Costly defense traits in structured populations
aa r X i v : . [ m a t h . P R ] M a y Altruistic defense traits in structured populations
Martin Hutzenthaler ∗ , Felix Jordan ∗ , and Dirk Metzler ∗ May 16, 2018
Abstract
We propose a model for the frequency of an altruistic defense trait. More precisely, we consider Lotka-Volterra-type models involving a host/prey population consisting of two types and a parasite/predatorpopulation where one type of host individuals (modeling carriers of a defense trait) is more effective indefending against the parasite but has a weak reproductive disadvantage. Under certain assumptions weprove that the relative frequency of these altruistic individuals in the total host population converges tospatially structured Wright-Fisher diffusions with frequency-dependent migration rates. For the many-demes limit (mean-field approximation) hereof, we show that the defense trait goes to fixation/extinctionif and only if the selective disadvantage is smaller/larger than an explicit function of the ecological modelparameters.
Contents ∗ Research supported by the DFG in the Priority Program ”Probabilistc Structures in Evolution” (SPP 1590)
AMS subject classifications
Key words and phrases altruistic defense, group selection, kin selection, interacting Wright-Fisher diffusions, local competi-tion, extinction, survival, Lotka-Volterra equations, McKean-Vlasov limit, many-demes-limit, host-parasite, predator-prey, parasitedefense, slave rebellion, slavemaker ants Introduction
Altruism refers to the behavior of an individual that decreases the reproductive success of the actor whileincreasing the reproductive success of one or more recipients. In most natural systems, non-altruistic individualsbenefit from altruistic individuals without suffering from the fitness disadvantage and, thus, have a directreproductive advantage. So how can genetically inherited selfless behavior be explained by natural selection?This problem has bothered biologists since Charles Darwin who reflected the puzzle of sterile social insects suchas the worker castes of ants in his famous book “The Origin of Species” [5].In the biology and game theory literature there exist several explanations for the emergence of altruism (alsoreferred to as cooperation in game theory). The central idea behind kin selection is that helping direct relativesbenefits the reproductive success of the altruists’ genes. This idea is formalized in Hamilton’s rule which statesthat traits increase in frequency if R · B > C where R is the genetic relatedness of the recipient and the actor,B is the additional reproductive benefit gained by the recipient, and C is the reproductive cost to the actor; seeHamilton [13]. Relatedness is frequently defined as the probability of sharing the same allele by descent, e.g.,1 / / We begin with a stochastic extension of the classical and long-established Lotka-Volterra model (see Lotka [22]and Volterra [39]) which can be obtained as an approximation of discrete Markov chains such as renormalized2wo-types birth and death processes in the case of large populations. To formulate these stochastic extensions,we consider the following setting (see Section 1.2 for notational conventions used throughout this article).Let (Ω , F , P ) be a probability space, let D be an at most countable set (the set of demes) and let m ∈ [0 , ∞ ) D×D satisfy for every i ∈ D that P k ∈D m ( k, i ) = P k ∈D m ( i, k ) = 1. We refer to m as the migrationmatrix or matrix of migration rates. Let λ, K, δ, ν, γ, η, ρ ∈ (0 , ∞ ) satisfy ρ < η . For every N ∈ N , let κ NH , κ NP , α N , β NH , β NP , ι NH , ι NP ∈ [0 , ∞ ), let W A,N ( i ) , W C,N ( i ) , W P,N ( i ) : [0 , ∞ ) × Ω → R , i ∈ D , be independentBrownian motions with continuous sample paths, let A N , C N , P N : [0 , ∞ ) ×D× Ω → [0 , ∞ ) be adapted processeswith continuous sample paths that for all t ∈ [0 , ∞ ) and all i ∈ D satisfy P -a.s. A Nt ( i ) = A N ( i ) + Z t κ NH X j ∈D m ( i, j ) (cid:0) A Ns ( j ) − A Ns ( i ) (cid:1) + A Ns ( i ) h λ (cid:16) − A Ns ( i )+ C Ns ( i ) K (cid:17) − δP Ns ( i ) − α N i ds + Z t ι NH A Ns ( i ) A Ns ( i )+ C Ns ( i ) ds + Z t q β NH A Ns ( i ) dW A,Ns ( i ) ,C Nt ( i ) = C N ( i ) + Z t κ NH X j ∈D m ( i, j ) (cid:0) C Ns ( j ) − C Ns ( i ) (cid:1) + C Ns ( i ) h λ (cid:16) − A Ns ( i )+ C Ns ( i ) K (cid:17) − δP Ns ( i ) i ds + Z t ι NH C Ns ( i ) A Ns ( i )+ C Ns ( i ) ds + Z t q β NH C Ns ( i ) dW C,Ns ( i ) ,P Nt ( i ) = P N ( i ) + Z t κ NP X j ∈D m ( i, j ) (cid:0) P Ns ( j ) − P Ns ( i ) (cid:1) ds + Z t P Ns ( i ) (cid:2) − ν − γP Ns ( i ) + ηC Ns ( i ) + ( η − ρ ) A Ns ( i ) (cid:3) + ι NP ds + Z t q β NP P Ns ( i ) dW P,Ns ( i ) , (1)let H N : [0 , ∞ ) × D × Ω → [0 , ∞ ) satisfy H N = A N + C N , and let F N : [0 , ∞ ) × D × Ω → [0 ,
