# Survival and extinction for a contact process with a density-dependent birth
rate

Jonas Köppl, Nicolas Lanchier and Max Mercer

###### Abstract

To study later spatial evolutionary games based on the multitype contact
process, we first focus in this paper on the conditions for
survival/extinction in the presence of only one strategy, in which case our
model consists of a variant of the contact process with a density-dependent
birth rate. The players are located on the $d$-dimensional integer lattice,
with natural birth rate $\lambda$ and natural death rate one. The process also
depends on a payoff $a_{11}=a$ modeling the effects of the players on each
other: while players always die at rate one, the rate at which they give birth
is given by $\lambda$ times the exponential of $a$ times the fraction of
occupied sites in their neighborhood. In particular, the birth rate increases
with the local density when $a>0$, in which case the payoff $a$ models mutual
cooperation, whereas the birth rate decreases with the local density when
$a<0$, in which case the payoff $a$ models intraspecific competition. Using
standard coupling arguments to compare the process with the basic contact
process (the particular case $a=0$), we prove that, for all payoffs $a$, there
is a phase transition from extinction to survival in the direction of
$\lambda$. Using various block constructions, we also prove that, for all
birth rates $\lambda$, there is a phase transition in the direction of $a$.
This last result is in sharp contrast with the behavior of the nonspatial
deterministic mean-field model in which the stability of the extinction state
only depends on $\lambda$. This underlines the importance of space (local
interactions) and stochasticity in our model.

††footnotetext: AMS 2000 subject classifications: Primary 60K35, 91A22
Keywords and phrases: Interacting particle systems; Contact process; Spatial
games; Block construction.

##  1 Introduction

The field of evolutionary game theory was developed by Maynard Smith [22], and
first appeared in his work with Price [23]. The basic idea in this field is to
reinterpret the different strategies as species and their payoff as fitness to
create realistic dynamical systems with density-dependent birth and/or death
rates. The most popular (nonspatial deterministic) model in evolutionary game
theory is the replicator equation. Having an $n\times n$ payoff matrix
$A=(a_{ij})$ where $a_{ij}$ represents the payoff a type $i$ player receives
from a type $j$ player, and letting $u_{i}$ denote the density of type $i$
players in the population, the payoff of each type $i$ player is given by

$\phi_{i}=\phi_{i}(u_{1},u_{2},\ldots,u_{n})=a_{i1}u_{1}+a_{i2}u_{2}+\cdots+a_{in}u_{n}.$

Reinterpreting the payoff as fitness (a birth rate when the payoff is positive
or minus a death rate when the payoff is negative) and assuming that each
individual produced replaces a player chosen uniformly at random, while each
individual removed is replaced by a player chosen uniformly at random, result
in the following so-called replicator equation [16]:

$\begin{array}[]{c}u_{i}^{\prime}=(\phi_{i}u_{i})(\sum_{j\neq
i}u_{j})-(\sum_{j\neq i}\phi_{j}u_{j})\,u_{i}=\sum_{j\neq
i}(\phi_{i}-\phi_{j})\,u_{i}u_{j}\quad\hbox{for}\quad
i=1,2,\ldots,n.\end{array}$

This system of coupled differential equations can be turned into a spatially
explicit stochastic process following the modeling approach of [26, 27]. More
precisely, to include a spatial structure in the form of local interactions,
we first assume that the players are located on the $d$-dimensional integer
lattice $\mathbb{Z}^{d}$, so the state at time $t$ is a configuration

$\xi_{t}:\mathbb{Z}^{d}\longrightarrow\\{1,2,\ldots,n\\}\quad\hbox{where}\quad\xi_{t}(x)=\hbox{strategy
of the player at site\leavevmode\nobreak\ $x$}.$

Then, writing $x\sim y$ to indicate that the two lattice points $x$ and $y$
are nearest neighbors (distance one apart), the payoff of the player at site
$x$ is defined as

$\begin{array}[]{c}\phi(x,\xi_{t})=\sum_{i,j}a_{ij}\,f_{j}(x,\xi_{t})\,\mathbf{1}\\{\xi_{t}(x)=i\\}\quad\hbox{where}\quad
f_{j}(x,\xi_{t})=\sum_{y\sim x}\mathbf{1}\\{\xi_{t}(y)=j\\}/2d\end{array}$ (1)

denotes the fraction of nearest neighbors of site $x$ following strategy $j$,
i.e., the payoff only depends on the strategy of the neighbors. The fitness is
then given by

$\Phi(x,\xi_{t})=(1-w)\times 1+w\times\phi(x,\xi_{t})=(1-w)\times
1+w\times\hbox{payoff},$ (2)

where the parameter $w\in(0,1]$ represents the strength of selection. Weak
selection refers to the case where $w$ is small, while strong selection means
$w=1$. The most popular models that fall under this framework are the birth-
death updating process and the death-birth updating process introduced in
[28]. In the birth-death updating process, the fitness is interpreted as a
birth rate, and offspring replace a neighbor of the parent’s site chosen
uniformly at random, so the rate at which site $x$ switches from strategy $i$
to strategy $j$ is given by

$\begin{array}[]{c}c_{i\to j}(x,\xi_{t})=\sum_{y\sim
x}\Phi(y,\xi_{t})\,\mathbf{1}\\{\xi_{t}(y)=j\\}/2d\quad\hbox{for all}\quad
i\neq j.\end{array}$ (3)

In contrast, in the death-birth updating process, players die at rate one and
are instantaneously replaced by the offspring of a neighbor chosen at random
with a probability proportional to its fitness, so the local transition rates
are now given by

$\begin{array}[]{c}c_{i\to j}(x,\xi_{t})=\sum_{y\sim
x}\Phi(y,\xi_{t})\,\mathbf{1}\\{\xi_{t}(y)=j\\}/\sum_{y\sim
x}\Phi(y,\xi_{t})\quad\hbox{for all}\quad i\neq j.\end{array}$ (4)

Taking $w=0$, the transition rates in (3)–(4) simplify to $f_{j}(x,\xi_{t})$,
showing that the birth-death and the death-birth updating processes both
reduce to the voter model [6, 17]. In the presence of weak selection, these
two processes were studied in [4, 5, 21] in the context of two-strategy games,
while [7, 10, 24] also considered games with more strategies. In the weak
selection limit $w\to 0$, voter model perturbations techniques developed in
[8] can be used to have a precise description of the phase structure of the
processes. In the presence of strong selection $w=1$, more qualitative aspects
such as the existence of phase transitions were proved in [12, 18]. Other
natural variants with discontinuous transition rates were also studied
rigorously in the presence of strong selection: the best-response dynamics
[11], and the death-birth of the fittest process [13].
In the previous models, each birth/death induces the instantaneous death/birth
of a neighbor so each site is occupied by exactly one player at all times. In
particular, these models, as well as all the models of interacting particle
systems of interest in evolutionary game theory that have been studied in the
probability literature, consist of variants of the voter model with density-
dependent birth and/or death rates in which all the sites are occupied. As far
as we know, the only exception is the variant of the multitype contact process
[25] introduced in [19]. This process, however, was only designed to model the
interactions among cooperators and defectors in the prisoner’s dilemma rather
than general games described by a payoff matrix. Our main objective is to
initiate the study of spatial evolutionary games based more realistically on
Neuhauser’s multitype contact process instead of the voter model. In
particular, the state at time $t$ is now

