Coexistence for a multitype contact process with seasons
aa r X i v : . [ m a t h . P R ] O c t The Annals of Applied Probability (cid:13)
Institute of Mathematical Statistics, 2009
COEXISTENCE FOR A MULTITYPE CONTACTPROCESS WITH SEASONS
By B. Chan, R. Durrett and N. Lanchier
Cornell University, Cornell University and Arizona State University
We introduce a multitype contact process with temporal hetero-geneity involving two species competing for space on the d -dimensionalinteger lattice. Time is divided into seasons called alternately season1 and season 2. We prove that there is an open set of the parame-ters for which both species can coexist when their dispersal range islarge enough. Numerical simulations also suggest that three speciescan coexist in the presence of two seasons. This contrasts with thelong-term behavior of the time-homogeneous multitype contact pro-cess for which the species with the higher birth rate outcompetes theother species when the death rates are equal.
1. Introduction.
Consider an ecosystem inhabited by multiple speciescompeting for a single resource represented by a function R . Let u i , i =1 , , . . . , K , denote the population density of species i . In the original modelsof Lotka (1925) and Volterra (1928), the per capita growth rate for eachspecies increases linearly with the amount of resource available:1 u i du i dt = β i R ( u , u , . . . , u K ) − δ i , (1)where β i and δ i , respectively, denote the birth and death rates of species i . When R is a decreasing function of population densities, Volterra (1928)showed that only the species with the highest β i to δ i ratio survives as t → ∞ .When R = 1 − ( u + u + · · · + u K ) denotes the density of space available,the Lotka–Volterra model (1) can be seen as the mean-field approximation,or nonspatial deterministic analogue, of the multitype contact process in-troduced by Neuhauser (1992). Assuming for simplicity that the number ofspecies is K = 2, her model is a continuous-time Markov process whose state Received August 2008; revised February 2009.
AMS 2000 subject classification.
Key words and phrases.
Coexistence, competition model, time-heterogeneous multi-type contact process.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Applied Probability ,2009, Vol. 19, No. 5, 1921–1943. This reprint differs from the original inpagination and typographic detail. 1
B. CHAN, R. DURRETT AND N. LANCHIER at time t is a function η t that maps the d -dimensional integer lattice Z d into { , , } , with 0 denoting empty sites, and 1 and 2 denoting sites occupiedby an individual of species 1 and 2, respectively. The evolution at site x ∈ Z d is described by the following transition rates:0 → β X y ∈ Z d p ( y, x ) { η ( y )=1 } , → δ , → β X y ∈ Z d p ( y, x ) { η ( y )=2 } , → δ , where p ( y, x ) is a translation invariant transition probability on Z d . This in-dicates that species of type i , i = 1 ,
2, produces offspring at rate β i and diesat rate δ i . Moreover, an offspring produced at site y ∈ Z d is sent to a site x chosen at random with probability p ( y, x ). If site x is vacant, it becomes oc-cupied by the offspring, otherwise the birth is suppressed. Neuhauser (1992)proved for the multitype contact process that if δ = δ then the specieswith the larger birth rate outcompetes the other species. She also conjec-tured that, as predicted by the model (1), the species with the larger β i to δ i ratio takes over. While it is believed that there is no coexistence in theNeuhauser’s multitype contact process, there are a number of interactingparticle systems for which coexistence has been proved rigorously.One mechanism that allows coexistence is when intraspecific competitionexceeds interspecific competition. This has been proved by Neuhauser andPacala (1999) for a voter type model in which each site of the lattice isoccupied by an individual of type 1 or 2. When both species have the samebirth rate, the state at site x ∈ Z d evolves according to the transition rates1 → f ( f + α f ) , → f ( f + α f ) , where f i = f i ( x ) denotes the fraction of neighbors of site x in state i . The pa-rameters α ij model the pressure of species i on species j . Except for the1-dimensional nearest neighbor case, in the absence of interspecific competi-tion, that is, α = α = 0, there is a stationary distribution in which sitesare independent [see Theorem 1 in Neuhauser and Pacala (1999)], indicatingthat both species coexist. While increasing α and α , spatial correlationsbuild up, but coexistence is maintained as long as the interspecific com-petition is sufficiently small. The inclusion of a spatial structure, however,translates into a reduction of the set of parameters for which coexistenceoccurs.A second mechanism that promotes coexistence is environmental hetero-geneity. Suppose that the regular lattice Z is inhabited by two types ofplants. The environment is static and described by a partition { H , H } ofthe space with H i denoting the set of sites occupied by plant i . Consider two OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS specialist symbionts; for i = 1 ,
2, symbiont of type i feeds on plant i . Thiscan be modeled by a multitype contact process in which births of symbiontsof type i are suppressed whenever the offspring is sent to a site x ∈ H j with i = j . Since the habitats H and H are disjoint and so symbionts 1 and 2 donot interact, the problem of coexistence reduces to the following percolationproblem: If for i = 1 ,