1] satisfy F N = A N H N . For every N ∈ N , the process H N describes the host (or prey) populations, A N describes thealtruists (or cooperators), C N describes the cheaters (or defectors), and P N describes the parasite (or predator)populations, each measured in units of N individuals. Existence of solutions to (1), which we assume here, canbe established in suitable Liggett-Spitzer spaces if D is an Abelian group and if m is translation invariant andirreducible; cf. Proposition 2.1 in [17].The central goal of this article is to prove convergence of the sequence (( A Nt A Nt + C Nt ) t ∈ [0 , ∞ ) ) N ∈ N and to derivethe diffusion equation which the limit solves. In other words, we will derive an analog of the Kimura steppingstone model (i.e., spatially structured Wright-Fisher diffusions) for altruistic defense against parasites. Since A N , C N , P N are measured in units of N individuals and the stochastic fluctuations scale with √ N as N → ∞ ,we need to assume that β NH and β NP are of order √ N for large N ∈ N . To get a nontrivial diffusion approximationwe additionally assume – as is usual in the derivation of the Kimura stepping stone model – slow migrationand weak selection in the sense that the sequences ( N κ NH ) N ∈ N , ( N κ NP ) N ∈ N and ( N α N ) N ∈ N converge. Thus therelative frequency of altruists A N A N + C N evolves on the time scale of order N as N → ∞ .In the special case that for some N ∈ N it holds that κ NH = κ NP = ι NH = ι NP = α N = β NH = β NP = A N = 0, then A N ≡ H N ( i ) , P N ( i )), i ∈ D , satisfy classical Lotka-Volterra equations. It is well known that if Kη > ν ,then the solutions of these equations converge to the nontrivial equilibrium (
Kδνλγ + Kδη , λKη − λνλγ + δKη ) ∈ (0 , ∞ ) in eachdeme. Since we assume that κ NH , κ NP , α N , ι NH , ι NP , β NH , β NP are of order o (1) as N → ∞ and since the altruistfrequencies evolve slowly, for every i ∈ D , the processes ( H N ( i ) , P N ( i )) should asymptotically be close to theequilibrium of the classical Lotka-Volterra equations with η being replaced by η − ρF N ( i ) as N → ∞ . Moreprecisely, we will prove in Theorem 1.2 below under further assumptions that if the local frequency of altruistsis q ∈ [0 , h ∞ ( q ) , p ∞ ( q )) where the functions h ∞ and p ∞ are defined by[0 , ∋ x h ∞ ( x ) := K ( δν + γλ ) λγ + δK ( η − ρx ) = b ( a − x ) ∈ (0 , ∞ )[0 , ∋ x p ∞ ( x ) := λK ( η − ρx ) − λνλγ + δK ( η − ρx ) = λδ (cid:16) − Kb ( a − x ) (cid:17) ∈ (0 , ∞ ) (2)and where a := λγ + δKηδKρ and b := δρδν + λγ . For these functions to be well defined we will assume that Kb ( a − > K ( η − ρ ) > ν .The above heuristic is incorrect if all populations go extinct by chance due to stochasticity in the offspringdistributions. To avoid this difficulty we will assume that there is sufficient immigration of hosts (2 ι NH ≥ β NH )and parasites (2 ι NP ≥ β NH ) in order that both host populations and parasite populations cannot go extinct;see Lemmas 2.2 and 2.3, respectively. However, note that both altruists and cheaters can locally die out. Forour proof, which is based on the Lyapunov function (67), we additionally require further restrictions on theparameters and on (inverse) moments of the initial configuration. Assumption 1.1.
In the setting of the first paragraph of Section 1.1 it holds that λ > ν , η − ρ > λK , γ ≥ δ ,for all N ∈ N it holds that α N + κ NH ≤ λ , ι NP ≤ λ ( ν + λ )8 δ , κ NP + κ NH + α N ≤ λ − ν , ι NH ≥ δκ NP ν + λ ) + β NH , ι NP ≥ β NP ,and there exist σ = ( σ i ) i ∈D ∈ (0 , ∞ ) D and c ∈ (0 , ∞ ) such that P i ∈D σ i < ∞ , such that for every j ∈ D itholds that X i ∈D σ i m ( i, j ) ≤ cσ j , (3) and such that sup N ∈ N E h(cid:13)(cid:13)(cid:13)(cid:16) H N + P N (cid:17) + H N ) + P N ( H N ) + P N + P N H N (cid:13)(cid:13)(cid:13) σ i < ∞ . The following theorem, which appears to be new even for non-spatial stochastic Lotka-Volterra stochastic dif-ferential equations (SDEs), proves for every t ∈ [0 , ∞ ) that the L ([0 , t ] × l σ × Ω; R )-distance between ( H N · N , P N · N )and ( h ∞ ( F N · N ) , p ∞ ( F N · N )) converges to 0 as N → ∞ at least with rate . Theorem 1.2 follows immediately fromTheorem 2.8 below together with a time substitution. Theorem 1.2.
