#  Extinction and coexistence in a binary mixture of proliferating motile
disks

Alejandro Almodóvar<EMAIL_ADDRESS>Tobias Galla
<EMAIL_ADDRESS>Cristóbal López<EMAIL_ADDRESS>IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB),
Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain

###### Abstract

A binary mixture of two-different-sizes proliferating motile disks is studied.
As growth is space-limited, we focus on the conditions such that there is
coexistence of both large and small disks, or dominance of the larger disks.
The study involves systematically varying some system parameters, such as
diffusivities, growth rates, and self-propulsion velocities. In particular, we
demonstrate that diffusing faster confers a competitive advantage, so that
larger disks can in the long time coexist or even dominate to the smaller
ones. In the case of self-propelled disks a coexistence regime is induced by
the activity where the two types of disks show the same spatial distribution:
both phase separated or both homogenously distributed in the whole system.

††preprint: APS/123-QED

##  I Introduction

Motivated by the collective dynamics in biological phenomena such as wound
healing [1, 2, 3], tissue formation [4], the expansion of tumours [5], or the
dynamics of bacterial populations [6], there is growing interest in the study
of proliferating motile matter [7, 8]. These are often modelled as interacting
particle systems [9, 10, 3, 5], and the number of particles may not be
constant in time due to processes such as birth and death. Coupled with
individual movement one can expect new emergent properties. For example, we
recently introduced a simple model of proliferating motile finite-size
particles. Specifically, in [11] we studied both systems of passive disks
(where motility has its origins in a thermal bath) and self-propelled disks
[12, 13, 14, 15, 16] subject to reproduction and death. We analysed different
emerging structural phases [liquid, hexatic, solid and motility-induced phase
separation (MIPS)] [17, 18, 19, 20, 21, 22, 16]. These phases result as a
consequence of the disks filling the available space, and the phase that is
realised depends on parameters such as the birth and death rates, and the
motility.

In [11] we focussed on a two-dimensional collection of disks (we will refer to
these also as particles), which were all taken to be identical, in particular,
they have the same size. This is not realistic in many biological
applications. In the current work we analyse the influence on some properties
of the system when relaxing this condition, and consider a binary mixture of
disks of two different sizes [23, 24]. In the standard equilibrium system of
hard disks (without birth-death events or activity) the two-sizes binary
mixture is known to have important effects leading for example to the
dissapearence of the hexatic phase even for very low concentration of small
disks [25]. Thus we expect new behavior in a non-equilibrium model of a binary
mixture with birth and death dynamics.

We start with a similar concentration for both types (or species) of disks,
initially randomly distributed in a two-dimensional space. As the system
evolves, particles move, die and reproduce, eventually filling up space. Due
to the death processes it is possible for one species to go extinct. In
reproduction events a new particle is created in the system, with the same
size as the parent, and placed close to it if there is enough room.

As in many real-world systems, reproduction in our model is limited by the
available space [26, 27, 28]. Under identical conditions, the smaller disks
have a higher a-priori chance to reproduce, and therefore to persist in the
long-run. Even though the availability of space puts the larger particles at a
disadvantage [29, 30], we find that, depending on motility and demography,
coexistence of both species can occur. In some other cases we find that the
larger particles can dominate the system, i.e. their number is much larger
than that of the smaller particles, typically more than $90\%$ of the total
particle number.

The main questions we address are what type of disk dominates the system in
the long-run, under what conditions both types co-exist, and, in the important
case of activity-induced phase separation, what the resulting spatial
structure is. We are interested in the influence of the birth and death rates
and the motilities of the particles on the outcome. In particular, we ask
which population survives when the two species compete (due to reproduction
limited by size) with different diffusivities [31, 32], and we show results
that support the prevalence of the fast diffusing type of particle. Our
overall aim is to characterise the system behaviour for different choices of
the model parameters.

In the case of active motion [33, 34, 35] we also study what the conditions
are under which the spatial distribution of the particles presents coexisting
dilute and dense phases, i.e., the MIPS regime. This is for example motivated
by studies hypothesing that phase heterogeneity due to MIPS can trigger a
transition from swarming behaviour to biofilm formation in some type of
bacteria [36]. We analyse the role of this heterogeneity for the dominance
dynamics of the binary mixture.

Throughout our analysis, we will mostly concentrate on systems in which the
diameter of one type of disk is $20\%$ larger than the other, but we will also
discuss other size ratios [37, 38, 39, 40].

The remainder of the paper is organised as follows. In Sec. II, we present the
model of mobile disks undergoing birth and death dynamics. The main outcomes
are presented in Sec. III. A summary and discussion of the findings is
contained in Sec. IV.