$\xi_{t}:\mathbb{Z}^{d}\longrightarrow\\{0,1,2,\ldots,n\\}\quad\hbox{where}\quad\xi_{t}(x)=\hbox{strategy
of the player at site\leavevmode\nobreak\ $x$},$

with the convention 0 = empty. The dynamics combines the multitype contact
process and the birth-death updating process (3) dynamics. Like in the neutral
multitype contact process, we assume that, regardless of their strategy, the
players have the same natural birth rate $\lambda$ and the same natural death
rate one, and that births onto already occupied sites are suppressed. Having a
payoff matrix $A=(a_{ij})$, each player now receives a payoff from its
occupied neighbors while empty neighbors have no effects, so the payoff can be
defined as in (1) assuming that $a_{i0}=0$, i.e., empty sites give a zero
payoff. Like in the birth-death updating process, the payoff of the players
affects their birth rate. To have a well-defined positive birth rate even when
the payoff is negative, instead of using the fitness function (2), we assume
that the natural birth rate $\lambda$ is multiplied by the exponential of the
payoff. In particular, the transition rates are given by

$\begin{array}[]{c}c_{0\to i}(x,\xi_{t})=\sum_{y\sim
x}\Phi(y,\xi_{t})\,\mathbf{1}\\{\xi_{t}(y)=i\\}/2d\quad\hbox{and}\quad c_{i\to
0}(x,\xi_{t})=1,\end{array}$

where the fitness function is defined as

$\begin{array}[]{c}\Phi(x,\xi_{t})=\lambda\exp(\phi(x,\xi_{t}))=\lambda\exp(\sum_{i,j\neq
0}a_{ij}\,f_{j}(x,\xi_{t})\,\mathbf{1}\\{\xi_{t}(x)=i\\}).\end{array}$

In particular, players with no neighbors give birth at rate $\lambda$, while
players with a positive payoff give birth at a higher rate and players with a
negative payoff give birth at a lower rate.
Before looking at the competition among multiple strategies, the first step is
to study the conditions for survival in the presence of only one strategy, say
strategy 1, which is the objective of this paper.

$\lambda=8$, $a=-1.75$$\lambda=2$, $a=0$$\lambda=0.25$, $a=1.16$

Figure 1: Snapshots at time 1000 of the two-dimensional single-type contact
process with various birth parameters $\lambda$ and payoffs $a$. Middle: When
$a=0$, the process reduces to the basic contact process. Left: When $a<0$,
small clusters have a reduced birth rate, which results in a more scattered
configuration at equilibrium. Right: In contrast, when $a>0$, small clusters
have an increased birth rate, which results in stronger spatial correlations.

In this case, the model depends on two parameters: the natural birth rate
$\lambda$, and the single payoff coefficient $a_{11}$ that we simply denote by
$a$. A player at $x$ gives birth at rate

$\begin{array}[]{c}\Phi(x,\xi_{t})=\lambda\exp(af_{1}(x,\xi_{t})),\end{array}$
(5)

and dies at rate one. See Figure 1 for simulation pictures. Note that this
function is nondecreasing with respect to the natural birth rate $\lambda$ and
the payoff coefficient $a$. In particular, one expects the probability of
survival, i.e., the probability that, starting with a single player, there is
at least one player at all times, to be nondecreasing with respect to the
parameters $\lambda$ and $a$. Standard coupling arguments, however, fail to
prove this result when $a<0$. Indeed, increasing the birth rate (5) increases
the local density of occupied sites, which decreases the birth rate of the
surrounding players. Coupling processes with different parameters/initial
configurations, we can prove the following result: letting
$\mathbb{P}_{\lambda,a}^{\xi}$ be the law of the process with parameters
$\lambda$ and $a$, starting from $\xi_{0}=\xi$,

$\begin{array}[]{rcl}\lambda_{1}\leq\lambda_{2},\ \ a_{2}\geq a_{1}\vee 0,\ \
\xi^{1}\subset\xi^{2}&\Longrightarrow&\mathbb{P}_{\lambda_{1},a_{1}}^{\xi^{1}}[\xi_{t}\neq\varnothing\
\forall t]\leq\mathbb{P}_{\lambda_{2},a_{2}}^{\xi^{2}}[\xi_{t}\neq\varnothing\
\forall t].\end{array}$ (6)

Looking at particular cases with the same initial configuration with a single
player and/or the same natural birth rate and/or the same payoff coefficient,
the implication in (6) shows that the process is attractive and monotone with
respect to $\lambda$ and $a$ in the parameter region where $a\geq 0$. In
particular, there is at most one phase transition from extinction to survival
in the direction of each of the two parameters.

$a$$\lambda$extinctionsurvival

contact process

contact process

0$+\infty$$-\infty$0$\lambda_{c}$$+\infty$TH 3TH 2TH 1.aTH 1.bTH 1.bTH
1.a$\lambda=\lambda_{c}\,e^{-a(1-1/2d)}$$\lambda=\lambda_{c}\,e^{-a(1-1/2d)}$one
phase transition (6)

Figure 2:  Phase structure of the contact process with a density-dependent
birth rate.

The implication also shows that the probability of survival for $a\geq 0$ is
no less than the probability of survival for $a\leq 0$. In addition, in the
special case $a=0$, the birth rate (5) becomes $\lambda$ regardless of the
configuration, therefore the process reduces to the basic contact process
[14]. In particular, letting $\lambda_{c}=\lambda_{c}(\mathbb{Z}^{d})$ denote
the critical value of the contact process on $\mathbb{Z}^{d}$, and using other
coupling arguments, we can prove the following result.

######  Theorem 1.

For every fixed $a\in\mathbb{R}$, the process

$\begin{array}[]{rrrclcrcl}\hbox{(a)}&\hbox{survives
when}&(a>0&\hbox{and}&\lambda>\lambda_{c})&\hbox{or}&(a<0&\hbox{and}&\lambda>\lambda_{c}\,e^{-a(1-1/2d)}),\vspace*{2pt}\\\
\hbox{(b)}&\hbox{dies out
when}&(a<0&\hbox{and}&\lambda\leq\lambda_{c})&\hbox{or}&(a>0&\hbox{and}&\lambda\leq\lambda_{c}\,e^{-a(1-1/2d)}).\end{array}$

The theorem shows that, for each fixed payoff $-\infty<a<\infty$, there exists
at least one (and exactly one when $a\geq 0$) phase transition in the
direction of the birth rate $\lambda$. Proving that, for each fixed birth rate
$0<\lambda<\infty$, there exists a phase transition in the direction of the
payoff $a$ is more complicated. Even when $\lambda>0$ is small, for $a<\infty$
large, adjacent players have a large birth rate due to cooperation. In
particular, it can be proved that, with high probability, a small block of
players quickly doubles in size. Using also attractiveness when the payoff is
positive, and a block construction, implies survival when $a$ is sufficiently
large.

######  Theorem 2 (Survival).

For every birth rate $\lambda>0$, there exists an
$a_{+}=a_{+}(\lambda,d)<\infty$ such that the process survives for all payoffs
$a>a_{+}$.

In contrast, even when $\lambda<\infty$ is large, for $a=-\infty$, adjacent
players cannot give birth before one of them dies due to competition. In
particular, it can be proved that the family generated by a single player
decays exponentially in space and time. Using another block construction and a
perturbation argument implies extinction when $a$ is sufficiently small.

######  Theorem 3 (Extinction).

For every birth rate $\lambda<\infty$, there exists an
$a_{-}=a_{-}(\lambda,d)>-\infty$ such that the process dies out for all
payoffs $a<a_{-}$.