Assume the setting of the first paragraph of Section 1.1, let Assumption 1.1 hold, assume that sup N ∈ N ( N max { κ NH , κ NP , α N , ι NH , ι NP , β NH , β NP } ) < ∞ and let h ∞ and p ∞ be given by (2) . Then we get for all sets ˆ D ⊆ D and all t ∈ [0 , ∞ ) that sup N ∈ N N Z t E X i ∈ ˆ D σ i (cid:0) H NuN ( i ) − h ∞ (cid:0) F NuN ( i ) (cid:1)(cid:1) + X i ∈ ˆ D σ i (cid:0) P NuN ( i ) − p ∞ (cid:0) F NuN ( i ) (cid:1)(cid:1) du < ∞ . (4)Knowing the asymptotic behavior of the host populations, we can formally replace the ( H N ) N ∈ N in thediffusion equation (13) of the altruist frequencies and, thereby, we arrive at the diffusion equation which thelimit of altruist frequencies solves. Our main result, Theorem 1.3, then proves that the altruist frequenciesconverge to the solution of the diffusion equation (5). The proof of Theorem 1.3 is deferred to Section 2.4.2below and is based on a general stochastic averaging result in Kurtz [21]. Theorem 1.3.
Assume the setting of the first paragraph of Section 1.1, let Assumption 1.1 hold, assume that P i ∈D sup N ∈ N σ i E h H N ( i ) i < ∞ , that sup N ∈ N ( N max { κ NP , ι NH , ι NP , β NP } ) < ∞ , that there exist κ, α, β ∈ [0 , ∞ ) such that lim N →∞ κ NH N = κ , lim N →∞ α N N = α and lim N →∞ β NH N b = β and assume that F N = ⇒ X as N → ∞ in l σ . Then the SDE dX t ( i ) = κ X j ∈D m ( i, j ) a − X t ( i ) a − X t ( j ) (cid:16) X t ( j ) − X t ( i ) (cid:17) dt − αX t ( i )(1 − X t ( i )) dt + p β ( a − X t ( i )) X t ( i )(1 − X t ( i )) dW t ( i ) , t ∈ (0 , ∞ ) , i ∈ D (5) (where { W ( i ) : i ∈ D} are independent standard Brownian motions) has a unique strong solution and (cid:0) F NtN (cid:1) t ∈ [0 , ∞ ) = ⇒ ( X t ) t ∈ [0 , ∞ ) (6) as N → ∞ in C ([0 , ∞ ) , l σ ) . An important problem is to derive conditions under which altruists persist, that is, to derive conditions on theparameters of the SDE (5) under which the process goes to fixation. Here we simplify this problem and considerthe many-demes-limit (also denoted as mean-field approximation) of the SDE (5). More precisely, for every D ∈ N , let X D : [0 , ∞ ) ×{ }× Ω → [0 ,
1] be the solution of the SDE (5) with D replaced by { , . . . , D } and4ith m replaced by ( D i = j ) i,j ∈{ ,...,D } . We will show in Proposition 3.1 together with Lemma 3.2 below that if,for every D ∈ N , ( X D ( i )) i ∈{ ,...,D } are exchangeable [0 , D ∈ N E [( X D (1)) ] < ∞ ,if Z : [0 , ∞ ) × Ω → [0 ,
1] is the solution of the SDE (8) below with respect to the Brownian motion W (1) and ifsup D ∈ N √ D E (cid:2)(cid:12)(cid:12) X D ( i ) − Z ( i ) (cid:12)(cid:12)(cid:3) < ∞ , then for all t ∈ [0 , ∞ ) it holds thatsup D ∈ N √ D E (cid:2) | X Dt (1) − Z t | (cid:3) < ∞ . (7)Thus the solution of the SDE (8) is the many-demes limit of the SDE (5). For this many-demes limit we derivea simple necessary and sufficient condition ( α < β ) under which the altruistic defense trait goes to fixationwhen starting with a positive frequency. The proof of Theorem 1.4 is deferred to Section 4.3. Theorem 1.4.
Let α, β, κ ∈ (0 , ∞ ) , let a ∈ (1 , ∞ ) , let (Ω , F , P , ( F t ) t ∈ [0 , ∞ ) ) be a filtered probability space,let W : [0 , ∞ ) × Ω → R be a standard ( F t ) t ∈ [0 , ∞ ) Brownian motion with continuous sample paths, and let Z : Ω → [0 , be an F / B ([0 , -measurable mapping. Then the SDE dZ t = κ ( a − Z t ) (cid:16) ( a − Z t ) E h a − Z t i − (cid:17) dt − αZ t (1 − Z t ) dt + p β ( a − Z t ) Z t (1 − Z t ) dW t (8) has a unique solution. Furthermore, if E [ Z ] = 1 , then P [ Z t = 1 for all t ∈ [0 , ∞ )] = 1 , if E [ Z ] = 0 , then P [ Z t = 0 for all t ∈ [0 , ∞ )] = 1 and if E [ Z ] ∈ (0 , , then lim t →∞ E (cid:2)(cid:12)(cid:12) Z t − (cid:12)(cid:12)(cid:3) = 0 , if α > β, lim