##  II Model and numerical algorithm

The model is similar to that in [11] but with particles of two different
sizes. We consider a two-dimensional system of $N(t)=N_{L}(t)+N_{S}(t)$
interacting disks with diameters $\sigma_{S}$ and $\sigma_{L}$ such that
$\sigma_{S}<\sigma_{L}$ ($S$ and $L$ stands for ‘small’ and ‘large’,
respectively). The particle numbers can change in time, due to birth and death
events, as explained below. We consider the overdamped limit and take the
friction coefficient to be equal to unity for both species. The motion of
disks is then as follows,

$\dot{{\bf r}_{i}}={{\bf F}}_{i}+{{\bf
F}}^{act}_{i}+\sqrt{2D_{i}}\bm{\zeta}_{i}(t),\ \ i=1,...,N(t).$ (1)

If disk $i$ is of the small type then $D_{i}=D_{S}$, and if it is of the large
type then $D_{i}=D_{L}$. The variables $\\{\bm{\zeta}_{i}\\}$ are independent
Gaussian noise vectors satisfying $\langle\bm{\zeta}_{i}\rangle=0$,
$\langle{\zeta}_{i,a}(t){\zeta}_{j,b}(t^{\prime})\rangle=\delta_{ij}\delta_{ab}\delta(t-t^{\prime})$
($a$ and $b$ are the entries of the two-component vectors $\bm{\zeta}_{i}$ and
$\bm{\zeta}_{j}$). No Einstein relation is assumed, and $D_{L}$ and $D_{S}$
are taken as parameters of the model. The finite size of the disks is
simulated using a truncated Lennard-Jones potential so that the force on
particle $i$ resulting from the interaction with the rest of particles is
${\bf F}_{i}=-{\bf\nabla}_{i}\sum_{i\neq j}U(|{\bf r}_{i}-{\bf r}_{j}|)$,
where the potential is given by (with $r=|{\bf r}_{i}-{\bf r}_{j}|$)

$U(r)=4\varepsilon[(\frac{\sigma_{ij}}{r})^{12}-(\frac{\sigma_{ij}}{r})^{6}]+\varepsilon,$
(2)

if $r<2^{1/6}\sigma_{ij}$, and $U(r)=0$ if $r>2^{1/6}\sigma_{ij}$. The
quantity $\sigma_{ij}>0$ is defined from the Lorentz-Berthelot rule as
$\sigma_{ij}=(\sigma_{i}+\sigma_{j})/2$, where $\sigma_{i}$ and $\sigma_{j}$
are the diameters of disks $i$ and $j$, respectively. Thus, $\sigma_{ij}$
reflects the effective distance between the two disks. The parameter
$\varepsilon$ is an energy scale [41, 42].

In some of our numerical experiments particles are self-propelled. We model
this using active forces

${\bf F}^{act}_{i}=v_{i}{\bf n}[\theta_{i}(t)]$ (3)

of constant modulus $v_{i}=\\{v_{L},v_{s}\\}$ (typically called activity or
velocity) and with a direction given by the unit vector ${\bf
n}(\theta_{i})=(\cos\theta_{i},\sin\theta_{i})$. The angle $\theta_{i}$ for
disk $i$ performs diffusive motion,
$\dot{\theta}_{i}(t)=\sqrt{2D_{r}}\eta_{i}(t)$. The term $\eta_{i}$ represents
a zero-mean Gaussian noise with
$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$.

In addition to movement and interaction, disks may randomly self-replicate or
die, so that the number of disks of each type, $N_{L}(t)$ and $N_{S}(t)$, can
change with time. These events occur as follows (see [11] for further details
of the algorithm):

  1. 1.

Death occurs as a Poisson process. Each existing particle dies with per capita
rate $\delta$. Particles that die are removed from the system.

  2. 2.

Potential births are triggered with per capita rate $\beta_{L}$ for large
disks, and $\beta_{S}$ for small disks. The diameter of the potential
offspring is identical to that of the parent. The birth event only occurs if
there is sufficient space around the parent particle to place the offspring
without overlapping with any other disk. If there is no space, no birth event
occurs. This means that not all potential reproduction events complete.

We always assume that the growth rates $\beta_{S}$ and $\beta_{L}$ are larger
than the death rate (the latter is taken to be equal for all the disks), i.e.
$\beta_{L},\beta_{S}>\delta$. At long times the system reaches a stationary
state. The numbers of disks of each type in this state are such that the mean
effective birth rate for each species is equal to the death rate.

In the next sections we study this steady state while varying one or two model
parameters at a time (e.g., the diffusivity of both types of particles, the
growth rates or activity). Our main objective is to study what type of disk
dominates, and under what conditions there is coexistence.