We refer to the phase diagram in Figure 2 for a summary/visualization of our
main results. Before going into the proofs, we point out that Theorem 3 cannot
be deduced from general ergodicity results like the $(M-\epsilon)$-criterion
[20, Theorem I.4.1] because we always have $\epsilon=1$ while $M$ is lower
bounded by the corresponding value for the contact process. We also note that,
although the technical details in the proofs of Theorems 2 and 3 differ
significantly, the intuition behind both results is the same. Even if
$\lambda$ is very small, once a player gives birth, this player and its
offspring form an adjacent pair with an arbitrarily large birth rate when $a$
is large. Similarly, even if $\lambda$ is very large, once a player gives
birth, this player and its offspring form an adjacent pair with an arbitrarily
small birth rate when $a$ is small. In both cases, the conclusion
(survival/extinction) is due to the presence of local interactions: the
players place their offspring in their neighborhood, while their payoff is
also determined by their neighbors. In particular, the two theorems are
expected to fail in the absence of local interactions. Indeed, a simple
analysis of the mean-field model shows that whether the trivial extinction
fixed point 0 is stable or unstable, and so whether the population dies out or
survives starting at low density, depends on $\lambda$ but not on $a$.
The rest of the paper is devoted to the proofs. Section 2 gives a brief
analysis of the mean-field model, focusing on the local stability of the
trivial fixed point, but also on the size of its basin of attraction. Section
3 relies on various coupling arguments to prove monotonicity and
attractiveness in the case where $a\geq 0$. Similar couplings are used to
compare the process with the basic contact process, and deduce Theorem 1.
Section 4 uses a block construction to prove Theorem 2. Finally, Section 5
establishes some exponential decay to deduce extinction of the process with
$a=-\infty$ from a block construction. Theorem 3 is then deduced by using a
perturbation argument.

##  2 Mean-field model

This section gives a brief analysis of the nonspatial deterministic mean-field
model, which describes the process in the large population limit when the
system is homogeneously mixing. Letting $u$ be the density of occupied sites,
the mean-field model reduces to the differential equation

$u^{\prime}=\phi(u)=\lambda e^{au}u(1-u)-u.$

Because of the exponential form of the birth rate, we cannot obtain the exact
expression of the fixed points. However, the stability of the trivial fixed
point 0, corresponding to the extinction state, as well as the existence and
stability of additional (interior) fixed points, can be studied. To begin
with, observe that, for all fixed $-\infty<a<\infty$,

$\begin{array}[]{rcl}u\approx 0&\Longrightarrow&\phi(u)\approx\lambda
u-u=(\lambda-1)u.\end{array}$ (7)

This shows that the stability of the trivial fixed point, and so whether the
population survives or dies out when starting at low density, depends on the
natural birth rate $\lambda$ but not on the payoff coefficient $a$, which is
in sharp contrast with Theorems 2 and 3.
Extinction phase. It follows from (7) that, when $\lambda<1$, the trivial
fixed point is locally stable, so the population dies out ($u\to 0$ starting
at low density) even when $a$ is very large. This contrasts with Theorem 2,
which states that, even starting with a finite number of players (density
zero), the population survives with positive probability. However, for all
$\lambda>0$ and $\bar{u}<1/2$,

$\begin{array}[]{rcl}a>\ln(2/\lambda)/\bar{u}&\Longrightarrow&\phi(\bar{u})>\lambda
e^{\ln(2/\lambda)}\,\bar{u}(1-\bar{u})-\bar{u}=2\bar{u}(1-\bar{u})-\bar{u}\geq
0.\end{array}$

This shows that $\phi(u)$ becomes positive at some unstable fixed point
$u_{*}\in(0,\bar{u})$ therefore, starting at a density $>u_{*}$, the
population converges to a limit $u^{*}>0$. In conclusion, when $\lambda<1$,
there is extinction in the sense that the trivial fixed point is locally
stable, but starting from a fixed positive density, the population survives
for all $a$ sufficiently large (depending on that density).
Survival phase. It follows from (7) that, when $\lambda>1$, the trivial fixed
point is unstable, so the population survives (starting from a positive
density, $u$ converges to a positive limit) even when $a$ is very small. This
contrasts with Theorem 3, which states that, regardless of the initial
configuration, the density of occupied sites vanishes to zero. However, for
all $\lambda<\infty$ and $\bar{u}<1/2$,

$\begin{array}[]{rcl}a<\ln(1/\lambda)/\bar{u}&\Longrightarrow&\phi(\bar{u})<\lambda
e^{\ln(1/\lambda)}\,\bar{u}(1-\bar{u})-\bar{u}=\bar{u}(1-\bar{u})-\bar{u}\leq
0.\end{array}$

This shows that $\phi(u)$ becomes negative at some stable fixed point
$u^{*}\in(0,\bar{u})$ therefore, starting at low density, the population
converges to a limit that cannot exceeds $\bar{u}$. In conclusion, when
$\lambda>1$, there is survival in the sense that the trivial fixed point is
unstable, but starting at low density, the limiting density can be made
arbitrarily small by taking $a$ sufficiently small.

##  3 Proof of Theorem 1 (monotonicity and attractiveness)

The proofs of Theorem 1 and (6) are based on standard coupling arguments.
Processes with different parameters and/or different initial configurations
can be coupled by constructing them jointly on the same graphical
representation [15]. However, the graphical representation of the contact
process with a density-dependent birth rate is somewhat complicated. Instead,
we use the classical comparison result [20, Theorem III.1.5], which we restate
for the reader’s convenience.

######  Theorem 4.

Let $\xi_{t}^{1}$ and $\xi_{t}^{2}$ be two interacting particle systems with
state space $\Omega=\\{0,1\\}^{\mathbb{Z}^{d}}$, and assume that, whenever
$\xi^{1}\subset\xi^{2}$, we have the inequalities

$c_{0\to 1}(x,\xi^{1})\leq c_{0\to 1}(x,\xi^{2})\quad\hbox{and}\quad c_{1\to
0}(x,\xi^{1})\geq c_{1\to 0}(x,\xi^{2}).$

Then, there is a coupling of the two processes such that

$\begin{array}[]{rcl}\xi^{1}\subset\xi^{2}&\Longrightarrow&\mathbb{P}^{(\xi^{1},\xi^{2})}[\xi_{t}^{1}\subset\xi_{t}^{2}\
\forall t]=1.\end{array}$

Using this result, we can prove (6) about the monotonicity and the
attractiveness of the contact process with a density-dependent birth rate.

###### Proof of (6).

For $i=1,2$, let $\xi_{t}^{i}$ be the process with natural birth rate
$\lambda_{i}$, payoff coefficient $a_{i}$, and initial configuration
$\xi^{i}$. Because the death rate of the processes is always equal to one, the
second inequality in Theorem 4, which is in fact an equality, is always
satisfied. To prove the first inequality, recall that the birth rate of the
process is of the form

$\begin{array}[]{c}c_{0\to 1}(x,\xi)=\sum_{y\sim
x}\Phi(y,\xi)\,\xi(y)/2d=\sum_{y\sim
x}\lambda\exp(af_{1}(y,\xi))\,\xi(y)/2d=\psi(\lambda,a,\xi).\end{array}$

The function $\psi(\lambda,a,\xi)$ is nondecreasing with respect to $\lambda$,
$a$, and $\xi$ when $a\geq 0$. In particular, according to Theorem 4, the
processes can be coupled in such a way that

$\begin{array}[]{rcl}\lambda_{1}\leq\lambda_{2},\ \ 0\leq a_{1}\leq a_{2},\ \
\xi^{1}\subset\xi^{2}&\Longrightarrow&\mathbb{P}^{(\xi^{1},\xi^{2})}[\xi_{t}^{1}\subset\xi_{t}^{2}\
\forall t]=1.\end{array}$ (8)