t →∞ E (cid:2)(cid:12)(cid:12) Z t − (cid:12)(cid:12)(cid:3) = 0 , if α < β,Z t t →∞ = ⇒ Z · m ( z ) dz, if α = β, (9) where m ( z ) = c z κβ ( aθ − − (1 − z ) κβ (1 − θ ( a − − ( a − z ) αβ − for z ∈ (0 , , where c ∈ (0 , ∞ ) is a normalizingconstant and where θ = E [ a − Z ] . Informally speaking, Theorem 1.4 asserts that an altruistic defense allele persists in an infinite dimensionalspace if α < β and if the mean frequency of altruists over all demes is positive. This does not imply that a newmutation resulting in altruistic defense behavior can establish itself on one island or even in the total population.Our final result partially closes this gap and considers a process which could be the limit lim D →∞ P Di =1 X D ( i )if for all D ∈ N and i ∈ { , . . . , D } it holds that X D ( i ) = Y i =1 for some [0 , P [ Y >
0] =1 that the process converges to 0 in probability as time goes to infinity if and only if α ≥ β . Informallyspeaking, Proposition 5.1 asserts that an altruistic defense allele has a positive invasion probability in aninfinite dimensional space if and only if α < β . Throughout this article, we will use the following notation. We define [0 , ∞ ] := [0 , ∞ ) ∪ {∞} . We will use theconventions that 0 = 1, 0 · ∞ = 0, and that for any x ∈ (0 , ∞ ) we have that x ∞ = 0 and x = ∞ . For all x, y ∈ R we define x + := max { x, } , sgn( x ) := x> − x< , and x ∧ y := min { x, y } . We define sup( ∅ ) := −∞ andinf( ∅ ) := ∞ . For a topological space ( E, E ) we denote by B ( E ) the Borel sigma-algebra of ( E, E ). Moreover weagree on the convention that zero times an undefined expression is set to zero. For every countable set D andevery σ = ( σ i ) i ∈D ∈ (0 , ∞ ) D define a function k · k σ : R D → [0 , ∞ ] by R D ∋ z = ( z i ) i ∈D
7→ k z k σ := P i ∈D σ i | z i | and define l σ := { z ∈ R D : k z k σ < ∞} . Assume the setting of the first paragraph of Section 1.1. Define ¯ κ H := sup N ∈ N κ NH , ¯ κ P := sup N ∈ N κ NP , ¯ β H :=sup N ∈ N β NH , ¯ β P := sup N ∈ N β NP , ¯ ι H := sup N ∈ N ι NH , ¯ ι P := sup N ∈ N ι NP , and β H := lim N →∞ β NH . For all z =5 z i ) i ∈D ∈ (0 , ∞ ) D and p ∈ R let z p = ( z pi ) i ∈D . Furthermore, let 1 := (1) i ∈D ∈ l σ . Define E := [0 , D and E := l σ ∩ [0 , ∞ ) D . For all i ∈ D and all N ∈ N let W H,N ( i ) : [0 , ∞ ) × Ω → R and W F,N ( i ) : [0 , ∞ ) × Ω → R be stochastic processes with continuous sample paths such that for every t ∈ [0 , ∞ ) it holds P -a.s. that dW H,Nt ( i ) = √ A Nt ( i ) dW A,Nt ( i )+ √ C Nt ( i ) dW C,Nt ( i ) √ H Nt ( i ) (10)and dW F,Nt ( i ) = √ C Nt ( i ) dW A,Nt ( i ) − √ A Nt ( i ) dW C,Nt ( i ) √ H Nt ( i ) , (11)respectively, with W H,N ( i ) = W F,N ( i ) = 0. Assume the setting of Section 2.1. In this section we collect some first results that are used in the proofs of thestatements in subsequent sections.
Lemma 2.1.
Assume the setting of Section 2.1. Then W H,N ( i ) and W F,N ( i ) , N ∈ N , i ∈ D , are independentBrownian motions and for all t ∈ [0 , ∞ ) , all i ∈ D , and all N ∈ N it P -a.s. holds that H Nt ( i ) = H N ( i ) + Z t κ NH X j ∈D m ( i, j ) H Ns ( j ) + ( λ − κ NH − α N F Ns ( i )) H Ns ( i ) − λK (cid:0) H Ns ( i ) (cid:1) (12) − δP Ns ( i ) H Ns ( i ) + ι NH ds + Z t q β NH H Ns ( i ) dW H,Ns ( i ) ,F Nt ( i ) = F N ( i ) + Z t κ NH X j ∈D m ( i, j ) (cid:0) F Ns ( j ) − F Ns ( i ) (cid:1) H Ns ( j ) H Ns ( i ) − α N F Ns ( i ) (cid:0) − F Ns ( i ) (cid:1) ds (13)+ Z t r β NH F Ns ( i ) ( − F Ns ( i ) ) H Ns ( i ) dW F,Ns ( i ) ,P Nt ( i ) = P N ( i ) + Z t κ NP X j ∈D m ( i, j ) P Ns ( j ) − ( κ NP + ν ) P Ns ( i ) − γ (cid:0) P Ns ( i ) (cid:1) + (cid:0) η − ρF Ns ( i ) (cid:1) P Ns ( i ) H Ns ( i ) (14)+ ι NP ds + Z t q β NP P Ns ( i ) dW P,Ns ( i ) . Proof.