##  III Results

We consider a two-dimensional box of length $L_{s}=150$ with periodic boundary
conditions, $\delta=0.01$, and for the most part $\sigma_{S}=1.0$ and
$\sigma_{L}=1.2$, supplemented by some discussion of other choices of the
diameters. For simplicity, we set $\varepsilon=1.0$ and $D_{r}=1.0$.
Simulation results are independent of $\delta$ whose role is mostly to set the
time scale needed to reach the steady state [11]. We start with $250$
particles of each size, together occupying around $8\%$ of the total area. We
compute the packing fraction for particles of type. We write these as
$\phi_{\alpha}(t)=N_{\alpha}(t)\pi(\sigma_{\alpha}/2)^{2}/L_{s}^{2}$ for
$\alpha\in\\{L,S\\}$, where $N_{\alpha}(t)$ is the number of particles of type
$\alpha$ at time $t$.

We study passive particles in Sec. III.1. Systems of active particles are
discussed in Sec. III.2.

###  III.1 Passive particles

We set $\mathbf{F}_{i}^{act}=0$, and study the effects of the parameters of
the demographic dynamics, and of the diffusivities separately.

####  III.1.1 Effects of the birth and death rates

Figure 1: Panels in the first row (a-c) show snapshots of the spatial
distribution of passive disks in space at long times (only a part of the full
size $L_{s}=150$ is shown). Panels in the second row (d-f) correspond to the
temporal evolution of the packing fraction starting with a random
configuration of $250$ disks of each type. In each column, we have used
different values of $\beta_{L}$, at fixed $\beta_{S}=0.1$, and
$\beta_{L}=0.114,0.118,0.13$ from left to right. The remaining parameters are
$\delta=0.01$, $D=0.001$, $\sigma_{L}/\sigma_{S}=1.2$.

We fix the diffusivities to low values (i.e. the time scale of a disk to move
a distance of the order of several diameters is large compared to its typical
lifetime) $D_{L}=D_{S}=0.001$, to focus on the effects of the birth and death
dynamics. Since reproduction is limited by the available space, we naively
expect only the smaller disks to be present in the long-run when the raw birth
rates of both types are similar. However, the outcome may be different if the
birth rate for the larger particles is much larger than that of the smaller
type. This is indeed what we observe in the numerical experiments shown in
Fig. 1.

In the upper row of Fig. 1 we plot snapshots of the spatial distribution of
disks in the stationary state. Smaller particles shown in red, larger ones in
green. The growth rate of the small particles is lower than that of the larger
particles in all three panels, but the ratio $\beta_{L}/\beta_{S}$ increases
from left to right. In panel a) we observe the extinction of the larger
particles, in panel b) we find balanced co-existence throughout the
simulation, and in panel c) the ratio $\beta_{L}/\beta_{S}$ is sufficiently
large for the larger disks to fully dominate (the smaller particles go
extinct). In the lower panels of Fig. 1 we plot the corresponding time
evolution of the packing fractions (the snapshots in the upper panels are
taken at the final time of the lower ones). These time series confirm the
extinction of the larger type of particle [panel d)], co-existence [panel e)]
and the extinction of the smaller type [panel f)], respectively. In all three
situations, and regardless of the final fate of the larger particles, there is
initially a faster increase of the packing of the larger particles. This is
because $\beta_{L}>\beta_{S}$ in all panels in Fig. 1

In Fig. 2a) we plot the phase diagram in the plane of birth rates for the two
species (each divided by the death rate $\delta=0.01$). The colored heatmap
indicates the normalised packing fraction $\phi_{S}/(\phi_{S}+\phi_{L}$) of
the smaller disks. In Fig. 2b) we show the respective packing fractions of
both types when varying $\beta_{L}$, for fixed $\beta_{S}$. This corresponds
to a vertical cut of the phase diagram in panel a). The dominance of the
smaller type at low $\beta_{L}$, co-existence at intermediate values, and
dominance of the larger disks at high $\beta_{L}$ are clearly visible.

Figure 2: Coexistence diagram for the system of passive particles in the plane
spanned by the birth rates $\beta_{S}$ and $\beta_{L}$. Panel a): color
indicates where in parameter space either type of particle dominates, or if
there is co-existence. Panel b): Vertical cut in the phase diagram in (a) for
$\beta_{S}/\delta=10$ showing the packing fractions for both types of disks.
The colors of the symbols represent the fraction of small particles in the
system. $\delta=0.01$, $\sigma_{S}=1.0$, and $\sigma_{L}=1.2$.