Note also that, for all $\lambda_{1}\leq\lambda_{2}$, $a_{1}\leq 0\leq a_{2}$,
and $\xi^{1}\subset\xi^{2}$,

$\begin{array}[]{rcl}\psi(\lambda_{1},a_{1},\xi_{1})\leq\sum_{y\sim
x}\lambda_{1}\,\xi^{1}(y)/2d\leq\sum_{y\sim
x}\lambda_{2}\,\xi^{2}(y)/2d\leq\psi(\lambda_{2},a_{2},\xi_{2}),\end{array}$

from which we deduce that

$\begin{array}[]{rcl}\lambda_{1}\leq\lambda_{2},\ \ a_{1}\leq 0\leq a_{2},\ \
\xi^{1}\subset\xi^{2}&\Longrightarrow&\mathbb{P}^{(\xi^{1},\xi^{2})}[\xi_{t}^{1}\subset\xi_{t}^{2}\
\forall t]=1.\end{array}$ (9)

The implication in (6) follows from the existence of the couplings (8) and
(9). ∎

Using again Theorem 4, we can also prove Theorem 1.

###### Proof of Theorem 1.

Let $\eta_{t}$ and $\zeta_{t}$ be the contact processes with parameter
$\lambda$ and $\lambda e^{-a(1-1/2d)}$, respectively, and let $\xi_{t}$ be the
density-dependent contact process. Observing that the density-dependent
contact process with payoff $a=0$ reduces to the basic contact process with
the same natural birth rate, it follows from (9) that the process $\xi_{t}$
dominates $\eta_{t}$ when $a\geq 0$ but is dominated by $\eta_{t}$ when $a\leq
0$. To compare the processes $\xi_{t}$ and $\zeta_{t}$, observe that, when
site $x$ is empty, its neighbors have at most $2d-1$ occupied neighbors,
therefore

$\begin{array}[]{rclcl}a\leq 0,\ \ \xi\supset\zeta&\Longrightarrow&c_{0\to
1}(x,\xi)&\geq&\sum_{y\sim x}\lambda e^{a(1-1/2d)}\,\xi(y)/2d=\lambda
e^{a(1-1/2d)}\,f_{1}(x,\xi)\vspace*{8pt}\\\ &&&\geq&\lambda
e^{a(1-1/2d)}\,f_{1}(x,\zeta)=c_{0\to 1}(x,\zeta),\vspace*{8pt}\\\ a\geq 0,\ \
\xi\subset\zeta&\Longrightarrow&c_{0\to 1}(x,\xi)&\leq&\sum_{y\sim x}\lambda
e^{a(1-1/2d)}\,\xi(y)/2d=\lambda e^{a(1-1/2d)}\,f_{1}(x,\xi)\vspace*{8pt}\\\
&&&\leq&\lambda e^{a(1-1/2d)}\,f_{1}(x,\zeta)=c_{0\to 1}(x,\zeta).\end{array}$

In particular, by Theorem 4, the process $\xi_{t}$ dominates $\zeta_{t}$ when
$a\leq 0$ but is dominated by $\zeta_{t}$ when $a\geq 0$. In conclusion, there
are couplings of the three processes such that

$\begin{array}[]{rcl}a\leq 0,\ \
\eta\supset\xi\supset\zeta&\Longrightarrow&\mathbb{P}^{(\eta,\xi,\zeta)}[\eta_{t}\supset\xi_{t}\supset\zeta_{t}\
\forall t]=1,\vspace*{4pt}\\\ a\geq 0,\ \
\eta\subset\xi\subset\zeta&\Longrightarrow&\mathbb{P}^{(\eta,\xi,\zeta)}[\eta_{t}\subset\xi_{t}\subset\zeta_{t}\
\forall t]=1.\end{array}$

Since [2, Theorem 1] implies that the contact process $\eta_{t}$ survives if
and only if $\lambda>\lambda_{c}$, while the contact process $\zeta_{t}$
survives if and only if $\lambda>\lambda_{c}\,e^{-a(1-1/2d)}$, the theorem
follows. ∎

##  4 Proof of Theorem 2 (survival for $a<\infty$ large)

This section is devoted to the proof of Theorem 2, which states that, for all
birth rates $\lambda>0$ even small, the process survives provided the payoff
coefficient $a$ is sufficiently large. The basic idea is that, though it might
be likely that isolated players die before they have a chance to give birth,
players with at least one neighbor have a large birth rate when $a$ is large,
so a small fully occupied cube will double in size with probability close to
one in a small deterministic time. This and a block construction imply
survival. More precisely, letting

$\Lambda_{-}=\\{0,1\\}^{d}\quad\hbox{and}\quad\Lambda_{+}=\\{-1,0,1,2\\}^{d},$

the goal is to prove that, if the small cube $\Lambda_{-}$ is initially fully
occupied then, with probability arbitrarily close to one when $a$ is large,
the larger cube $\Lambda_{+}$ will be fully occupied at some fixed
deterministic time $\tau$. The density-dependent process with $a\geq 0$ is
attractive according to (6), and the birth rate of players with at least one
occupied neighbor is larger than $\lambda e^{a/2d}$. In particular, it
suffices to prove the result for the process $\bar{\xi}_{t}$ with transition
rates

$\begin{array}[]{c}c_{0\to 1}(x,\bar{\xi}_{t})=\sum_{y\sim x}\lambda
e^{a/2d}\,\mathbf{1}\\{f_{1}(y,\bar{\xi}_{t})\neq
0\\}\,\bar{\xi}_{t}(y)/2d\quad\hbox{and}\quad c_{1\to
0}(x,\bar{\xi}_{t})=1,\end{array}$

modified so that births outside $\Lambda_{+}$ are suppressed. The transition
rates indicate that isolated players cannot give birth, while players with at
least one occupied neighbor give birth at rate $\lambda e^{a/2d}$, and that
players die at rate one. This process can be constructed graphically as
follows:

  * •

Births. Equip each $\vec{xy}$, $x\sim y$, with an exponential clock with rate
$\lambda e^{a/2d}/2d$. At the times $t$ the clock rings, draw an arrow
$(x,t)\to(y,t)$ to indicate that, if $x$ is occupied and has at least one
occupied neighbor, and $y$ is empty, then $y$ becomes occupied.

  * •

Deaths. Equip each $x$ with an exponential clock with rate one. At the times
$t$ the clock rings, put a cross $\times$ at $(x,t)$ to indicate that, if $x$
is occupied, then it becomes empty.

We denote by $\bar{\mathbb{P}}_{\lambda,a}$ the law of this process. To prove
survival, we first show that, with high probability, there are no death marks
$\times$ in $\Lambda_{+}$ by some small time $\tau$.