2, there exists an infinite sequence of sites in H i alongwhich symbiont i can spread then coexistence is possible. This depends onboth the spatial arrangement of H i and the dispersal range of symbiont i .Lanchier and Neuhauser (2006) proved that coexistence occurs in a non-trivial case in which the habitats of both symbionts overlap. Their modelis referred to as the static-host model. Symbiont 1 is a specialist feeding onplant 1 while symbiont 2 is a generalist feeding on both plants. The universeis tiled with N × N squares. Whether a square is occupied by plants of type1 or 2 is determined by flipping independent fair coins. In this situation,coexistence occurs when the birth rate of the specialist is larger than thatof the generalist and the square size N is much larger than the dispersalrange of the symbionts. In the mean-field approximation of the static-hostmodel, the birth rate of the specialist needs to exceed twice the birth rateof the generalist so that coexistence occurs. Lanchier and Neuhauser (2006)also introduced a stochastic process in which spatial heterogeneities are nolonger static but generated by the symbionts. This model is referred to asthe dynamic-host model. In the absence of symbionts, plants of type 1 and2 compete according to a biased voter model with a selective advantage forthe 1’s. In particular, plants of type 1 outcompete plants of type 2. Intro-ducing a specialist pathogen feeding on plants of type 1 whose harmful effectis modeled by an increase of the death rate of infected plants, Durrett andLanchier (2008) proved that coexistence may occur provided the dispersalrange of the species is sufficiently large, but conjectured that coexistenceis not possible if the symbiont is a specialist mutualist feeding on plants oftype 2.In some cases, the presence of two types of space is not immediately obvi-ous. Chan and Durrett (2006) considered a generalization of the multitypecontact process in which species 1 is a good competitor with a short rangedispersal kernel, whereas species 2 has a long range dispersal kernel whichmakes it a good colonizer. In this situation, occurrences of forest fires (i.e.,removal of all the individuals contained in a large square) at an appropriaterate allows species 2 to survive by migrating to newly created patches. Inthis model, there are two types of space: blocks freshly cleared by a firewhich are occupied by individuals of species 2 and older blocks which areoccupied by individuals of species 1.The aim of this article is to prove that temporal heterogeneity can alsopromote coexistence in a stochastic spatial model. Armstrong and McGehee(1976) have shown in an ODE model that coexistence can occur for species B. CHAN, R. DURRETT AND N. LANCHIER living in a periodic environment with seasonal changes in birth rates: wheneach species has a specific growing season disjoint from all others, and allresources are conservative (meaning it is a linear function of populationdensities), a system with four resources can support more than four specieswithout any one going into extinction. Our objective is to extend their ob-servation to competition models based on interacting particle systems.Recall that when two contact processes with equal death rates δ andequal dispersal kernels compete with one another on Z , one of the twospecies outcompetes the other one. To search for coexistence, we follow thework of Armstrong and McGehee and consider an environment with seasonalchanges in birth rates: time will be divided into seasons of length D , calledalternately season 1 and season 2. For simplicity and because this is the caseof interest in spatial ecology, we assume that d = 2. However, all our resultsextend easily to any dimension d ≥
1. Since we will deal with long rangeinteractions, it is convenient to assume that particles evolve on the rescaledlattice Z /L where L is a large integer which is referred to as the rangeof the interactions. Our model is a continuous-time Markov process whosestate at time t is a function ξ t that maps the rescaled lattice into { , , } .To formulate the dynamics, we let N ( x ) = { y ∈ Z /L : 0 < k x − y k ∞ ≤ } be the interaction neighborhood of site x ∈ Z /L , and f i ( x, ξ ) = |{ y ∈ N ( x ) : ξ ( y ) = i }||N ( x ) | be the fraction of neighbors of site x in state i . The state of site x ∈ Z /L flips according to the following transition rates:transition season 1 season 20 → β f ( x, ξ ) β f ( x, ξ )0 → β f ( x, ξ ) β f ( x, ξ )1 → δ δ → δ δ This indicates that each species behaves individually like a contact process.Their associated birth rates, however, depend on the season, making ourprocess a generalization of the Neuhauser’s competing model in which thebirth rates are functions of time. We call the process ξ t the periodic compe-tition model . We assume for simplicity that each of the species has the samedeath rate in both seasons. However, the reduction to equal death rates isnot important for the proofs. In particular, Theorem 1 and Corollary 3 be-low can be extended to the case when the death rates may vary provided OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS one replaces the death rate by the average of the death rates in season 1 andseason 2, respectively. We now give a sufficient condition for coexistence.To begin with, we observe that when the dispersal range L tends to infin-ity, the population dynamics (in a finite volume) converges to the followingmean-field model: du dt = β ( t ) u (1 − u − u ) − δ u , (2) du dt = β ( t ) u (1 − u − u ) − δ u , where u i , i = 1 ,
2, denotes the population density of species i and β i ( t ) areperiodic step functions with period 2 D given by β i ( t ) = (cid:26) β i , when 0 ≤ t < D , β i , when D ≤ t < D .In the absence of species 1, species 2 evolves according to the pair of logisticequations du dt = (cid:26) β u (1 − u ) − δ u , if t ∈ [2 nD, (2 n + 1) D ), β u (1 − u ) − δ u , if t ∈ [(2 n + 1) D, (2 n + 2) D ).(3)Denoting by u ( t ) the solution, the function t u (2 nD + t ) for 0 ≤ t ≤ D converges uniformly to a piecewise-smooth periodic curve ¯ u . This followsfrom the monotonicity of the solution with respect to the initial condition.Moreover, since ¯ u is periodic and continuous, standard topological argu-ments imply that the convergence is uniform on the real line. We call ¯ u the equilibrium curve for the 2’s . One can define ¯ u for the 1’s in a similar way.The top panel of Figure 1 gives an illustration of the density of 2’s in theabsence of 1’s for the interacting particle system which, in view of the sizeof the universe and the range of the interactions, is a good approximationof ¯ u .We now return to the spatial model. When the birth rates β ij are chosenappropriately, one can compare the 1’s (resp., the 2’s) with a supercriticalbranching process and show that the 1’s and 2’s coexist. Based on Armstrongand McGehee (1976), we expect that any number of species can coexistprovided there are as many seasons as there are species, but for simplicitywe restrict ourselves to a system with only two species. Theorem 1.
Suppose that D Z D β ( t )(1 − ¯ u ( t )) dt > δ and D Z D β ( t )(1 − ¯ u ( t )) dt > δ . Then, when L is large, coexistence occurs. B. CHAN, R. DURRETT AND N. LANCHIER
Fig. 1.