Coexistence of both species occurs in a very limited region of the diagram,
around a small area near a specific line in the $(\beta_{S},\beta_{L})$ plane.
An approximate characterisation of this coexistence curve can be obtained from
a simple description of the dynamics in terms of rate equations. To construct
these equations with proper parameters, we first consider only one type of
disk in isolation, neglect fluctuations and any notion of space. We use a
Lotka–Volterra model to describe the combination of death events and growth
limited by volume exclusion [43],

$\frac{dN}{dt}=-\delta N+{\hat{\beta}}N\left(1-\frac{N}{N_{\rm max}}\right),$
(4)

where the term in the brackets ensures that the growth dynamics stops as $N$
reaches $N_{\rm max}$; $\delta$ is the death rate, and $\hat{\beta}$ is an
effective growth rate.

This effective growth rate is a fitting parameter introduced in this very
simple description to take into account that the actual birth rate is limited
by the available space, and therefore not equal to the raw birth rate $\beta$.
We take $\hat{\beta}=a\beta$, where $a$ is an unknown adimensional
coefficient. In [11] we observed that the steady number of disks in a single-
species model depends in a non-trivial way on the birth and death rates, the
size of the particles, and on the diffusion coefficient. Thus, in this
description we use $a>0$ and $N_{\rm max}$ as fitting parameters.

We rewrite Eq. (4) in the standard form

$dN/dt=rN(1-N/K),$ (5)

with $r=a\beta-\delta$. The carrying capacity $K=N_{\rm
max}(1-\frac{\delta}{a\beta})$ is the effective long-time number of disks in
the system.

To verify the validity of the logistic-growth approach we have carried out
simulations of the model in which all particles have the same size, using a
small diffusion coefficient, $D=0.001$. In the long-time we compute the
packing fraction $\phi_{\infty}=N(t\to\infty)\pi(\sigma/2)^{2}/L_{s}^{2}$. We
do this for several different disk sizes ($\sigma=1.0,1.2,1.4$) and for some
values of $\beta$ to check the relationship $\phi_{\rm
stat}=K\pi(\sigma/2)^{2}/L_{s}^{2}=N_{\rm
max}\pi(\sigma/2)^{2}/L_{s}^{2}(1-\frac{\delta}{a\beta})$ (the stationary
packing fraction of the Lotka–Volterra equation). The results are shown in
Fig. 3. There is a good collapse of all the plots for the different values of
$\sigma$, and the best fit to the expression of $\phi_{\infty}$ is obtained
for $\phi_{\rm max}=N_{\rm max}\pi(\sigma/2)^{2}/L_{s}^{2}=0.7$, $a=1.5$. This
confirms that the logistic description with suitable effective parameters can
be used to describe the stationary state with a single type of particle.

Figure 3: Long-time average packing fraction, $\phi_{\infty}$, for a system of
only one type of disks as a function of the rate $\beta$ and $\delta=0.01$.
The plots correspond to different disk sizes $\sigma$ as the symbols indicate.
The black line corresponds to the best fitted curve for $a=1.5$, and
$\phi_{\rm max}=0.7$.

The Lotka–Volterra model can be generalised to the case of a binary mixture of
disks. This approximate description can be useful to study the influence of
one type of disks on the other, in particular, attending to their different
sizes and growth rates. We assume the following Lotka–Volterra competition
dynamics between the two species (which we label again $S$ and $L$ for ‘small’
and ‘large’ disks, respectively) [44]:

$\displaystyle\frac{dN_{L}}{dt}=-\delta
N_{L}+a\beta_{L}\left(1-\frac{N_{L}+\alpha_{L}N_{S}}{N_{L,{\rm
max}}}\right)N_{L},$ $\displaystyle\frac{dN_{S}}{dt}=-\delta
N_{S}+a\beta_{S}\left(1-\frac{N_{S}+\alpha_{S}N_{L}}{N_{S,{\rm
max}}}\right)N_{S}.$ (6)

The term $\alpha_{L}N_{S}$ ($\alpha_{S}N_{L}$) is the decline of the growth
rate of $N_{L}$ ($N_{S}$) due to the presence of the smaller (larger) disks.
The quantities $N_{S,{\rm max}}$ and $N_{L,{\rm max}}$ are the number of
particles for either species at which no further growth can occur.