######  Lemma 5.

For all $\epsilon>0$, there exists $\tau>0$ such that

$\bar{\mathbb{P}}_{\lambda,a}[\hbox{no death marks\leavevmode\nobreak\
$\times$ in\leavevmode\nobreak\ $\Lambda_{+}\times[0,\tau]$}]\geq
1-\epsilon/2.$

###### Proof.

The number $D_{\tau}$ of death marks in the space-time box
$\Lambda_{+}\times[0,\tau]$ is Poisson distributed with parameter $4^{d}\tau$.
In particular, the probability of no death marks satisfies

$\begin{array}[]{rcl}\mathbb{P}[D_{\tau}=0]=e^{-4^{d}\tau}=1-\epsilon/2&\Longleftrightarrow&\tau=-\ln(1-\epsilon/2)/4^{d}>0.\end{array}$
(10)

This completes the proof. ∎

The next step is to prove invasion $\Lambda_{-}\to\Lambda_{+}$ by time $\tau$.
Because the players in the small cube $\Lambda_{-}$ cannot immediately give
birth onto the corners of the larger cube $\Lambda_{+}$ when $d>1$, we divide
the problem into $d$ steps by considering the sequence of spatial regions

$\begin{array}[]{c}\Lambda_{i}=\\{y\in\Lambda_{+}:\min_{x\in\Lambda_{-}}|\\!|x-y|\\!|_{1}=i\\}\quad\hbox{for}\quad
i=0,1,\ldots,d.\end{array}$

Figure 3 shows a picture in the $d=2$ case. Note that
$\Lambda_{-}=\Lambda_{0}$ and $\Lambda_{+}=\cup_{i}\Lambda_{i}$. In addition,
each site in one region has at least one neighbor in the previous region, so a
fully occupied region can immediately invade the next region. We are now ready
to prove invasion.

$\Lambda_{-}$$\Lambda_{1}$$\Lambda_{2}$

Figure 3:  Illustration of the $d$-step process used in the proof of Theorem
2. The region delimited with bold lines represents $\Lambda_{-}$ on the left,
$\Lambda_{1}$ in the middle, and $\Lambda_{2}$ on the right. In all three
pictures, the dashed box is $\Lambda_{+}$.

######  Lemma 6.

For all $\epsilon>0$ and $\tau>0$ like in (10), there exists $a<\infty$ large
such that

$\bar{\mathbb{P}}_{\lambda,a}^{\Lambda_{-}}[\bar{\xi}_{\tau}=\Lambda_{+}]\geq
1-\epsilon.$

###### Proof.

As previously mentioned, we proceed in $d$ steps, and prove that, conditional
on no deaths, each invasion $\Lambda_{i}\to\Lambda_{i+1}$ occurs in less then
$\tau/d$ units of time with high probability. Given that there are no deaths
by time $\tau$, and that $\Lambda_{i}$ is fully occupied, the probability that
$\Lambda_{i+1}$ becomes fully occupied in less than $\tau/d$ units of time is
larger than the probability that, for each site $y\in\Lambda_{i+1}$, there is
at least one birth arrow $x\to y$ for some $x\in\Lambda_{i}$ in this time
window. Because each region has less than $4^{d}$ sites, and birth arrows
occur along each directed edge at rate $\lambda e^{a/2d}/2d$, this is larger
than the probability that $X_{1},X_{2},\ldots,X_{4^{d}}$ = independent
exponential random variables with that rate are all less than $\tau/d$. In
particular, for all $i=0,1,\ldots,d-1$,

$\begin{array}[]{l}\bar{\mathbb{P}}_{\lambda,a}[\bar{\xi}_{t}\supset\Lambda_{i+1}\
\forall
t\in[(i+1)\tau/d,\tau]\,|\,\bar{\xi}_{i\tau/d}=\Lambda_{i},D_{\tau}=0]\vspace*{6pt}\\\
\hskip
34.0pt=\bar{\mathbb{P}}_{\lambda,a}^{\Lambda_{i}}[\bar{\xi}_{\tau/d}\supset\Lambda_{i+1}\,|\,D_{\tau}=0]\geq\mathbb{P}[X_{1},X_{2},\ldots,X_{4^{d}}\leq\tau/d]\vspace*{6pt}\\\
\hskip 34.0pt=(1-\exp(-\lambda\tau e^{a/2d}/2d^{2}))^{4^{d}}.\end{array}$ (11)

Some basic algebra shows that this is $\geq 1-\epsilon/2d$ for all $a$ larger
than

$a_{+}=2d\,\ln(-2d^{2}\,\ln(1-(1-\epsilon/2d)^{4^{-d}})/\lambda\tau)<\infty.$
(12)

In particular, for all $a\geq a_{+}$, it follows from (11) and Lemma 5 that

$\begin{array}[]{rcl}\bar{\mathbb{P}}_{\lambda,a}^{\Lambda_{-}}[\bar{\xi}_{\tau}\neq\Lambda_{+}]&\leq&\bar{\mathbb{P}}_{\lambda,a}^{\Lambda_{-}}[\bar{\xi}_{\tau}\neq\Lambda_{+}\,|\,D_{\tau}=0]+\mathbb{P}[D_{\tau}\neq
0]\vspace*{6pt}\\\
&\leq&\sum_{i<d}\bar{\mathbb{P}}_{\lambda,a}^{\Lambda_{i}}[\bar{\xi}_{\tau/d}\not\supset\Lambda_{i+1}\,|\,D_{\tau}=0]+\mathbb{P}[D_{\tau}\neq
0]\vspace*{6pt}\\\ &\leq&d(1-(1-\epsilon/2d))+\epsilon/2=\epsilon,\end{array}$

which proves the lemma. ∎

The rest of the proof of Theorem 2 is standard and relies on a block
construction. This technique first appeared in [3] and is explained in detail
in [9]. The basic idea is to compare the process properly rescaled in space
and time with supercritical oriented site percolation. Let

$\mathscr{L}=\\{(m,n)\in\mathbb{Z}^{d}\times\mathbb{N}:m_{1}+\cdots+m_{d}+n\
\hbox{is even}\\},$

and turn $\mathscr{L}$ into a directed graph $\vec{\mathscr{L}}$ by placing an
edge

$(m,n)\to(m^{\prime},n^{\prime})\quad\hbox{if and only if}\quad
m_{i}^{\prime}=m_{i}\pm 1\ \hbox{for all}\ i\ \hbox{and}\ n^{\prime}=n+1.$

Fix $\epsilon>0$, let $\tau$ as in (10), and call $(m,n)\in\mathscr{L}$ a good
site whenever

$E_{m,n}=\\{\hbox{the box\leavevmode\nobreak\ $m+\Lambda_{-}$ is fully
occupied at time\leavevmode\nobreak\ $n\tau$}\\}\quad\hbox{occurs}.$

######  Lemma 7.

Let $a_{+}$ as in (12). Then, for all $a\geq a_{+}$, the set of good sites
dominates stochastically the set of wet sites in an oriented site percolation
process on $\vec{\mathscr{L}}$ with parameter $1-\epsilon$.

###### Proof.

Because the process $\xi_{t}$ dominates the process $\bar{\xi}_{t}$, and the
evolution rules of the process $\xi_{t}$ are translation invariant, it follows
from Lemma 6 that, for all sites $(m^{\prime},n^{\prime})\leftarrow(m,n)$,

$\begin{array}[]{l}\mathbb{P}_{\lambda,a}[\xi_{(n+1)\tau}\supset
m^{\prime}+\Lambda_{-}\,|\,\xi_{n\tau}\supset m+\Lambda_{-}]\vspace*{4pt}\\\
\hskip 50.0pt\geq\mathbb{P}_{\lambda,a}[\xi_{(n+1)\tau}\supset
m+\Lambda_{+}\,|\,\xi_{n\tau}\supset m+\Lambda_{-}]\geq
1-\epsilon\quad\hbox{for all}\quad a\geq a_{+}.\end{array}$

This shows the existence of a collection of good events $G_{m,n}$ such that

$\hbox{(a)}\ \ \mathbb{P}_{\lambda,a}[G_{m,n}]\geq
1-\epsilon\quad\hbox{and}\quad\hbox{(b)}\ \ E_{m,n}\cap G_{m,n}\subset
E_{m^{\prime},n^{\prime}}\ \ \hbox{for all}\ \
(m^{\prime},n^{\prime})\leftarrow(m,n).$

The lemma then follows from [9, Theorem A.4] ∎

To deduce the theorem, we start the process $\xi_{t}$ with a single player at
the origin. Because $\lambda>0$, there is a positive probability that
$\Lambda_{-}$ is fully occupied at time one, so we may assume that the process
starts from $\xi_{0}=\Lambda_{-}$ instead. In addition, because Lemma 6
applies to the modified process $\bar{\xi}_{t}$, the events $G_{m,n}$ in Lemma
7 can be made measurable with respect to the graphical representation in a
bounded space-time box. This shows that the range of dependence of the
percolation process is finite, so we can fix $\epsilon>0$ small to make the
percolation process supercritical. Since in addition the set of good sites
dominates the set of wet sites for all $a\geq a_{+}$, the process survives.