For all t ∈ [0 , ∞ ), all N ∈ N , and all i ∈ D we get (cid:10) W H,N ( i ) (cid:11) t = (cid:10) W F,N ( i ) (cid:11) t = t as well as (cid:10) W H,N ( i ) , W F,N ( i ) (cid:11) t = Z t √ A Ns ( i ) C Ns ( i ) − √ A Ns ( i ) C Ns ( i ) H Ns ( i ) ds = 0 . (15)Hence, we see that W H,N ( i ) and W F,N ( i ), N ∈ N , i ∈ D , are independent Brownian motions. Equation (12)follows from Itˆo’s lemma (e.g., Klenke [19]) and rearranging terms. Furthermore, applying Itˆo’s lemma we see6or all t ∈ [0 , ∞ ), all i ∈ D , and all N ∈ N that P -a.s. it holds that F Nt ( i ) = F N ( i ) + Z t C Ns ( i )( H Ns ( i )) κ NH X j ∈D m ( i, j ) (cid:0) A Ns ( j ) − A Ns ( i ) (cid:1) + A Ns ( i ) (cid:0) λ (cid:16) − H Ns ( i ) K (cid:17) − δP Ns ( i ) − α N (cid:1) + ι NH A Ns ( i ) H Ns ( i ) ! ds + Z t C Ns ( i )( H Ns ( i )) q β NH A Ns ( i ) dW As ( i ) − Z t A Ns ( i )( H Ns ( i )) κ NH X j ∈D m ( i, j ) (cid:0) C Ns ( j ) − C Ns ( i ) (cid:1) + C Ns ( i ) (cid:16) λ (cid:16) − H Ns ( i ) K (cid:17) − δP Ns ( i ) (cid:17) + ι NH C Ns ( i ) H Ns ( i ) ! ds − Z t A Ns ( i )( H Ns ( i )) q β NH C Ns ( i ) dW Cs ( i ) − Z t C Ns ( i )( H Ns ( i )) β NH A Ns ( i ) + A Ns ( i )( H Ns ( i )) β NH C Ns ( i ) ds = F N ( i ) + Z t κ NH H Ns ( i ) X j ∈D m ( i, j ) (cid:0)(cid:0) − F Ns ( i ) (cid:1) F Ns ( j ) H Ns ( j ) − F Ns ( i ) (cid:0) − F Ns ( j ) (cid:1) H Ns ( j ) (cid:1) − α N F Ns ( i ) (cid:0) − F Ns ( i ) (cid:1) ds + Z t r β NH F Ns ( i ) ( − F Ns ( i ) ) H Ns ( i ) dW F,Ns ( i ) (16)and (13) follows. Finally, we obtain (14) from the definition of (cid:0) H N (cid:1) N ∈ N and (cid:0) F N (cid:1) N ∈ N . Lemma 2.2.
Assume the setting of Section 2.1 and assume that for all N ∈ N we have ι NH ≥ β NH . Furthermore,assume that we have for all N ∈ N and all i ∈ D that P -a.s. H N ( i ) > . Then we have P (cid:2) H Nu ( i ) > , for all u ∈ [0 , ∞ ) , all N ∈ N , and all i ∈ D (cid:3) = 1 . (17) Proof.
For every
N, M ∈ N let ˆ H N,M : [0 , ∞ ) × D × Ω → [0 , ∞ ) be an adapted process with continuous samplepaths that for all t ∈ [0 , ∞ ) and all i ∈ D satisfies P -a.s.ˆ H N,Mt ( i ) = ˆ H N,M ( i ) + Z t h ˆ H N,Ms ( i ) (cid:16) λ − α N − κ NH − λK ˆ H N,Ms ( i ) − δM (cid:17) + ι NH i ds + Z t q β NH ˆ H N,Ms ( i ) dW H,Ns ( i ) (18)with ˆ H N,M ( i ) = H N ( i ). Due to Feller’s boundary classification (e.g., p. 366 in Ethier and Kurtz [7]) with theassumption that for all N ∈ N it holds that ι NH ≥ β NH we have for every N, M ∈ N and all i ∈ D that P h ˆ H N,Mt ( i ) > , for all t ∈ [0 , ∞ ) i = 1 . (19)For all N, M ∈ N , all i ∈ D , and all t ∈ [0 , ∞ ) consider the event A NM ( i ) := ( sup s ∈ [0 ,t ] P Ns ( i ) ≤ M ) . We have forall N, M ∈ N , all i ∈ D , and all t ∈ [0 , ∞ ) that A NM ( i ) ⊆ A NM +1 ( i ) , P " [ M ∈ N A NM ( i ) = P " sup s ∈ [0 ,t ] P Ns ( i ) < ∞ = 1 . (20)Using a comparison result due to Ikeda and Watanabe (see e.g., Theorem V.43.1 in Rogers and Williams [34]),we get for all N, M ∈ N , all i ∈ D , and all t ∈ [0 , ∞ ) that P " ∃ u ∈ [0 , t ] : H Nu ( i ) < ˆ H N,Mu ( i ) , sup s ∈ [0 ,t ] P Ns ( i ) ≤ M = 0 . (21)7hus, combining (19), (20), and (21) we obtain for all N ∈ N , all i ∈ D , and all t ∈ [0 , ∞ ) that1 ≥ P (cid:2) H Nu ( i ) > , for all u ∈ [0 , t ] (cid:3) = 1 − P (cid:2) ∃ u ∈ [0 , t ] : H Nu ( i ) = 0 (cid:3) ≥ − X M ∈ N P " ∃ u ∈ [0 , t ] : H Nu ( i ) = 0 , sup s ∈ [0 ,t ] P Ns ( i ) ≤ M ≥ − X M ∈ N P " ∃ u ∈ [0 , t ] : H Nu ( i ) < ˆ H N,Mu ( i ) , sup s ∈ [0 ,t ] P Ns ( i ) ≤ M = 1 . (22)This implies for all N ∈ N , all i ∈ D , and all t ∈ [0 , ∞ ) that P (cid:2) H Nu ( i ) > , for all u ∈ [0 , t ] (cid:3) = 1, which in turnimplies (17). This finishes the proof of Lemma 2.2. Lemma 2.3.