The dependence of the interaction terms on the ratio of disk sizes becomes
more apparent if we write this equation in terms of packing fractions:

$\displaystyle\frac{d\phi_{L}}{dt}=-\delta\phi_{L}+a\beta_{L}\left(1-\frac{\phi_{L}+\alpha_{L}\frac{\sigma_{L}^{2}}{\sigma_{S}^{2}}\phi_{S}}{\phi_{L,{\rm
max}}}\right)\phi_{L},$
$\displaystyle\frac{d\phi_{S}}{dt}=-\delta\phi_{S}+a\beta_{S}\left(1-\frac{\phi_{S}+\alpha_{S}\frac{\sigma_{S}^{2}}{\sigma_{L}^{2}}\phi_{L}}{\phi_{S,{\rm
max}}}\right)\phi_{S}.$ (7)

Now $\phi_{\alpha,{\rm max}}=N_{\alpha,{\rm
max}}\pi(\sigma_{\alpha}/2)^{2}/L_{s}^{2}$ ($\alpha=L,S$). We assume that
$\alpha_{L}$ and $\alpha_{S}$ depend on the demographic parameters of the
system $(\beta_{L},\beta_{S})$, and as detailed in Appendix A. Fitting to
simulation data we find

$\displaystyle\alpha_{L}$ $\displaystyle\approx$
$\displaystyle\left(\frac{\beta_{S}}{\beta_{L}}\right)^{1/4},$
$\displaystyle\alpha_{S}$ $\displaystyle\approx$
$\displaystyle\left(\frac{\beta_{L}}{\beta_{S}}\right)^{1/4}.$ (8)

These relations characterise the tradeoff between geometry and demography
(reproduction) in the competition dynamics. For example the coefficient in the
growth rate for the larger disks
$\alpha_{L}\frac{\sigma_{L}^{2}}{\sigma_{S}^{2}}=\frac{\sigma_{L}^{2}}{\sigma_{S}^{2}}\left(\frac{\beta_{S}}{\beta_{L}}\right)^{1/4}$
increases proportionally to $(\sigma_{L}/\sigma_{S})^{2}$, while its
dependence on the relative birth rates is much weaker and only scales as
$(\beta_{S}/\beta_{L})^{1/4}$.

####  III.1.2 The role of diffusion

Figure 4: Panels in the first row [a)-d)] show snapshots of the spatial
distribution of passive disks space at long times (graphs only show a part of
the full system, which has lateral size $L_{s}=150$). Panels in the second row
[e)-h)] show the corresponding time evolution of the packing fractions for
both species, starting with a random configuration of $250$ disks of each
type. In the different columns we have used different values of $D_{L}$ and
$D_{S}=0.001$: $D_{L}$ = 0.01, 0.02, 1.0 from left to right. Remaining
parameters are $\delta=0.01$, $\beta=0.05$, $\sigma_{L}/\sigma_{S}=1.2$.

In this subsection we study the case in which both types of passive disks have
the same birth and death rates but where their diffusion coefficients are
different. In biological applications situations of this type have given rise
to contradictory results (albeit under very different settings) so that
diffusing faster provides a competitive advantage [31] or can be detrimental
[32]). Thus, one of the main questions we address is if high or low
diffusivity can provide a selective advantage for either of the two competing
populations. In our model diffusivity introduces a spatial scale increasing
the effective size of the disks, and hence affecting the dynamics of space-
limited growth.

In the upper row of Fig. 4 we show the spatial distribution of disks at long
times, for different $D_{L}$ and leaving all other model parameters fixed. In
the leftmost panel (small ratio $D_{L}/D_{S}$) the smaller particles dominate
the system. In the central panel (intermediate $D_{L}/D_{S}$), there is a
balance of both types of disks, and when $D_{L}/D_{S}$ is sufficiently high
(rightmost panel), the larger disks occupy most of the space. Figs. 4 d)-f)
show the time evolution of the packing fractions of both types in simulations
starting with $250$ particles of each size. The corresponding phase diagram,
for $\sigma_{L}/\sigma_{S}=1.2$, is shown in Fig. 5a). Increasing the
diffusivity of either species promotes an increased relative abundance of that
species. This is because larger diffusivity for a given particle means it
effectively occupies more space where others cannot place their offspring. In
particular we observe a transition between phases in which the system is
predominantly filled by the smaller particles to one in which it is mostly
filled by larger particles.

Figure 5: Phase diagram in the space of diffusivities, $D_{S}$ and $D_{L}$.
Panel a): Background color indicates the relative filling fraction of the two
type of disks. In the red region the smaller particles fill most of the space,
in the green region the larger particles dominate. Panel b): Long-time average
packing fractions for both species as a function of $D_{L}$, for fixed
$D_{S}=0.001$. The colors of the symbols indicate the relative amount of space
occupied by the small particles. Model parameters are $\delta=0.01$,
$\beta_{L}=\beta_{S}=0.05$, $\sigma_{S}=1.0,\sigma_{L}=1.2$.