##  5 Proof of Theorem 3 (extinction for $a>-\infty$ small)

This section is devoted to the proof of Theorem 3, which states that, for all
$\lambda<\infty$ even large, the process dies out provided the payoff
coefficient $a$ is sufficiently small. Like in Theorem 2, the basic idea is
that isolated players and players with at least one neighbor may have
fitnesses that differ strongly, except that the effects are now reversed:
though it might be likely that isolated players give birth quickly, once they
give birth and have one neighbor (their offspring), they are no longer likely
to give birth when $a$ is small. The proof again relies on a block
construction, but the technical details are somewhat more complicated. To
begin with, we consider the process starting with a single individual in the
limit $a=-\infty$. In this case, adjacent individuals cannot give birth and
the process essentially behaves like a symmetric random walk that dies after a
geometric number of jumps. This implies that the length of the invasion path
decays exponentially in both space and time (radius and time to extinction).
Because the birth rate decreases with the local density of individuals, the
process starting from a general configuration is dominated stochastically by a
system of such independent random walks starting with one particle per site.
The exponential decay implies that, regardless of the configuration outside a
large space-time box and with probability close to one, a large space-time
region around the center of the box is not reached by any of the random
walks/invasion paths (dead region). This is used to prove percolation of the
dead regions under a suitable space-time rescaling. Once the space and time
scales are fixed, we can use a perturbation argument to prove that percolation
still occurs for all $a>-\infty$ sufficiently small. Extinction of the process
then follows from the fact that the percolation parameter can be chosen close
enough to one to ensure the lack of percolation of the closed sites
(corresponding to potentially occupied blocks), and the fact that individuals
cannot appear spontaneously.

Fast extinction for $a=-\infty$. We first study the process in the limiting
case $a=-\infty$ starting with a single player at the origin, whose law is
denoted by $\mathbb{P}_{\lambda,-\infty}^{0}$. Let

$T=\inf\\{t:\xi_{t}=\varnothing\\}=\hbox{time to extinction}.$

######  Lemma 8.

For every $\lambda\geq 0$, there exists a constant $\beta=\beta(\lambda,d)>0$
such that

$\mathbb{P}_{\lambda,-\infty}^{0}[T\geq t]\leq e^{-\beta t}\quad\hbox{for
all}\quad t\geq 0.$

###### Proof.

Since $a=-\infty$, two adjacent players cannot give birth, so there can be at
most two players alive at the same time. By the superposition property, the
time until one of the two dies is distributed as an exponential random
variable with parameter two. Now, if there is just a single player, the
probability that it dies before it reproduces is equal to $1/(1+\lambda)$.
Therefore, the total number of generations $N$ in which exactly two players
are alive is geometrically distributed:

$\mathbb{P}_{\lambda,-\infty}^{0}[N>n]\leq\left(1-\frac{1}{1+\lambda}\right)^{n}=\left(\frac{\lambda}{1+\lambda}\right)^{n}=\bigg{(}1+\frac{1}{\lambda}\bigg{)}^{-n},\quad
n\in\mathbb{N}.$ (13)

In particular, $T$ can be written as a sum of independent exponential random
variables:

$T=\sum_{i=1}^{N}\ (T_{i}+S_{i})+U_{N+1}$

where the $(T_{i})_{i\in\mathbb{N}}$ are exponential random variables with
parameter two, the $(S_{i})_{i\in\mathbb{N}}$ are exponential random variables
with parameter $\lambda$, and the $(U_{i})_{i\in\mathbb{N}}$ are exponential
random variables with parameter one. In particular, $T$ is a hypoexponential
random variable. Since we are interest in the case where $\lambda$ is large
(and so typically $\lambda>1$), and we do not need very precise bounds, we can
upperbound the time to extinction $T$ by the sum $\tilde{T}_{N}$ of $2N+1$
exponential random variables with parameter one. Then, $\tilde{T}_{N}$ is a
$\text{Gamma}(2N+1,1)$ random variable. For a deterministic $n$, one can use
the following Chernoff bound for the tails of the Gamma distribution:

$\mathbb{P}_{\lambda,-\infty}^{0}[\tilde{T}_{n}\geq t]\leq\frac{e^{-\theta
t}}{(1-\theta)^{2n+1}}.$ (14)

Combining (13) and (14) implies that, for any $\theta<1$ and $m\in\mathbb{N}$,

$\mathbb{P}_{\lambda,-\infty}^{0}[T\geq t]\leq\sum_{n=1}^{m}\
\mathbb{P}[\tilde{T}_{n}\geq t]+\mathbb{P}[N>m]\leq m\bigg{(}\frac{e^{-\theta
t}}{(1-\theta)^{2m+1}}\bigg{)}+e^{-m\ln(1+1/\lambda)}.$

Taking for example $\theta=1/2$ and $m=\lfloor tr\rfloor$ with $r<1/4\ln(2)$
gives the result. ∎

Looking closely at the proof of the previous lemma, one easily deduces the
following result, which also gives us an exponential decay of the radius of
the invasion paths.

######  Lemma 9.

Let $\xi^{0}\subset\mathbb{Z}^{d}$ be the (random) subset of vertices that are
ever occupied. Then, for every birth rate $\lambda\geq 0$, there exists a
constant $\delta=\delta(\lambda,d)>0$ such that

$\mathbb{P}_{\lambda,-\infty}^{0}[|\xi^{0}|>n]\leq e^{-\delta n}\quad\hbox{for
all}\quad n\in\mathbb{N}.$

###### Proof.

Recall from the proof of Lemma 8 that the number of generations $N$ for which
our process survives is a geometric random variable with parameter
$1/(1+\lambda)$. Since we can only visit at most one new site per generation,
it follows from (13) that

$\mathbb{P}_{\lambda,-\infty}^{0}[|\xi^{0}|>n]\leq\mathbb{P}[N>n]=(1+1/\lambda)^{-n}=e^{-n\ln(1+1/\lambda)},$

which is exactly the exponential decay we were looking for. ∎

Comparison to non-interacting copies. In order to use union bounds, it will be
convenient to compare the process with arbitrary initial condition to a family
of non-interacting copies of the process started with a single occupied site.
For all $A\subset\mathbb{Z}^{d}$, we let $\xi_{t}^{A}$ denote the process
starting from the set $A$ occupied and, as previously, simply write
$\xi_{t}^{z}$ when $A=\\{z\\}$. Having a collection $\\{(\xi^{x}_{t})_{t\geq
0}:\ x\in A\\}$ of non-interacting copies of our process, we let

$\begin{array}[]{c}\Xi_{t}^{A}(x)=\sum_{z\in A}\xi_{t}^{z}(x)\quad\hbox{for
all}\quad x\in\mathbb{Z}^{d}.\end{array}$

With this notation at hand, we are ready to state the following comparison
result.