Evolution of the densities of species for the interacting particle system along 4periods = × lattice with periodic boundary condi-tions. The length of each season is D = 10 , the range of the interactions L = 200 and thedeath rates δ i = 1 . Our pictures show species 2 alone, species 1 and 2, and species 1, 2 and3 in competition, respectively. In the previous theorem and after, coexistence for the interacting particlesystem means that starting from any initial configuration with infinitelymany 1’s and infinitely many 2’s,lim inf t →∞ P ( ξ t ( x ) = i ) > i = 1 , x ∈ Z /L .Note that due to temporal heterogeneities there is no stationary distribution,but there is an initial distribution so that the states at times 0 and 2 D havethe same distribution. It seems reasonable to call this a periodic distribution.Since R (1 − ¯ u ) / D dt is the average number of empty sites available for
OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS invasion, the first integral in Theorem 1 represents the growth rate of 1’swhen rate averaged over time. Therefore, if the average growth rate is greaterthan the death rate then the 1’s can invade a community of 2’s. Similarly,the second inequality says that if the average growth rate for species 2 isgreater than the death rate then the 2’s can invade a community of 1’s.As previously mentioned, it has already been proved by Armstrong andMcGehee (1976) that temporal heterogeneity promotes coexistence. Theirresult, however, was based on a deterministic nonspatial model. It is im-portant to extend their result to a model including both temporal hetero-geneity and spatial structure in the form of local interactions because thespatial component is identified in ecology as an important factor in howcommunities are shaped. Even though the sufficient conditions we found forcoexistence of two species in the presence of two seasons turn out to bethe same for both the interacting particle system and its nonspatial ana-log, our main result is interesting for ecologists since it is known from pastresearch that spatial models can result in predictions that differ from non-spatial models. For instance, it has been proved theoretically in Neuhauserand Pacala (1999) and Lanchier and Neuhauser (2009) that the inclusion oflocal interactions may translate into a reduction of the coexistence region.Their models were based on particle systems with short range interactions.Interestingly, Neuhauser (1994) also proved that a sexual reproduction pro-cess with a long range dispersal kernel does not exhibit the same featuresas the corresponding mean-field model, namely, for intermediate values ofthe birth rate survival occurs for the mean-field model while the interactingparticle system exhibits a metastable behavior and eventually goes extinct.In other words, even in the presence of long range interactions, spatial andnonspatial models may have significantly different behaviors.The second panel of Figure 1 shows the evolution of the densities of species1 and 2 in competition, which illustrates Theorem 1. The bottom panel sug-gests that temporal heterogeneity can also promote coexistence of threespecies even in an environment with only two seasons. Interestingly, numer-ical simulations also suggest that, with the birth rates indicated in Figure1, the three species coexist whereas in the absence of species 2, species 3outcompetes species 1. This indicates that the inclusion of an additionalspecies may promote coexistence. In conclusion, we state the following. Conjecture 2.
In the presence of two seasons, three species can coex-ist.
The conjecture is probably very difficult to prove. In principle it canbe approached with invadability ideas as in Durrett (2002). Suppose forsimplicity that we have found three species that can coexist in pairs. To showthat all three can coexist, we need to (i) show that the two species systems
B. CHAN, R. DURRETT AND N. LANCHIER converge to equilibrium, and that (ii) in all three cases the third speciescan invade the other two in their equilibrium. Problem (i) seems difficult.Indeed, we do not see how to prove that the mean field ODE converges toequilibrium. Due to the time inhomogeneity, one does not have Lyapunovfunctions.We now derive from the two integrals in Theorem 1 a (stronger but)more explicit sufficient condition for the coexistence of both species in thespatially explicit model. This condition is given in Corollary 3 below. Firstof all, we set p j = lim n →∞ u (2 nD + ( j − D ) while holding u ≡ j = 1 ,
2. That is, p j is the limiting density of the 2’s at the beginningof season j in the absence of 1’s. Since the 2’s grow according to the pair oflogistic equations (3), the densities p and p are related by ρ ( p , D ; β , δ ) = p and ρ ( p , D ; β , δ ) = p , where the function ρ is defined by ρ ( u , t ; β, δ ) = u exp( rt )1 − K − u (1 − exp( rt )) , where r = β − δ and K = 1 − δ/β are the intrinsic rate of growth and the carrying capacity, respectively. Byletting j = ( n mod 2) + 1, one can define ¯ u recursively, by¯ u (0) = p and ¯ u ( nD + θ ) = ρ (¯ u ( nD ) , θ ; β j , δ )for 0 ≤ θ < D . One can define p , p and ¯ u for the 1’s in a similar way.Now, assuming that season 2 is the growing season for species 2, that is, β < β , the limiting curve ¯ u is convex and decreasing in season 1 andconcave and increasing in season 2. (See the top panel of Figure 1 for anapproximation of ¯ u .) In particular,1 D Z D − ¯ u ( t ) dt ≥ − p + p D Z DD − ¯ u ( t ) dt ≥ − p . This indicates that the densities of empty sites available for the 1’s in seasons1 and 2 are bounded from below by 1 − ( p + p ) / − p , respectively.Therefore, the 1’s dominate a contact process with birth rate β (cid:18) − p + p (cid:19) and β (1 − p ) OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS in seasons 1 and 2, respectively. Similarly, assuming that β > β , the 2’sdominate a contact process with birth rate β (1 − p ) and β (cid:18) − p + p (cid:19) in seasons 1 and 2, respectively. The symmetry is flipped because the 2’shave a different growing season. By applying Theorem 1, we finally obtain Corollary 3.
Suppose that β > β and β > β , and that (cid:20) β (cid:18) − p + p (cid:19) + β (1 − p ) (cid:21) > δ , (cid:20) β (1 − p ) + β (cid:18) − p + p (cid:19)(cid:21) > δ , where p ij is the density of species i in the absence of the other species at thebeginning of season j in the mean-field equilibrium. Then, when L is large,coexistence occurs. In the previous result, each species does better than the other in oneseason, but this is not necessary for coexistence. For a concrete example,suppose that D = 1 , β = 5 . , β = 1 . δ = δ = 2 . . Solving the equations numerically shows that the integral of the density¯ u ( t ) over the first season is approximately 0 . β = 10000 , β = 0 , δ = 6000 and δ = 100 , then we have β /δ = 5 / < / β /δ but12 D Z D β ( t )(1 − ¯ u ( t )) dt = β Z (1 − ¯ u ( t )) dt > > δ + δ , so the 1’s can invade the 2’s. In other respects,¯ u ( t ) ≤ − δ /β = 0 . t ≤ , ¯ u ( t ) ≤ . − t − ≤ t ≤ , which implies that12 D Z D β ( t )(1 − ¯ u ( t )) dt = β Z (1 − ¯ u ( t )) dt + β Z (1 − ¯ u ( t )) dt ≥ β + β (cid:18) − Z exp( − t ) dt (cid:19) ≈ β + 4981000 β = 2 . > δ + δ , B. CHAN, R. DURRETT AND N. LANCHIER so the 2’s can invade the 1’s as well. The fact that coexistence may occureven though species 2 is a better competitor than species 1 in both seasons isdue to the combination of the presence of seasons (temporal heterogeneity)and the fact that species 1 evolves at a much faster rate than species 2. Inour example, the density of 1’s immediately goes to the carrying capacity inseason 1 and then drop quickly to 0 in season 2. Individuals of species 2, onthe contrary, evolve slowly so they never really get to full strength: Theireffective average density is less than in the homogeneous case, which allowsthe 1’s to invade the 2’s, even thought the 2’s are superior competitors.