We conclude that increasing diffusivity of the larger disks can reverse the
competitive advantage of the smaller ones. At difference with the results of
the previous subsection (Fig. 1), the dominance of larger disks for high
$D_{L}/D_{S}$ is not complete though. Instead, we find a remaining population
of smaller disks occupying about $10\mathchar 45\relax 15\%$ of space until
the end of our simulations (i.e., $\phi_{S}=0.1\mathchar 45\relax 0.15$), see
Fig. 5b). We attribute this to interstitial holes, i.e., empty space between
the disks, noting that the effect is more pronounced (i.e.,
$\phi_{S}/(\phi_{S}+\phi_{L})$ becomes larger) when the ratio
$\sigma_{L}/\sigma_{S}$ is increased (see Fig. 6).

Figure 6: Normalized packing fraction of small particles versus $D_{L}$ for
different values of $\sigma_{L}/\sigma_{S}$ as indicated. Same remaining
parameters as in Fig. 5

.

An exception to the survival of the smaller disks can be observed in a region
in $(D_{S},D_{L})$ space in which the larger disks almost completely dominate
the system [dark green to the lower left in Fig. 5a)]. This dominance only
occurs when $D_{S}$ is small enough so that the movement of the smaller disks
is negligible with respect to their lifetime. This region disappears as the
ratio $\sigma_{L}/\sigma_{S}$ becomes larger (see Fig. 6), so that we think it
is because small disks may occupy the interstitial holes.

###  III.2 Active particles

We consider systems in which both types of disks are self-propelled, possibly
with different values of the propulsion velocities $v_{L}$ and $v_{S}$. The
remaining parameters are set to the same: $\beta_{L}=\beta_{S}=\beta$,
$D_{L}=D_{S}=D$, and $\delta=0.01$.

We first take a birth rate of $\beta=0.05$, and diffusivity of $D=0.05$. Under
these conditions, without activity, the system is in a liquid state, which
means that the packing fraction of both types is low and there is no ordering.
Also the system reaches the steady state faster. Our motivation for this
choice is that we aim at analysing MIPS, which is typically not observed for
large values of the packing fraction (solid phase) [11, 15].

In the previous section we have shown that Brownian mobility provides an
advantage in the competition for space between the two types of disks. Given
that self-propulsion yields comparable effects (at low velocities) to
diffusivity [45, 46] we expect in this low activity regime similar outcomes
for active particles. However, MIPS typically appears when activity is large,
and thus we study this regime in detail by varying the self-propulsion
coefficients of both types of disks, and compute the average packing fraction
of each species at long times.

The resulting phase diagram, obtained for $\sigma_{L}/\sigma_{S}=1.2$, is
shown in Fig. 7a). When the activity of both types of disks is sufficiently
low the smaller disks dominate the system. As $v_{L}$ increases (keeping
$v_{S}$ fixed at a sufficiently small value) a transition to coexistence is
found. This coexistence remains as the activity of the larger particles is
further increased. We find interesting behaviour for sufficiently large
$v_{S}$. This is the region of MIPS. We observe that coexistence can be
reached for low and intermediate values of $v_{L}$ (see Fig. Fig. 7 b)). In
between there is a range of values where large particles dominate, and for
very high $v_{L}$ small particles dominate the system.

Figure 7: Panel (a): Phase diagram of the system with active disks in the
$(v_{L},v_{S})$ plane. Background color represents the normalised packing
fraction of the smaller particles (darker red color indicates dominance of the
smaller type, darker green dominance of the larger). Panel (b): Long-time
average packing fractions for both species as function of $v_{L}$ for fixed
$v_{S}=50$. The colors of the symbols indicate the relative space occupied by
the small particles. Model parameters are $\delta=0.01$,
$\beta_{L}=\beta_{S}=0.05$, $\sigma_{S}=1.0,\sigma_{L}=1.2$.

Which type of particle survives (or the presence of coexistence) results from
a complicated interplay between the separation in dense and diluted phases
formed of both species in non-steady conditions because of activity, and the
birth/death dynamics which is itself mediated by the spatial distribution of
disks. This is shown in the upper panels of Fig. 8 with some examples of the
disk distribution at long times for $v_{S}=50$, and different values of
$v_{L}$. In panel a) there is coexistence for sufficiently small $v_{L}$. As
we increase $v_{L}$ the smaller species becomes extinct (dark green region in
the phase diagram) [Fig. 7b)]. As $v_{L}$ is increased further, coexistence is
observed again and both species show MIPS [see Fig. 8c)]. For even higher
values of $v_{L}$, the large disks become extinct [see Fig. 8d)]. In the lower
row we show the corresponding time evolutions of the packing fractions of both
types of particles starting from a configuration with $250$ particles of each.

Figure 8: Panels in the first row (a-d) show snapshots of the spatial
distribution of active disks space at long times (only a part of the full size
$L_{s}=150$). Panels in the second row (e-h) correspond to the temporal
evolution of the packing fraction starting with a random configuration of
$250+250$ disks. In addition, for each column, we have used different values
of $v_{L}$ and $v_{S}=50$. The values used are $v_{L}$ = 25, 40, 56 and 70
from left to right.

