails

$\mathbb{E}\Bigl{[}\sup_{u\leq T}\lvert
R_{u}^{(z,r)}-R_{u}^{(\infty,r)}\rvert^{2}\Bigr{]}\leq
3\bigl{(}K_{1}(T,z)+K_{2}(T,z)+K_{3}(T,z)\bigr{)}\exp\\{3K_{4}(T)T\\}.$

The proof concludes by noting that
$\bigl{(}K_{1}(T,z)+K_{2}(T,z)+K_{3}(T,z)\bigr{)}\exp\\{3K_{4}(T)T\\}\to 0$ as
$z\to\infty$. ∎

Given Theorem 5.1, we now proceed to explain two particular cases where
coexistence arises in the dynamical system described by (5.2) by different
mechanisms: mutation and logistic growth. The analysis done in section 5.2 and
section 5.3 is based on the techniques contained in Chapter 2 of [20], see in
particular section 2.2 and section 2.4 of [20].

###  5.2. Linear coefficients

For the first example we consider $b_{ij}^{(z)}(x)=a_{ij}x$, where $a_{ij}$ is
constant for $i,j\in\\{1,2\\}$ such that $a_{ij}\geq 0$ whenever $i\neq j$.
Under this assumption (5.2) becomes

(5.5)
$dR_{t}^{(\infty,r)}=d_{1}R_{t}^{(\infty,r)}(1-R_{t}^{(\infty,r)})dt+d_{2}(1-R_{t}^{(\infty,r)})^{2}dt-
d_{3}(R_{t}^{(\infty,r)})^{2}dt,$

where $d_{1},d_{2}$ and $d_{3}$ are defined as:

$\displaystyle d_{1}$ $\displaystyle=a_{11}-a_{22},$ $\displaystyle d_{2}$
$\displaystyle=a_{12}+\int_{U_{2}}w_{1}\mu_{2}(dw),$ $\displaystyle d_{3}$
$\displaystyle=a_{21}+\int_{U_{2}}w_{2}\mu_{1}(dw).$

It is not difficult to see that the equilibria that arise in the dynamical
system modeled through (5.5) are described by:

  1. (1)

When $d_{2}-d_{1}-d_{3}\neq 0$, the equilibria of the system will be given by

$\frac{2d_{2}-d_{1}\pm\sqrt{d_{1}^{2}+4d_{2}d_{3}}}{2(d_{2}-d_{1}-d_{3})},$

of which

$\frac{2d_{2}-d_{1}-\sqrt{d_{1}^{2}-4d_{2}d_{3}}}{2(d_{2}-d_{1}-d_{3})},$

will be the stable one. A complete description of the equilibria is given
below:

    1. (a)

There is a stable equilibrium at $0$ and an unstable equilibrium at $1$ if
$d_{2}=d_{3}=0$ and $d_{1}<0$.

    2. (b)

There is an unstable equilibrium at $0$ and a stable equilibrium at $1$ if
$d_{2}=d_{3}=0$ and $d_{1}>0$.

    3. (c)

There is a unique equilibrium, which will be stable, at $0$ if $d_{2}=0$,
$d_{1}\leq 0$ and $d_{3}>0$.

    4. (d)

There is a unique equilibrium, which will be stable, at $1$ if $d_{3}=0$,
$d_{1}\geq 0$ and $d_{2}>0$.

    5. (e)

There is an unstable equilibrium at $0$ and a stable equilibrium in $(0,1)$ if
$d_{2}=0$, $d_{1}>0$ and $d_{3}>0$.

    6. (f)

There is an unstable equilibrium at $1$ and a stable equilibrium in $(0,1)$ if
$d_{3}=0$, $d_{1}<0$ and $d_{2}>0$.

    7. (g)

There is a unique equilibrium in $(0,1)$, which will be stable, if
$d_{2},d_{3}>0$.