2. Mean-field model.
In this section, we prove a coexistence result forthe mean-field model which is analogous to Theorem 1. A common way forshowing coexistence of two species models is to prove that both species aremutually invadable, that is, species 1 can invade a community of 2’s when the1’s are small in number, and vice versa. This implies that their densities arebounded away from the axes on the u – u plane. In Proposition 2.1, below,coexistence for the mean-field model means that starting with a positivedensity of type 1 and type 2,lim inf t →∞ u i ( t ) > i = 1 , . Note that, similarly to the interacting particle system, due to temporal het-erogeneities, the mean-field model (2) has no fixed point but instead con-verges to a limit cycle.
Proposition 2.1 (Coexistence).
Suppose that D Z D β ( t )(1 − ¯ u ( t )) dt > δ and D Z D β ( t )(1 − ¯ u ( t )) dt > δ . Then there is coexistence in the mean-field model (2).
Proof.
By symmetry, we only need to prove survival of the 1’s. Theidea is to show the existence of a small ε >
0, such that if u ( t ) ∈ (0 , ε ) forall t ∈ [0 , D ] and Z D β ( t )(1 − ¯ u ( t )) − δ dt > , then u (2 D ) > u (0). Assuming that u ( t ) ∈ (0 , ε ), we obtain du dt > β ( t ) u (1 − ε − u ) − δ u . Dividing both sides by u and integrating from 0 to 2 D giveslog u (2 D ) − log u (0) > (1 − ε )( β + β ) D − Z D β ( t ) u ( t ) dt − δ D. OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS Therefore, if u satisfies the condition( β + β ) D − Z D β ( t ) u ( t ) dt > δ D (4)one can choose ε > u (2 D ) > u (0). If u (0) ≤ p then u ( t ) ≤ ¯ u ( t ) at all times, which implies that( β + β ) D − Z D β ( t ) u ( t ) dt ≥ Z D β ( t )(1 − ¯ u ( t )) dt > δ D, so (4) is satisfied. This proves the result when u (0) ≤ p . If on the contrary u (0) > p , then, by monotonicity of the solution with respect to the initialcondition, survival of type 2 holds as well. This completes the proof. (cid:3) Note that taking ¯ u ( t ) = 0 in Proposition 2.1 gives the following sufficientcondition for the survival of species 2 in the absence of species 1. Corollary 4 (Survival of a single species).
In the absence of 1’s, lim inf t →∞ u ( t ) > whenever β + β > δ .
3. Preliminaries on periodic contact processes.
From now on, we calla periodic contact process a contact process on Z /L in which particlesgive birth at rate α ( t ) and die at rate δ , where α ( t ) is positive, piecewisecontinuous and periodic. A particle produced at site x ∈ Z /L is sent toa site chosen uniformly from N ( x ). The birth occurs if the target site isempty, and is suppressed otherwise. This process is denoted by A t ( α, δ ) andthought of as a set valued process, that is, A t ( α, δ ) is the subset of Z /L corresponding to the set of sites which are occupied at time t . The dualprocess of the periodic contact process starting at site x at time t is denotedby ˆ A x,ts and represents the descendants of ( x, t ) at time t − s . Note thatcontrary to the basic contact process, the periodic contact process is notself-dual since particles in the dual process ˆ A x,ts give birth at rate α ( t − s ),which a priori is different from α ( s ).Similarly, we call a periodic branching random walk on R a branchingrandom walk in which particles give birth at rate α ( t ) and die at rate δ .A particle produced at x ∈ R is sent to a site chosen uniformly at randomfrom x + [ − , . This process is denoted by Z t ( α, δ ). For both the periodiccontact process and the periodic branching random walk, we use a super-script to indicate the initial configuration. We first prove some convergenceproperties of A t ( α, δ ). Lemma 3.1 (Convergence to branching random walk).
Let A = { } and T > . Then, as the range L → ∞ , the periodic contact process A At ( α, δ ) B. CHAN, R. DURRETT AND N. LANCHIER converges weakly to Z At ( α, δ ) in the Skorohod space of c`adl`ag functions from [0 , T ] to the subsets of R . Proof.