Let us next characterise the emerging spatial structures when there is
coexistence. To do this we compute the distribution of local packing fractions
of large disks, $P(\phi_{L})$, and small ones, $P(\phi_{S})$. These are
obtained from the local packing fraction of either species at each location in
space. MIPS is characterised by a double-peaked distribution, with the two
peaks corresponding to the dense and diluted phases, respectively [15]. In
Fig. 9a) we plot $P(\phi_{S})$ and $P(\phi_{L})$ corresponding to two
situations: i) small particles with very low $v_{S}=0.1$, and large particles
with low $v_{L}=10$ (the two distributions are plotted in red); ii) small
particles again with $v_{S}=0.1$, but large particles with high $v_{L}=150$.
These scenarios correspond, respectively, to the cases where (if the two types
were in isolation without the presence of the other) a) neither small nor
large disks would form MIPS, and b) small disks would not form MIPS, but large
ones would because they have a high $v_{L}$. However when both types of
particles are present in the system, the distribution of local packing
fractions is single-peaked, and there is no separation between dilute and
dense phases.

Instead, in Fig. 9b) we use a higher value of $v_{S}=50$, and two different
values of $v_{L}$: $v_{L}=25$ for which there would be no MIPS for the large
particles in isolation, and $v_{L}=56$, for which there would be MIPS. Now
$P(\phi_{L})$ and $P(\phi_{S})$ are double-peaked, that is both types of disks
have a dense and a dilute phase. These results indicate that the two species
arrange in the same spatial distribution when they coexist. Either both
display a single phase, or both species show MIPS.

Figure 9: Distribution of the local packing fraction for different values of
activity in configurations where both disks types coexist. Panel (a):
$v_{S}=0.1$. Panel (b): $v_{S}=50$. The $v_{L}$ value is shown in the legend.

Finally, in Fig. 10 we plot the normalized packing fraction for the smaller
types of disks when $v_{L}$ varies (keeping $v_{S}=50$ fixed, i.e., within the
MIPS region) and for several values of the ratio of size particles. When we
increase $\sigma_{L}/\sigma_{S}$, the effect of the interstices becomes more
pronounced which causes the small particles packing fraction to increase. For
the same reason the region in Fig. 10 in which the smaller particles go
extinct reduces. In other words, as $\sigma_{L}/\sigma_{S}$ increases, the
large particles need a higher value of activity to form MIPS. This results in
a higher value of $v_{L}$ for the small particles to go extinct.

Figure 10: Normalized packing fraction of the smaller type of particle versus
$v_{L}$ for different values of $\sigma_{L}/\sigma_{S}$ as indicated in the
legends. $v_{S}=50$, same remaining parameters as in Fig. 7.

##  IV Summary and discussions

The interplay between motility and proliferation is fundamental for many
biological processes. In this work we have studied a binary mixture of two
types of motile disks with different sizes, and undergoing birth and death
dynamics. At difference with our previous work [11] where we concentrated on
the spatial structure formed by a single population of identical disks, in
this work we have studied the conditions for coexistence and which of the two
types does not go extinct in the long-time. Since the birth probability is
limited by size, smaller disks have larger chances to reproduce (when the
remaining characteristics are identical). Thus we focused on the conditions
under which this is reversed, i.e., larger disks survive, or there is
coexistence of both types of disks. We have considered two main cases: passive
and self-propelled disks. For passive particles we have discussed the role of
growth rates and diffusivities, and have obtained typical phase diagrams when
the ratio of particles sizes is $\sigma_{L}/\sigma_{s}=1.2$, but discussed
also it for other ratios, unveiling the role of interstices. We observe that
Brownian mobility provides with an advantage such that the larger particles,
with less chances to reproduce, can coexist with the smaller or even dominate
in the steady state when they diffuse faster.

We have analysed the role of activity and have shown that, as expected, it is
similar similar to that of diffusivity when self-propelled velocities have low
values. However, when MIPS is present (typically obtained for larger values of
activities) the coexistence dynamics changes, and the typical situation is a
coexistance of both species, and both showing the same diluted and dense
phases.

Further work should be devoted to analyse the spatial structure in the
different situations we have described, and in particular, the influence on
the hexatic and solid phases, if any, of the binary mixture with demography.
Concerning biological applications, the main motivation for our work, where
populations are characterized by large diversity, it is of great interest to
consider not only two disks sizes but a whole size distribution, but also of
the other parameters like the growth rates and the self-propulsion velocities.