######  Lemma 10.

There exists a coupling such that
$\mathbb{P}_{\lambda,-\infty}[\xi^{A}_{t}\leq\Xi^{A}_{t}\ \forall t]=1$.

In particular, the set of sites occupied by the original process at time $t$
is always contained in the set of sites occupied by at least one of the non-
interacting copies.

###### Proof.

To compare the two processes, we differentiate the players assuming that, for
all $z\in A$, site $z$ is initially occupied by a type $z$ player. Then, in
the limiting case $a=-\infty$, we can construct all the processes from the
following graphical representation:

  * •

Births. For each $z\in A$, equip each $\vec{xy}$, $x\sim y$, with a rate
$\lambda/2d$ exponential clock. At the times $t$ the clock rings, draw an
arrow $(x,t)\overset{z}{\longrightarrow}(y,t)$.

  * •

Deaths. For each $z\in A$, equip each $x$ with a rate one exponential clock.
At the times $t$ the clock rings, put a cross $\times_{z}$ at the space-time
point $(x,t)$.

The crosses have the same effects on both processes: a cross $\times_{z}$ at
site $x$ kills a type $z$ particle at that site. The arrows, however, have
different effects. The process $\xi_{t}^{A}$ is constructed by assuming that,
if the tail $x$ of a type $z$ arrow is occupied by a type $z$ player, and none
of the neighbors of $x$ is occupied, then the head $y$ of the arrow becomes
occupied by a type $z$ player. In contrast, the system of non-interacting
copies $\Xi^{A}_{t}$ is constructed by assuming that, if the tail of a type
$z$ arrow is occupied by a type $z$ player, and none of the neighbors of site
$x$ is occupied by a type $z$ player, then the head $y$ of the arrow becomes
occupied by a type $z$ player. Because the condition for giving birth in the
first process is more restrictive (no players of any type in the neighborhood
as opposed to no type $z$ players), if there is a type $z$ player at $(x,t)$
in the first process then there is a type $z$ player at $(x,t)$ in the second
process, which proves the lemma. Note that there is at most one type $z$
player at each site in both processes, but sites can be occupied by multiple
players with different types in the system of non-interacting copies. ∎

Block construction. Using the exponential decay (in space and time) of the
invasion paths and the stochastic domination in the previous two sections, we
can now use a block construction to prove extinction (and more importantly
control the rate of extinction) of the process. Let
$\mathscr{L}_{d}=\mathbb{Z}^{d}\times\mathbb{N}$, which we turn into a
directed graph $\vec{\mathscr{L}}_{d}$ by placing an edge

$(m,n)\to(m^{\prime},n^{\prime})\quad\hbox{if and only
if}\quad|m_{1}-m_{1}^{\prime}|+\cdots+|m_{d}-m_{d}^{\prime}|+|n-n^{\prime}|=1\
\hbox{and}\ n\leq n^{\prime}.$

In other words, starting from each site $(m,n)$, there are $2d$ “horizontal”
arrows that we can think of as potential invasions in space, and one
“vertical” arrow that we can think of as a potential persistence in time. To
rescale the interacting particle system in space and time, we let $L$ be a
large integer, and define the space-time blocks

$\begin{array}[]{rcl}A_{m,n}&=&(2mL,nL)+[-2L,2L]^{d}\times[0,2L],\vspace*{4pt}\\\
B_{m,n}&=&(2mL,nL)+[-L,L]^{d}\times[L,2L],\end{array}$

for all $(m,n)\in\mathscr{L}_{d}$. See Figure 4 for a picture. We call $(m,n)$
a good site whenever

$E_{m,n}=\\{\hbox{the space-time block\leavevmode\nobreak\ $B_{m,n}$ is
empty}\\}\quad\hbox{occurs}.$

We now prove that, when $a=-\infty$ and regardless of the configuration
outside the space-time box $A_{m,n}$, the event $E_{m,n}$ occurs with
probability close to one when $L$ is large.

$(2m-2)L$$(2m+2)L$$A_{m,n}$$(2m-1)L$$(2m+1)L$$B_{m,n}$$(n+2)L$$(n+1)L$$nL$in
this regionno playersbottom $\Lambda_{-}$bottom $\Lambda_{-}$

periphery $\Lambda_{+}$

periphery $\Lambda_{+}$

periphery $\Lambda_{+}$

periphery $\Lambda_{+}$

Figure 4:  Illustration of the event $E_{m,n}$ when $d=1$. With high
probability, the invasion paths starting from the bottom or the periphery of
the big space-time box $A_{m,n}$ do not reach the smaller space-time box
$B_{m,n}$.

######  Proposition 11.

Let $\epsilon>0$. Then, $\mathbb{P}_{\lambda,-\infty}[E_{m,n}]\geq
1-\epsilon/2$ for sufficiently large $L\in\mathbb{N}$.

###### Proof.

In view of the stochastic domination in Lemma 10, it suffices to prove the
result for the process that evolves according to $\xi_{t}$ outside the space-
time block $A_{m,n}$ but according to $\Xi_{t}$ inside the block. The basic
idea is to use Lemma 8 to prove that the invasion paths starting from the
bottom of the block cannot live too long, and Lemma 9 to prove that the
invasion paths starting from the periphery cannot go too far. More precisely,
let

$\begin{array}[]{rcl}\Lambda_{-}&=&\\{(x,t)\in A_{m,n}:t=nL\\}=\hbox{bottom
of\leavevmode\nobreak\ $A_{m,n}$},\vspace*{4pt}\\\ \Lambda_{+}&=&\\{(x,t)\in
A_{m,n}:|\\!|x-2mL|\\!|_{\infty}=2L\\}=\hbox{periphery of\leavevmode\nobreak\
$A_{m,n}$},\end{array}$

and write $\Lambda_{\pm}\to B_{m,n}$ to indicate an invasion path going from
$\Lambda_{\pm}$ to the block $B_{m,n}$. Because players cannot appear
spontaneously, we have

$\mathbb{P}_{\lambda,-\infty}[E_{m,n}^{c}]\leq\mathbb{P}_{\lambda,-\infty}[\Lambda_{-}\to
B_{m,n}]+\mathbb{P}_{\lambda,-\infty}[\Lambda_{+}\to B_{m,n}].$ (15)

Now, by the domination in Lemma 10 and the exponential decay in Lemma 8,

$\mathbb{P}_{\lambda,-\infty}[\Lambda_{-}\to
B_{m,n}]\leq|\Lambda_{-}|\times\mathbb{P}_{\lambda,-\infty}[\xi_{L}^{0}\neq\varnothing]\leq(4L+1)^{d}\,e^{-\beta
L}.$ (16)

Dealing with the invasion paths coming in the block $A_{m,n}$ through its
periphery $\Lambda_{+}$ is more complicated because the number of such paths
is random. Note that the number paths is dictated by the process $\xi_{t}$
whereas their length is upperbounded by their counterpart for $\Xi_{t}$. To
deal with the number of invasion paths coming in through the periphery, let

$\begin{array}[]{l}H_{m,n}=\\{\hbox{there are less than\leavevmode\nobreak\
$2e\,4dL(4L+1)^{d-1}\lambda$}\vspace*{2pt}\\\ \hskip 80.0pt\hbox{players
outside\leavevmode\nobreak\ $A_{m,n}$ giving birth onto\leavevmode\nobreak\
$\Lambda_{+}$}\\}.\end{array}$

Because the births in the process $\xi_{t}$ occur at rate at most $\lambda$,
the number of births onto $\Lambda_{+}$ is dominated by the Poisson random
variable with parameter