This is a classical result when the birth rate α ( t ) is a constant[see, e.g., Durrett (1991) for a proof]. Let b = max α ( t ). Then, for any t ≤ T ,the number of births by time t in the contact process is dominated by thenumber of particles in Z At ( b,
0) from which it follows that E | A At ( α, δ ) | ≤ E | Z At ( b, | = e bt ≤ e bT . Then, by Markov’s inequality, we have P ( | A At ( α, δ ) | > L / ) ≤ e bT L / → L → ∞ . Now, conditional on E = {| A At ( α, δ ) | ≤ L / for all t ≤ T } , at each birth,the probability that the offspring is sent to a site already occupied is lessthan L / · L / (2 L + 1) → L → ∞ since the size of the interaction neighborhood is bounded by (2 L + 1) . Wedenote by E the event that, up to time T , all the offspring of the periodiccontact process are sent to empty sites. Then, on the event E ∩ E , one cancreate a coupling in which all the particles in A At ( α, δ ) are within distance L / /L in the uniform norm of their counterparts in the branching randomwalk. For both processes, we call particle 0 the particle at site 0 at time 0,and particle i , i ≥
1, the particle produced at the i th birth event. For x ∈ R ,let π L ( x ) be the (unique) vector such that k x − π L ( x ) k ∞ < / L and π L ( x ) ∈ Z /L. The processes are coupled as follows. If particle i in the periodic branchingrandom walk dies at time t i , then particle i in the periodic contact processdies at time t i as well. Now, assume that particle i in the periodic branchingrandom walk is located at x and that particle i in the periodic contactprocess is located at x L . If particle i in the periodic branching random walkproduces an offspring which is sent to y at time t i , then particle i in theperiodic contact process produces an offspring sent to x L + π L ( y − x ) at time t i . Since there are at most L / birth events by time T , adding the errors wefind that, on the event E ∩ E , the distance between particles in the contactprocess and their counterparts in the branching random walk is bounded by L / sup x π L ( x ) = L / / L → L → ∞ . OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS Since P ( E ∩ E ) → L → ∞ , we conclude that the periodic contactprocess A At ( α, δ ) converges weakly to the periodic branching random walk Z At ( α, δ ) as L → ∞ . (cid:3) We now prove that the number of particles in the periodic contact process A t ( α, δ ) is almost deterministic when L is large: If we focus our attention toa space–time box of finite size, one can show that the number of particlesdoes not deviate much from its expected value. Lemma 3.2 (Convergence to expected value).
Assume that α ( t ) is piece-wise constant and periodic. For A ⊂ R , let A At denote the periodic contactprocess starting from A ∩ ( Z /L ) . For any subset A , any site x on the rescaledlattice Z /L and any time t ≥ , let N ( x, t ) = | A At ∩ { x + [0 , }| . Fix
T > and S > . Then for all ε > , P ( | N ( x, t ) − EN ( x, t ) | > L ε for some x ∈ [ −√ T , √ T ] and some t ≤ S ) → as L → ∞ . Proof.
We follow the proof of Lemma 3.5 in Durrett and Lanchier(2008), in which the result is proved when the birth rate α ( t ) is a constant.Let N ε ( x, t ) = | A At ∩ { x + [0 , ε ] }| and y, z ∈ x + [0 , ε ] ∩ ( Z /L ) . First, we observe that the number of descendants in the dual processes ˆ A y,tt and ˆ A z,tt depends on t and the birth and death rates, but not on L . Inparticular, since the interaction neighborhood contains (2 L + 1) − ≤ C /L . Now, notingthat A At ( y ) = ( , if ˆ A y,tt ∩ A = ∅ ,1 , if ˆ A y,tt ∩ A = ∅ and A At ( z ) = ( , if ˆ A z,tt ∩ A = ∅ ,1 , if ˆ A z,tt ∩ A = ∅ ,a standard construction shows that the covariance of A At ( y ) and A At ( z ) canbe bounded by the probability that the dual processes hit [see page 21 inGriffeath (1979)]: cov( A At ( y ) , A At ( z )) ≤ C /L . (5)Since N ε ( x, t ) = P y ∈ x +[0 ,ε ] A At ( y ), inequality (5) implies thatVar N ε ( x, t ) ≤ ε L + ( ε L ) C /L ≤ C L ε . B. CHAN, R. DURRETT AND N. LANCHIER
Then, by Chebyshev’s inequality, we obtain P ( | N ε ( x, t ) − EN ε ( x, t ) | > L ε ) ≤ C /ε L → L → ∞ . (6)This holds for all x ∈ [ −√ T , √ T + 1] ∩ ( Z /L ) and all t ≤ S . To deducethe result, we first use a discretization of space and time. The idea is torely on the convergence in (6) to control the densities N ε ( x, t ) at a finitenumber (that does not depend on L ) of space–time points. More precisely,we let τ > N ε ( x, t ) forspace–time points ( x, t ) belonging to a subset of ε Z × τ Z + . As L → ∞ , P ( | N ε ( x, nτ ) − EN ε ( x, nτ ) | > L ε for some 0 ≤ n ≤ m andsome x ∈ [ −√ T , √ T + 1] ∩ ε Z ) ≤ C T ( m + 1) /ε L → , where m = min { n ≥ nτ ≥ S } . Using large deviation estimates for the Bi-nomial distribution and taking τ > x + [0 , ε ] that flip between nτ and ( n + 1) τ is smaller than 2 L ε with probability at least 1 − C exp( − αL ) so that P ( | N ε ( x, t ) − EN ε ( x, t ) | > L ε for some t ≤ S and some x ∈ [ −√ T , √ T + 1] ∩ ε Z ) → L → ∞ . To conclude, we observe that if | N ε ( x, t ) − EN ε ( x, t ) | ≤ L ε for all x ∈ [ −√ T , √ T + 1] ∩ ε Z , then we have that | N ( x, t ) − EN ( x, t ) | ≤ L ε /ε + 2 L ε = 5 L ε for all x ∈ [ −√ T , √ T ] ∩ ( Z /L ) since there are at most ε − squares withlength side ε included in the unit square x + [0 , and at most 2 L ε re-maining sites. (cid:3) Lemma 3.3 (Convergence to ODE).
Let A t = A t ( β , δ ) be the periodiccontact process starting from the “all occupied” configuration. For any site x on the rescaled lattice Z /L , we set u L ( x, t ) = P ( x ∈ A t ( β , δ )) . Let
T > . Then, as the range of the interactions L → ∞ , the function t u L ( x, t ) converges uniformly on [0 , T ] to a solution u ( t ) of the ODE dudt = β ( t ) u (1 − u ) − δu with u (0) = 1 . OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS Proof.