  2. (2)

When $d_{2}-d_{1}-d_{3}=0$ there is a unique equilibrium in $[0,1]$, given by

$\frac{d_{2}}{2d_{2}-d_{1}},$

which will be stable. Explicitly, the equilibrium will be:

    1. (a)

$0$ if $d_{2}=0$ and $d_{1}=-d_{3}$;

    2. (b)

$1$ if $d_{1}=d_{2}$ and $d_{3}=0$;

    3. (c)

not trivial, i.e. it is located in $(0,1)$, in any other case.

Because we are looking for non-trivial equilibria, the cases that we are
interested in are 1e, 1f, 1g and 2c. Figure 1 shows phase diagrams of the
behavior of the dynamical system in each of these cases. Before proceeding to
the next example, notice that we can rewrite (5.5) as

$dR_{t}^{(\infty,r)}=(d_{1}-d_{2}+d_{3})R_{t}^{(\infty,r)}(1-R_{t}^{(\infty,r)})dt+d_{2}(1-R_{t}^{(\infty,r)})dt-
d_{3}R_{t}^{(\infty,r)}dt.$

This form shows that the non-trivial equilibrium that is in the system is due
to the mutation terms $d_{2}(1-R_{t}^{(\infty,r)})$ and
$d_{3}R_{t}^{(\infty,r)}$, that counterbalance the effect that the “classical
selection” term $(d_{1}-d_{2}+d_{2})R^{(\infty,r)}(1-R^{(\infty,r)})$ has.

(a) Case $d_{2}=0$, $d_{1}>0$ and $d_{3}>0$.

(b) Case $d_{3}=0$, $d_{1}<0$ and $d_{2}>0$.

(c) Case $d_{2},d_{3}>0$.

(d) Case $d_{2}-d_{1}-d_{3}=0$.

Figure 1. Examples of non-trivial equilibrium in the linear coefficients case.

###  5.3. Logistic growth and no inter-type branching

Let us take $b_{ii}^{(z)}(x)=c_{ii}x^{2}/z+a_{ii}x$ for $i\in\\{1,2\\}$,
$b_{ij}^{(z)}\equiv 0$ is $i\neq j$, and $\mu_{i}(\\{w_{j}>0\\})=0$ for $i\neq
j$. Under this assumptions, $\beta_{ii}(x)=c_{ii}x^{2}+a_{ii}x$ for
$i\in\\{1,2\\}$ and $\beta_{ij}\equiv 0$, so (5.2) can be rewritten as

(5.6)
$dR_{t}^{(\infty,r)}=d_{1}R_{t}^{(\infty,r)}(1-R_{t}^{(\infty,r)})\biggl{(}R_{t}^{(\infty,r)}-\frac{d_{2}}{d_{1}}\biggr{)}dt,$

where

$\displaystyle d_{1}$ $\displaystyle=c_{11}+c_{22},$ $\displaystyle d_{2}$
$\displaystyle=c_{22}-a_{11}+a_{22}.$

Notice that the term $R_{t}^{(\infty,r)}(1-R_{t}^{(\infty,r)})$ corresponds to
classical selection, so if there exists a non-trivial equilibrium in $(0,1)$,
it will be due to the _balancing selection_ term
$R_{t}^{(\infty,r)}-d_{2}/d_{1}$. As we are interested in the non-trivial
equilibrium, we will only consider the case where $d_{1}d_{2}>0$ and $\lvert
d_{1}\rvert>\lvert d_{2}\rvert$, because only under those conditions there
will be a non-trivial equilibrium, given by $d_{2}/d_{1}$. It can be shown
that this equilibrium will be stable if $d_{2}<0$ and unstable if $d_{2}>0$.
Both of these cases are shown graphically, using phase diagrams, in figure 2.

(a) Case $d_{2}>0$.

(b) Case $d_{2}<0$.

Figure 2. Examples of non-trivial equilibrium in the logistic growth case.

What this example shows is that even if there is no multi-type branching,
there might exist a stable equilibrium in the large population limit by
considering logistic growth for each of the two types within the population.

Both examples point to the fact that the inclusion of typed branching and
general Malthusians, such as the logistic one, may lead to coexistence in the
stochastic model. We leave this as a future venue for research.

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