$\lambda|\Lambda_{+}|\leq 2d(4L+1)^{d-1}\times 2L\lambda.$

Using the standard tail bound for the Poisson distribution, we get

$\mathbb{P}_{\lambda,-\infty}[H_{m,n}^{c}]\leq
2\exp(-4Ld(4L+1)^{d-1}\lambda-2e\,4Ld(4L+1)^{d-1}\lambda\log(2)).$

This, together with Lemmas 9 and 10 and a union bound, implies that

$\begin{array}[]{rcl}\mathbb{P}_{\lambda,-\infty}[\Lambda_{-}\to
B_{m,n}]&\leq&\mathbb{P}_{\lambda,-\infty}[\Lambda_{-}\to
B_{m,n}\,|\,H_{m,n}]+\mathbb{P}_{\lambda,-\infty}[H_{m,n}^{c}]\vspace*{4pt}\\\
&\leq&2e\,4dL(4L+1)^{d-1}\lambda\,\mathbb{P}_{\lambda,-\infty}^{0}[|\xi^{0}|>L]+\mathbb{P}_{\lambda,-\infty}[H_{m,n}^{c}]\vspace*{4pt}\\\
&\leq&2e\,4dL(4L+1)^{d-1}\lambda\,e^{-\delta L}\vspace*{4pt}\\\ &&+\
2\exp(-4dL(4L+1)^{d-1}\lambda-2e\,4dL(4L+1)^{d-1}\lambda\log(2)).\end{array}$
(17)

Finally, plugging (16) and (17) into (15), and noticing that both can be made
arbitrarily small by choosing $L$ sufficiently large yield the claim. ∎

Perturbation argument. It follows from the block construction that the set of
good sites = empty blocks $B_{m,n}$ dominates the set of wet sites $(m,n)$ in
the percolation process. Because $\epsilon$ can be chosen arbitrarily small,
in which case the set of dry (not wet) sites does not percolate, and players
cannot appear spontaneously, this shows extinction of the process. To complete
the proof of Theorem 3, the last step is to transport this result from the
limiting case $a=-\infty$ to the case where the payoff coefficient $a$ is
small but finite. In the latter case, an individual with a neighbor can give
birth, but it is unlikely, so (the graphical representations of) the two
processes in a space-time block should agree with high probability, as long as
one chooses $a$ sufficiently small, depending on the size of the block.
Instead of working with the graphical representation in Lemma 10, which was
designed to compare our process with the system of non-interacting copies, we
construct the processes with $a>-\infty$ and $a=-\infty$ using the following
graphical representation:

  * •

Births. For each $i=0,1,\ldots,2d$, equip each $\vec{xy}$, $x\sim y$, with a
rate $\lambda e^{ia/2d}/2d$ exponential clock. At the times $t$ the clock
rings, draw an arrow $(x,t)\overset{i}{\longrightarrow}(y,t)$.

  * •

Deaths. Equip each $x$ with a rate one exponential clock. At the times $t$ the
clock rings, put a cross $\times$ at the space-time point $(x,t)$.

The crosses have the same effects on both processes: a cross $\times$ at site
$x$ kills a particle/player at that site. The process with $a>-\infty$ is
constructed by assuming that, if the tail $x$ of a type $i$ arrow is occupied,
the head $y$ is empty, and exactly $i$ of the neighbors of $x$ are occupied,
then the head $y$ of the arrow becomes occupied. The process with $a=-\infty$
is constructed similarly but using only the type 0 arrows, since the other
arrows occur at rate zero in the limit. Using this graphical representation,
we can now extend Proposition 11 to the process with $a>-\infty$ small.

######  Lemma 12.

Let $\epsilon>0$. Then, $\mathbb{P}_{\lambda,a}[E_{m,n}]\geq 1-\epsilon$ for
all $L\in\mathbb{N}$ large and $a>-\infty$ small.

###### Proof.

In view of Proposition 11, it suffices to show that, with probability
arbitrarily close to one, the process with $a=-\infty$ and the process with
$a>-\infty$ small agree in the block $A_{m,n}$, i.e., there are no type $i$
arrows for $i\neq 0$ in the block, which prevents players with at least one
neighbor to give birth. The overall rate of all the type $i$ arrows for $i\neq
0$ starting at a given site is

$\lambda e^{a/2d}+\lambda e^{2a/2d}+\lambda e^{3a/2d}+\cdots+\lambda
e^{2da/2d}\leq 2\lambda e^{a/2d}$

for all $a>-\infty$ small. This implies that the number of such arrows that
point at the space-time block $A_{m,n}$ is dominated by the Poisson
distribution $X$ with parameter

$|A_{m,n}|\times 2\lambda e^{a/2d}=2L(4L+1)^{d}\times 2\lambda e^{a/2d}.$

Now, the scale parameter $L\in\mathbb{N}$ being fixed as in Proposition 11, we
define

$a_{-}=2d\ln\bigg{(}\frac{-\ln(1-\epsilon/2)}{4L(4L+1)^{d}\lambda}\bigg{)}>-\infty.$

Finally, since the event $E_{m,n}$ occurs for the process with $a<a_{-}$
whenever it occurs for the process with $a=-\infty$ and the two processes
agree in the block, it follows from Proposition 11 that

$\begin{array}[]{rcl}\mathbb{P}_{\lambda,a}[E_{m,n}]&\geq&\mathbb{P}_{\lambda,-\infty}[E_{m,n}]\times\mathbb{P}[X=0]\vspace*{4pt}\\\
&\geq&(1-\epsilon/2)\times\exp(-2L(4L+1)^{d}\times 2\lambda
e^{a_{-}/2d})=(1-\epsilon/2)^{2}\geq 1-\epsilon\end{array}$

for all $a\leq a_{-}$. This completes the proof. ∎

Using Lemma 12 and an idea of [1], we can now conclude the proof of the
theorem.

###### Proof of Theorem 3.

It follows from Lemma 12 that there is a collection of good events $G_{m,n}$
that only depend on the graphical representation in the slightly enlarged
space-time blocks

$A^{+}_{m,n}=[-2L-1,2L+1]^{d}\times[0,2L],\quad(m,n)\in\mathscr{L}_{d},$

such that, for every $\epsilon>0$, we can choose the scale parameter
$L\in\mathbb{N}$ large, then the payoff coefficient $a>-\infty$ small, to
guarantee

$\mathbb{P}_{\lambda,a}[G_{m,n}]\geq 1-\epsilon\quad\hbox{and}\quad
G_{m,n}\subset E_{m,n}.$

This implies that the set of good sites dominates stochastically the set of
wet sites in the oriented site percolation process on $\vec{\mathscr{L}}_{d}$
with parameter $1-\epsilon$. If $\epsilon>0$ is small enough, not only the set
of wet sites percolates, but also the probability of a path of closed sites
with length at least $n$ starting from $(0,0)$ decays exponentially with $n$
(see [1, Section 8] for a proof). Because the players cannot appear
spontaneously, the presence of a player in a space-time block $B_{m,n}$
implies the existence of a path of closed sites to $(m,n)$, which shows
extinction of our process. ∎

Acknowledgments. Jonas Köppl acknowledges the financial support of the Leibniz
Association within the Leibniz Junior Research Group on Probabilistic Methods
for Dynamic Communication Networks as part of the Leibniz Competition and the
German Research Foundation under Germany’s Excellence Strategy MATH+: The
Berlin Mathematics Research Center. Nicolas Lanchier was partially supported
by NSF grant CNS-2000792.

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