Since the evolution rules are translation invariant, the probabil-ity of { x ∈ A t } does not depend on x . Therefore, we can set u L ( x, t ) = u L ( t ).Let T > S ( t ) be the semigroup that generates the process A t . Forany cylindric function f , that is any function that depend on finitely manycoordinates, the Hille–Yosida theorem states that ddt S ( t ) f = S ( t )Ω f, (7)where Ω is the Markov generator of S ( t ). Now, letting f ( A t ) = { x ∈ A t } , wehave S ( t ) f = Ef ( A t ) = P ( x ∈ A t ) = u L ( t ) . (8)Combining (7) and (8) implies that ddt P ( x ∈ A t ) = X y ∈N ( x ) β ( t ) |N ( x ) | P ( x / ∈ A t and y ∈ A t ) − δP ( x ∈ A t ) . (9)We claim that { x ∈ A t } and { y ∈ A t } are asymptotically independent forany t ≤ T , that is, P ( x / ∈ A t and y ∈ A t ) → P ( x / ∈ A t ) · P ( y ∈ A t ) as L → ∞ . (10)To see this, we first observe that (10) is equivalent to P ( ˆ A x,tt = ∅ and ˆ A y,tt = ∅ ) → P ( ˆ A x,tt = ∅ ) · P ( ˆ A y,tt = ∅ ) as L → ∞ . (11)Since t ≤ T , the number of particles in ˆ A x,ts and ˆ A y,ts , s ≤ t , depends on T andthe birth and death rates but not on L so, with probability arbitrarily closeto 1 when L is sufficiently large, the number of particles is bounded by L / (see the proof of Lemma 3.1). Moreover, on such an event, the probabilitythat the dual processes hit is less than 2 L / / L →
0. In particular, thedual processes are independent in the limit as L → ∞ which establishes (11)and (10). Finally, combining (9) and (10) gives the desired result. (cid:3)
4. Proof of Theorem 1.
Block construction.
The first step to prove Theorem 1 is to constructthe periodic competition model ξ t from a Harris’ graphical representation[Harris (1972)]. For x, y ∈ Z /L , we let q ( x, y ) = (cid:26) |N ( x ) | − , if 0 < k x − y k ∞ ≤ , otherwise.For any ordered pair of sites ( x, y ) and any i, j ∈ { , } , we introduce thefollowing collections of independent Poisson processes:arrival times rates symbols T ijn ( x, y ) β ij q ( x, y ) x i −→ yU in ( x ) δ i × at x B. CHAN, R. DURRETT AND N. LANCHIER that is, we draw an i -arrow from site x to site y at time T ijn ( x, y ) to indicatethat if x is occupied by species i , site y is empty, and ⌈ T ijn ( x, y ) /D ⌉ := min { a ∈ Z : a ≥ T ijn ( x, y ) /D } ≡ j (mod 2)(12)then site y becomes occupied by an individual of species i . Note that thecondition in (12) means that the arrival time T ijn ( x, y ) occurs in season j . Inaddition, we put a × at site x at time U in ( x ) to indicate that an individualof species i at x dies. A result of Harris (1972) implies that the previousconstruction induces a well-defined Markov process. Moreover, the flip ratesindicate that this process is the periodic competition model.Our proof of Theorem 1 relies on the use of a block construction, whichis now a standard technique. The idea is to show that, when viewed onsuitable length and time scales, our process dominates the set of wet sitesof a M -dependent oriented percolation process on L = { ( m, n ) ∈ Z : m + n is even and n ≥ } in which sites are open with probability p close to 1. More precisely, eachsite ( m, n ) ∈ L is associated with a random variable ω ( m, n ) ∈ { , } whichindicates whether the site is open (state 1) or closed (state 0). The M -dependency means that P ( ω ( m i , n i ) = 1 for 1 ≤ i ≤ k ) = p k , whenever k ( m i , n i ) − ( m j , n j ) k > M for i = j . A site ( m, n ) is said to be wetif there exists a sequence of integers m , m , . . . , m n = m such that:1. For i = 0 , , . . . , n −
1, we have k m i +1 − m i k = 1.2. For i = 0 , , . . . , n , site ( m i , i ) is open, that is ω ( m i , i ) = 1.See Durrett (1995) for more details. To compare the periodic competitionmodel viewed on suitable length and time scales with oriented percolation,we let γ be a large parameter to be fixed later, and let N i ( x, t ) denote thenumber of type i particles in x + [0 , at time t . Assuming that N i (0 , > L exp( − γT ) for i = 1 , , (13)we will prove that with probability close to 1 when the parameters T and L are large, N i (( √ T , , T ) > L exp( − γT ) for i = 1 , . (14)In particular, if we call ( m, n ) ∈ L an occupied site whenever N i ( m ( √ T , , nT ) > L exp( − γT ) for i = 1 , , then the set of occupied sites dominates the set of wet sites of an orientedpercolation process in which sites are open with probability p close to 1. OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS The construction of the percolation process relies on a coupling argument inwhich the random variables ω ( m, n ) are defined by induction over the level n . This technique is now standard and we refer the reader to the Appendixof Durrett (1995) for a detailed construction of the process in the generalcase. Since percolation occurs whenever the density p is sufficiently close to1 (i.e., starting with one open site at level 0, there is a positive probabilitythat the cluster of wet sites is infinite), Theorem 1 follows from the compar-ison with oriented percolation. In conclusion, the aim is to prove that theconditional probability of the events in (14) given the events in (13) can bemade arbitrarily close to 1.Since the proofs of the inequalities in (14) for i = 1 and i = 2 are identical,we will only show the first one ( i = 1). There are three steps in showingthe survival of species 1 (see Figure 2 for an illustration of the three stepsdescribed below):(i) The density of type 2 particles in [ −√ T , √ T ] between times √ T and T is (almost) bounded by the mean-field equilibrium ¯ u ( t ).(ii) If the unit square [0 , contains at least L exp( − γT ) type 1 particlesat time 0, then it will contain at least L exp( − γT + cT ) type 1 particlesat time T − √ T for some c > , contains at least L exp( − γT + cT ) type 1particles at T − √ T , then at least L exp( − γT ) of them will “invade”the unit square ( √ T ,
0) + [0 , at time T .4.2. Proofs of ( i )–( iii ). We start with condition (i) which is proved inLemma 4.1.
Lemma 4.1.