###### Acknowledgements.

A.A and C.L. acknowledge grant LAMARCA PID2021-123352OB-C32 funded by
MCIN/AEI/10.13039/501100011033323 and FEDER “Una manera de hacer Europa”. T.G.
acknowleges partial financial support from the Agencia Estatal de
Investigación and Fondo Europeo de Desarrollo Regional (FEDER, UE) under
project APASOS (PID2021-122256NB-C21, PID2021-122256NB-C22), and the Maria de
Maeztu programme for Units of Excellence, CEX2021-001164-M funded by
MCIN/AEI/10.13039/501100011033.

## Appendix A

Figure 11: Long-time average packing fractions for both species as a function
of $\beta_{L}/\delta$, for fixed $\beta_{S}=0.1$ (left) and $\beta_{S}=0.85$
(middle and right), and $\delta=0.01$. $\phi_{\alpha}^{*}$ obtained from Eq.
(12) and $\phi_{\alpha}$ from particle simulations. From left to right:
$\sigma_{L}/\sigma_{S}=1.2,1.3,1.4$.

The parameters for which both species coexist can be approximated from the
non-zero fixed points of Eqs. (7):

$\displaystyle\phi^{*}_{S}=\kappa_{S}-\alpha_{S}\phi^{*}_{L}\left(\frac{\sigma_{S}}{\sigma_{L}}\right)^{2},$
$\displaystyle\phi^{*}_{S}=\frac{1}{\alpha_{L}}(\kappa_{L}-\phi^{*}_{L})\left(\frac{\sigma_{S}}{\sigma_{L}}\right)^{2},$
(9)

with

$\displaystyle\kappa_{S}$ $\displaystyle=$
$\displaystyle\frac{a\beta_{S}-\delta}{a\beta_{S}}\phi_{S,{\rm max}},$
$\displaystyle\kappa_{L}$ $\displaystyle=$
$\displaystyle\frac{a\beta_{L}-\delta}{a\beta_{L}}\phi_{L,{\rm max}}.$ (10)

Both equalities are fulfilled at the same time in the coexistence region.
Furthermore, recall that $\phi_{S,\rm max}=\phi_{L,\rm max}$, since the
maximum value does not depend on either the type of particle or the size of
the system. Now, we make the ansatz:

$\displaystyle\alpha_{S}=c_{S}(\beta_{L}/\beta_{S})^{\gamma},$
$\displaystyle\alpha_{L}=c_{L}(\beta_{S}/\beta_{L})^{\gamma},$ (11)

where $c_{S}$, $c_{L}$ and $\gamma$ are constants.

We have assumed symmetry in the interaction of the species, so that both
species have the same exponent $\gamma$. Inserting the ansatz (11) in Eq. (9)
we obtain an expression for the packing fraction with explicit dependence on
the demographic rates:

$\displaystyle\phi^{*}_{S}=\frac{1}{1-c_{L}c_{S}}\left(\kappa_{S}-\kappa_{L}c_{S}\left(\frac{\beta_{L}}{\beta_{S}}\right)^{\gamma}\left(\frac{\sigma_{S}}{\sigma_{L}}\right)^{2}\right),$
$\displaystyle\phi^{*}_{L}=\frac{1}{1-c_{L}c_{S}}\left(\kappa_{L}-\kappa_{S}c_{L}\left(\frac{\beta_{S}}{\beta_{L}}\right)^{\gamma}\left(\frac{\sigma_{L}}{\sigma_{S}}\right)^{2}\right).$
(12)

These expressions predict the value of the steady packing fractions at each
point of the diagram $(\beta_{L},\beta_{S})$ inside the coexistence region. In
the region where only large (small) particles are found, we have
$\phi^{*}_{L}=\kappa_{L}$ and $\phi^{*}_{S}=0$ ($\phi^{*}_{L}=0$ and
$\phi^{*}_{S}=\kappa_{S}$).

We have carried out a series of numerical experiments for various values of
$\delta$, $\sigma_{L}$, $\sigma_{S}$, (not shown) beyond those used in Fig. 2
and have observed that the phase diagram is almost independent of the value of
$\delta$ (always assuming $\beta>\delta$), and the ratio
$\sigma_{L}/\sigma_{S}$. The coefficients $c_{L}$, $c_{S}$ and $\gamma$ in Eq.
(12) can be obtained from fits of the packing fraction (see Fig. 11). We
compute the difference between the packing fraction obtained from Eq. (12),
and the one obtained from the numerical simulations of the particle model for
different values of $\delta$, $\sigma_{S}$, $\sigma_{L}$, $\beta_{S}$,
$\beta_{L}$. Then, we minimize the sum of all the differences.

From these simulations, the best fit is obtained for $c_{L}=0.69$,
$c_{S}=1.34$ and $\gamma=1/8$.

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