For any ε > and for T sufficiently large, P ( N ( x, t ) > L (¯ u ( t ) + 7 ε ) for some x ∈ [ −√ T , √ T ] and some t ∈ [ √ T , T ]) → as L → ∞ . Proof.
Let A t be the periodic contact process with birth rate β ( t ),death rate δ , and initial configuration A ≡
2, and let ζ t = { x : ξ t ( x ) = 2 } denote the set of sites occupied by a type 2 particle at time t in the periodiccompetition model. Then A t and ξ t can be coupled in such a way that ζ t ⊂ A t for all t ≥
0. In particular, it suffices to prove the result for N ( x, t ) = | A t ∩ { x + [0 , }| instead of N ( x, t ). Let u ( t ) be the solution of the ODE dudt = β ( t ) u (1 − u ) − δ u with u (0) = 1 . B. CHAN, R. DURRETT AND N. LANCHIER
Fig. 2.
Boxes in the block construction for Theorem 1.
Then there exists T sufficiently large such that | u ( t ) − ¯ u ( t ) | < ε for all t ≥ √ T . (15)Moreover, by Lemma 3.3, P ( x ∈ A t ) → u ( t ) uniformly on [0 , T ] as L → ∞ which, together with inequality (15), implies that EN ( x, t ) < L (¯ u ( t ) + 2 ε )(16) for all ( x, t ) ∈ [ −√ T , √ T ] × [ √ T , T ] as L → ∞ . In other respects, by Lemma 3.2, P ( | N ( x, t ) − EN ( x, t ) | > L ε for some(17) x ∈ [ −√ T , √ T ] and some t ∈ [ √ T , T ]) → L → ∞ . The result follows by combining (16) and (17). (cid:3)
Let ¯ Z t = ¯ Z t ( α, δ ) be the restriction of the branching random walk withbirth rate α ( t ) and death rate δ to the square I T = [ −√ T , √ T ] , that is,particles landed outside I T are killed. In preparation for proving (ii), weneed to show that if the averaged birth rate α ( t ) along one period is greaterthan the death rate δ then the number of particles in ¯ Z t grows exponentiallyfast. OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS Lemma 4.2.
Assume that ¯ Z = ¯ Z x = { x } . For s > r > fixed and T sufficiently large E | ¯ Z xt ∩ { y + [0 , }| ≥ exp (cid:18) Z t ( α ( θ ) − δ ) dθ (cid:19) for all x, y ∈ [ −√ T / , √ T / and t ∈ [ rT, sT ] . Proof.
Let S xt = S xt ( α ) be the random walk that jumps at rate α ( t )and has kernel the uniform distribution on the square [ − , , that is, thenew position of the process is chosen uniformly at random from S xt + [ − , ,and denote by ¯ S xt its restriction to the square I T . Let m ( t, x, A ) = E | ¯ Z xt ( α, δ ) ∩ A | denote the mean number of particles in the set A at time t . Then we claimthat m ( t, x, A ) = exp (cid:18)Z t ( α ( θ ) − δ ) dθ (cid:19) P ( ¯ S xt ( α ) ∈ A ) . (18)This holds because both sides of (18) satisfy the differential equation ∂m ( t, x, A ) ∂t = − δm ( t, x, A ) + Z α ( t ) m ( t, x, dy ) { y ∈ [ −√ T , √ T ] } ν ( A − y ) , where ν is the uniform probability measure on [ − , . Since 0 < min α ( t ) ≤ max α ( t ) < ∞ , one can rescale time so that ¯ S xt has constant jump ratemin α ( t ). Let t = uF ( T ) , x = v q F ( T ) , y = w q F ( T ) , v, w ∈ [ − , ] . Here F ( T ) = O ( T ) is piecewise continuous with 1 ≤ F ( T ) /T ≤ max α ( t ). Anapplication of the invariance principle shows that the process ¯ S xt converges inprobability to a Brownian motion when time and space are properly scaled.More precisely, sending T → ∞ we have F ( T ) · P ( ¯ S xt ∈ y + [0 , ) → ¯ p u ( v, w ) , where ¯ p u ( v, w ) is the time- u transition probability of a 2-dimensional Brow-nian motion from v to w that is killed outside [ − , . Since | v − w | ≤ p u ( v, w ) is bounded from below which implies that, for T sufficiently large, P ( ¯ S xt ∈ y + [0 , ) = O ( T − ) . It follows that m ( t, x, y + [0 , ) = exp (cid:18)Z t ( α ( θ ) − δ ) dθ (cid:19) ·O ( T − ) ≥ exp (cid:18) Z t ( α ( θ ) − δ ) dθ (cid:19) B. CHAN, R. DURRETT AND N. LANCHIER for all t ∈ [ rT, sT ] with T sufficiently large. (cid:3) In the next lemma, we establish (ii). The idea is that, since the densityof type 2 particles is bounded by ¯ u + 7 ε , the 1’s grow exponentially fastfrom time √ T to time T − √ T so that they can regain all possible lossesoccurring by time √ T . Lemma 4.3.
Assume that N (0 , > L exp( − γT ) . Then, for γ and T sufficiently large, P ( N (0 , T − √ T ) ≤ L exp( − γT + cT )) → as L → ∞ for a suitable constant c > . Proof.
First of all, we observe that the probability that a type 1 particleremains alive √ T units of time is given by p = exp( − δ √ T ). Therefore, anapplication of the large deviation inequality for the Binomial distributionimplies that as L → ∞ , P ( N (0 , √ T ) ≤ ( L /
2) exp( − γT − δ √ T )) ≤ exp( − e − γT − δ √ T L / → . In particular, since the inclusion relation ⊂ on the set of sites occupied bytype 1 particles is preserved by the dynamics, it suffices to prove the resultwhen N (0 , √ T ) = ( L /
2) exp( − γT − δ √ T ) and(19) ξ √ T ( x ) = 1 for all x / ∈ [0 , . The assumptions of Theorem 1 allow to fix ε > Z D β ( t )(1 − ¯ u ( t ) − ε ) − δ dt > . (20)Applying (18) to the restricted branching random walk ¯ Z t with birth rate α ( t ) = β ( t ) and death rate δ = δ , which dominates the number of type 1particles in the square I T , and taking (19) into account, we obtain that, forall sites x ∈ I T and all times t ∈ [ √ T , T ], EN ( x, t ) ≤ N (0 , √ T ) exp (cid:18)Z t √ T ( β ( θ ) − δ ) dθ (cid:19) ≤ L exp( − γT + max( β , β ) T ) . In particular, there exists a sufficiently large γ , fixed from now on, such that P ( N ( x, t ) ≤ L ε for all x ∈ I T and all t ∈ [ √ T , T − √ T ]) → L → ∞ . OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS By Lemma 4.1, the density of type 2 particles inside I T = [ −√ T , √ T ] isbounded by ¯ u + 7 ε between time √ T and time T − √ T . This, togetherwith (21), implies that, with probability arbitrarily close to 1 when L islarge, the number of empty sites in each square x + [0 , is larger than L (1 − ¯ u ( t ) − ε ). So, in a given interval, as L → ∞ , the number of 1’sdominates the number of particles in a branching random walk with birthrate α ( t ) = β ( t )(1 − ¯ u ( t ) − ε )(22)and death rate δ , and with ( L /
2) exp( − γT − δ √ T ) particles in [0 , attime √ T . To see this, note first that (22) is a lower bound of the rate ofsuccessful birth of type 1, that is, the rate at which type 1 particles areactually created, in the periodic contact process when (21) holds. Moreover,the arguments of Lemma 3.1 imply that both processes can be coupled insuch a way that type 1 particles in the periodic contact process are arbitrarilyclose to their counterparts in the branching random walk. In particular, itsuffices to prove the result for the branching random walk described aboverestricted to the square I T . We denote this process by ¯ Z t ( α, δ ) and let¯ N ( x, t ) = | ¯ Z t ( α, δ ) ∩ { x + [0 , }| . By Lemma 4.2 and (20), for T sufficiently large, E ¯ N (0 , T − √ T ) > ( L /
2) exp( − γT − δ √ T ) × exp (cid:18) Z T − √ T √ T β ( t )(1 − ¯ u ( t ) − ε ) − δ dt (cid:19) > L exp( − γT + 2 cT )for a suitable constant c >
0. Finally, by Lemma 3.2 (which applies to theperiodic branching random walk as well),¯ N (0 , T − √ T ) > L exp( − γT + 2 cT ) − L ε > L exp( − γT + cT )with probability arbitrarily close to 1 for L and T sufficiently large. (cid:3) To complete the proof of Theorem 1, it remains to “move” our parti-cles from the square [0 , to the square ( √ T ,
0) + [0 , and show that L exp( − γT ) of them will live until time T . Our last lemma shows (iii) inthe list. Lemma 4.4.
Assume that N (0 , T − √ T ) > L exp( − γT + cT ) and that N ( x, t ) ≤ L (¯ u ( t ) + 7 ε ) for all x ∈ [ −√ T , √ T ] and t ∈ [ T − √ T , T ] . Then, for T large, P ( N (( √ T , , T ) ≤ L exp( − γT )) → as L → ∞ . B. CHAN, R. DURRETT AND N. LANCHIER
Proof.
Fix a >
0. For any integer m ∈ Z and any time t ≥
0, we let B m = ( m/ ,
0) + [0 , / and H m,t = |{ x ∈ B m : ξ t ( x ) = 1 }| . Assume first that H m +1 ,t ≥ L exp( − aT ) for some t ∈ ( s, s + 1). The prob-ability that a type 1 particle remains alive at least one unit of time is e − δ .Since these events are independent, the random variable H m +1 ,s +1 domi-nates a Binomial random variable with parameters n ≥ L exp( − aT ) and p = e − δ . An application of the large deviation result for the Binomial P ( X ≤ n ( p − z )) ≤ exp( − nz / p )with z = p/ P ( H m +1 ,s +1 ≤ L exp( − aT − δ ) | H m +1 ,t ≥ L exp( − aT )for some t ∈ ( s, s + 1))(23) ≤ exp( − np/ ≤ exp( − e − aT − δ L / . Now, assume on the contrary that H m +1 ,t < L exp( − aT ) for all t ∈ ( s, s +1). Then the probability that a type 1 particle in B m at time s gives birthby time s + 1 to an offspring which is sent to an empty site in B m +1 andthat both the parent and the offspring live until time s + 1 is bounded frombelow by p = e − δ (1 − e − b )(1 − max( p , p ) − ε − − aT )) , where b = min( β , β ). The terms on the right-hand side are lower boundsfor the probabilities that (i) the offspring is sent to B m +1 , (ii) particles liveuntil time s + 1, (iii) the birth occurs by time s + 1, and (iv) the landingsite is empty (since ¯ u ( t ) ≤ max( p , p )). This implies that P ( H m +1 ,s +1 ≤ ( p / H m,s | H m +1 ,t < L exp( − aT )(24) for all t ∈ ( s, s + 1) and H m,s > L exp( − γT )) → L → ∞ . Applying (23) and (24) with m = 2 √ T − n and s = T − n gives H √ T − n +1 ,T − n +1 >
12 min( p , e − δ ) H √ T − n,T − n = p H √ T − n,T − n with probability close to 1 when L is large on the event that H m,s > L exp( − γT ).Assuming that one fourth of the particles of N (0 , T − √ T ) are found in H ,T − √ T and observing that14 (cid:18) p (cid:19) √ T L exp( − γT + cT ) > L exp( − γT ) OEXISTENCE FOR A MULTITYPE CONTACT PROCESS WITH SEASONS for T large enough, a simple induction allows us to conclude that H √ T ,T > p H √ T − ,T − > (cid:18) p (cid:19) √ T H ,T − √ T > (cid:18) p (cid:19) √ T L exp( − γT + cT )with probability close to 1 when L is large. Recalling the definition of H m,t ,we obtain that, for T sufficiently large, and with probability arbitrarily closeto 1 when L is large, N (( √ T , , T ) > L exp( − γT ) . This completes the proof. (cid:3)
Acknowledgment.
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