Assume the setting of Section 2.1 and assume that for all N ∈ N it holds that ι NP ≥ β NP .Furthermore, assume that we have for all N ∈ N and all i ∈ D that P -a.s. P N ( i ) > . Then we have P (cid:2) P Nt ( i ) > , for all t ∈ [0 , ∞ ) , all N ∈ N , and all i ∈ D (cid:3) = 1 . (23) Proof.
Analogous to the proof of Lemma 2.2.
Lemma 2.4.
Assume the setting of Section 2.1. For all x = ( x i ) i ∈D ∈ E , all p ∈ [1 , ∞ ) , and all sets D ′ ⊆ D it holds that X i ∈D ′ σ i (cid:16) X j ∈D m ( i, j ) x j (cid:17) p ≤ X i ∈D cσ i x pi . (24) Proof.
For any x = ( x i ) i ∈D ∈ E , any p ∈ [1 , ∞ ), and any set D ′ ⊆ D we obtain from Jensen’s inequality and(3) that P i ∈D ′ σ i (cid:16) P j ∈D m ( i, j ) x j (cid:17) p ≤ P i ∈D σ i P j ∈D m ( i, j ) x pj ≤ P i ∈D cσ i x pi . In this section we will show the convergence of the time-rescaled Lotka-Volterra processes as given in (12) and(14). In Lemmas 2.5, 2.6, and 2.7 we will provide bounds for the expected value of the sum (over sets of demes)of functionals of the processes weighted by σ . These are then used in Theorem 2.8 to show a result on thebehavior of a spatial analogue of a well-known Lyapunov function (e.g., Dobrinevski and Frey [6]). From thatthe convergence of the processes follows immediately in Theorem 1.2. Lemma 2.5.
Assume the setting of Section 2.1 and let p ∈ { } ∪ [2 , ∞ ) . Then we have sup N ∈ N sup t ∈ [0 , ∞ ) E h(cid:13)(cid:13)(cid:13)(cid:0) ηH Nt + δP Nt (cid:1) p (cid:13)(cid:13)(cid:13) σ i ≤ sup N ∈ N E h(cid:13)(cid:13)(cid:13)(cid:0) ηH N + δP N (cid:1) p (cid:13)(cid:13)(cid:13) σ i + k k σ λ + ( − p + cp ) (¯ κ H +¯ κ P )2 min (cid:26) η λK , , δ γ (cid:27) p vuut (cid:26) η λK , , δ γ (cid:27)h η ¯ ι H + δ ¯ ι P +( p − (cid:16) η ¯ β H + 12 δ ¯ β P (cid:17)i ( λ + ( − p + cp ) (¯ κ H +¯ κ P ) ) p . (25) Proof.
If we assume sup N ∈ N E (cid:2)(cid:13)(cid:13)(cid:0) H N + P N (cid:1) p (cid:13)(cid:13) σ (cid:3) = ∞ , then the claim trivially holds. For the remainderof the proof assume sup N ∈ N E (cid:2)(cid:13)(cid:13)(cid:0) H N + P N (cid:1) p (cid:13)(cid:13) σ (cid:3) < ∞ . Define D := ∅ and for every n ∈ N let D n ⊆ D be a set with |D n | = min { n, |D|} and D n ⊇ D n − . Define real numbers c := min n η λK , , δ γ o , c := p (cid:2) η ¯ ι H + δ ¯ ι P + ( p − (cid:0) η ¯ β H + δ ¯ β P (cid:1)(cid:3) ∈ (0 , ∞ ), c := λp +( p − c )(¯ κ H +¯ κ P ) ∈ (0 , ∞ ), c := c p (cid:0)P k ∈D σ k (cid:1) − p ∈ (0 , ∞ ), and c := c (cid:0)P k ∈D σ k (cid:1) p ∈ (0 , ∞ ). For all N ∈ N , t ∈ [0 , ∞ ) define Y Nt := 2 ηH Nt + δP Nt and for all N, n ∈ N and all t ∈ [0 , ∞ ) let M N,nt be a real-valued random variable such that P -a.s. it holds that M N,nt = X i ∈D n σ i Z t ηp (cid:0) Y Nu ( i ) (cid:1) p − q β NH H Nu ( i ) dW H,Nu ( i ) + Z t δp (cid:0) Y Nu ( i ) (cid:1) p − q β NP P Nu ( i ) dW P,Nu ( i ) ! . (26)8pplying Itˆo’s lemma we get for all N, n ∈ N and all t ∈ [0 , ∞ ) that P -a.s. X i ∈D n σ i (cid:0) Y Nt ( i ) (cid:1) p − X i ∈D n σ i (cid:0) Y N ( i ) (cid:1) p = X i ∈D n σ i Z t ηp (cid:0) Y Nu ( i ) (cid:1) p − κ NH X j ∈D m ( i, j ) H Nu ( j ) + ( λ − κ NH − α N F Nu ( i )) H Nu ( i ) − λK (cid:0) H Nu ( i ) (cid:1) − δP Nu ( i ) H Nu ( i ) + ι NH ! + δp (cid:0) Y Nu ( i ) (cid:1) p − κ NP X j ∈D m ( i, j ) P Nu ( j ) − ( κ NP + ν ) P Nu ( i ) − γ (cid:0) P Nu ( i ) (cid:1) + (cid:0) η − ρF Nu ( i ) (cid:1) P Nu ( i ) H Nu ( i ) + ι NP ! + η p ( p − (cid:0) Y Nu ( i ) (cid:1) p − β NH H Nu ( i ) + δ p ( p − (cid:0) Y Nu ( i ) (cid:1) p − β NP P Nu ( i ) du + M N,nt . (27)Because 1 ≥ c λK ≥ c η , and γ ≥ δc we get for all N, n ∈ N and all t ∈ [0 , ∞ ) that P -a.s. X i ∈D n σ i (cid:0) Y Nt ( i ) (cid:1) p − X i ∈D n σ i (cid:0) Y N ( i ) (cid:1) p ≤ X i ∈D n σ i Z t p (cid:0) Y Nu ( i ) (cid:1) p − η ¯ κ H X j ∈D m ( i, j ) H Nu ( j ) + 2 ηλH Nu ( i ) − c (2 ηH Nu ( i )) − [ ηδ + c ηδ ] P Nu ( i ) H Nu ( i ) + 2 η ¯ ι H ! + p (cid:0) Y Nu ( i ) (cid:1) p − δ ¯ κ P X j ∈D m ( i, j ) P Nu ( j ) + λδP Nu ( i ) − c ( δP Nu ( i )) + ηδP Nu ( i ) H Nu ( i ) + δ ¯ ι P ! + p ( p − (cid:0) Y Nu ( i ) (cid:1) p − (cid:0) η ¯ β H + δ ¯ β P (cid:1) ηH Nu ( i ) + (cid:0) δ ¯ β P + 2 η ¯ β H (cid:1) δP Nu ( i ) ! du + M N,nt = Z t X i ∈D n σ i p (cid:0) Y Nu ( i ) (cid:1) p − η ¯ κ H X j ∈D m ( i, j ) H Nu ( j ) + X i ∈D n σ i p (cid:0) Y Nu ( i ) (cid:1) p − λ (cid:0) Y Nu ( i ) (cid:1) − c (cid:0) Y Nu ( i ) (cid:1) + (2 η ¯ ι H + δ ¯ ι P ) ! + X i ∈D n σ i p (cid:0) Y Nu ( i ) (cid:1) p − δ ¯ κ P X j ∈D m ( i, j ) P Nu ( j )+ X i ∈D n σ i p ( p − (cid:0) η ¯ β H + δ ¯ β P (cid:1) (cid:0) Y Nu ( i ) (cid:1) p − du + M N,nt . (28)Using Young’s inequality and Lemma 2.4 we get for all N, n ∈ N and all t ∈ [0 , ∞ ) that P -a.s. X i ∈D n σ i (cid:0) Y Nt ( i ) (cid:1) p − X i ∈D n σ i (cid:0) Y N ( i ) (cid:1) p ≤ Z t X i ∈D n σ i p − p p (cid:0) Y Nu ( i ) (cid:1) p ¯ κ H + X i ∈D σ i p p ¯ κ H c (cid:0) ηH Nu ( i ) (cid:1) p + X i ∈D n σ i λp (cid:0) Y Nu ( i ) (cid:1) p + X i ∈D n σ i c (cid:0) Y Nu ( i ) (cid:1) p − − X i ∈D n σ i c p (cid:0) Y Nu ( i ) (cid:1) p +1 + X i ∈D n σ i p − p p (cid:0) Y Nu ( i ) (cid:1) p ¯ κ P + X i ∈D σ i p p ¯ κ P c (cid:0) δP Nu ( i ) (cid:1) p du + M N,nt ≤ Z t X i ∈D σ i c (cid:0) Y Nu ( i ) (cid:1) p + X i ∈D n σ i c (cid:0) Y Nu ( i ) (cid:1) p − − X i ∈D n σ i c p (cid:0) Y Nu ( i ) (cid:1) p +1 du + M N,nt . (29)9or N, n, l ∈ N define [0 , ∞ ]-valued stopping times τ N,nl := inf ( t ∈ [0 , ∞ ) : X i ∈D n σ i (cid:0) Y Nt ( i ) (cid:1) p > l ) ∪ ∞ ! . (30)We now get for all N, n, l ∈ N and all t ∈ [0 , ∞ ) that E "Z t ∧ τ N,nl X i ∈D n σ i "(cid:18) ηp (cid:0) Y Nu ( i ) (cid:1) p − q β NH H Nu ( i ) (cid:19) + (cid:18) δp (cid:0) Y Nu ( i ) (cid:1) p − q β NP P Nu ( i ) (cid:19) du = E "Z t ∧ τ N,nl X i ∈D n σ i p " ηβ NH (cid:18)(cid:0) Y Nu ( i ) (cid:1) p − q ηH Nu ( i ) (cid:19) + δβ NP (cid:18)(cid:0) Y Nu ( i ) (cid:1) p − q δP Nu ( i ) (cid:19) du ≤ (cid:0) ηβ NH + δβ NP (cid:1) E "Z t ∧ τ N,nl X i ∈D n σ i p "(cid:18) (cid:0) Y Nu ( i ) (cid:1) p − (cid:19) du (31)Using Young’s inequality, we obtain for all N, n, l ∈ N and all t ∈ [0 , ∞ ) that E "Z t ∧ τ N,nl X i ∈D n σ i "(cid:18) ηp (cid:0) Y Nu ( i ) (cid:1) p − q β NH H Nu ( i ) (cid:19) + (cid:18) δp (cid:0) Y Nu ( i ) (cid:1) p − q β NP P Nu ( i ) (cid:19) du ≤ (cid:0) ηβ NH + δβ NP (cid:1) E "Z t ∧ τ N,nl X i ∈D n σ i (cid:20) p (cid:16) p − p (cid:0) Y Nu ( i ) (cid:1) p + p (cid:17) (cid:21) du ≤ (cid:0) ηβ NH + δβ NP (cid:1) E "Z t ∧ τ N,nl X i ∈D n σ i min k ∈D n σ k (cid:20)(cid:16) (2 p − (cid:0) Y Nu ( i ) (cid:1) p + 1 (cid:17) (cid:21) du ≤ ηβ NH + δβ NP min k ∈D n σ k E Z t ∧ τ N,nl X i ∈D n σ i (cid:16) (2 p − (cid:0) Y Nu ( i ) (cid:1) p + 1 (cid:17)! du ≤ ηβ NH + δβ NP min k ∈D n σ k E Z t (2 p − X i ∈D n σ i (cid:16) Y Nu ∧ τ N,nl ( i ) (cid:17) p + k k σ ! du ≤ ηβ NH + δβ NP min k ∈D n σ k t h ((2 p − l + k k σ ) i < ∞ . (32)Hence, we get for all N, n, l ∈ N and all t ∈ [0 , ∞ ) that E h M N,nt ∧ τ N,nl i = 0. From this and (29) and using Tonelli’stheorem we see for all N, n, l ∈ N and all t ∈ [0 , ∞ ) that E " X i ∈D n σ i (cid:16) Y Nt ∧ τ N,nl ( i ) (cid:17) p + Z t ∧ τ N,nl c p X i ∈D n σ i (cid:0) Y Nu ( i ) (cid:1) p +1 du ≤ E h(cid:13)(cid:13)(cid:13)(cid:0) Y N (cid:1) p (cid:13)(cid:13)(cid:13) σ i + E "Z t ∧ τ N,nl c (cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p (cid:13)(cid:13)(cid:13) σ + c (cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p − (cid:13)(cid:13)(cid:13) σ du ≤ E h(cid:13)(cid:13)(cid:13)(cid:0) Y N (cid:1) p (cid:13)(cid:13)(cid:13) σ i + Z t c E h(cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p (cid:13)(cid:13)(cid:13) σ i + c E h(cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p − (cid:13)(cid:13)(cid:13) σ i du. (33)For every N, n ∈ N the map [0 , ∞ ) ∋ t P i ∈D n σ i (cid:0) Y Nt ( i ) (cid:1) p ∈ R is P -a.s. continuous which implies for all N, n ∈ N and all t ∈ [0 , ∞ ) that P h lim l →∞ τ N,nl < t i = 0. From Tonelli’s theorem and monotone convergence,10hen using Fatou’s lemma, and finally applying (33) we see for all N ∈ N and all t ∈ [0 , ∞ ) that E "X i ∈D σ i (cid:0) Y Nt ( i ) (cid:1) p + Z t c p E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p +1 du = lim n →∞ E " X i ∈D n σ i (cid:0) Y Nt ( i ) (cid:1) p + Z t c p X i ∈D n σ i (cid:0) Y Nu ( i ) (cid:1) p +1 du = lim n →∞ E " lim l →∞ X i ∈D n σ i (cid:16) Y Nt ∧ τ N,nl ( i ) (cid:17) p + Z t ∧ τ N,nl c p X i ∈D n σ i (cid:0) Y Nu ( i ) (cid:1) p +1 du ! ≤ lim n →∞ lim inf l →∞ E " X i ∈D n σ i (cid:16) Y Nt ∧ τ N,nl ( i ) (cid:17) p + Z t ∧ τ N,nl c p X i ∈D n σ i (cid:0) Y Nu ( i ) (cid:1) p +1 du ≤ E h(cid:13)(cid:13)(cid:13)(cid:0) Y N (cid:1) p (cid:13)(cid:13)(cid:13) σ i + Z t c E h(cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p (cid:13)(cid:13)(cid:13) σ i + c E h(cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p − (cid:13)(cid:13)(cid:13) σ i du. (34)This implies using Jensen’s inequality for all N ∈ N and all t ∈ [0 , ∞ ) that we get E h(cid:13)(cid:13)(cid:13)(cid:0) Y Nt (cid:1) p (cid:13)(cid:13)(cid:13) σ i − E h(cid:13)(cid:13)(cid:13)(cid:0) Y N (cid:1) p (cid:13)(cid:13)(cid:13) σ i ≤ Z t c E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p + c E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p − − c p E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p +1 du = Z t c E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p + P k ∈D σ k P l ∈D σ l c E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p − − c p E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p +1 du ≤ Z t c E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p + c E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p p − p − c E "X i ∈D σ i (cid:0) Y Nu ( i ) (cid:1) p p +1 p du = Z t (cid:16) E h(cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p (cid:13)(cid:13)(cid:13) σ i(cid:17) p − p (cid:26) c + c (cid:16) E h(cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p (cid:13)(cid:13)(cid:13) σ i(cid:17) p − c (cid:16) E h(cid:13)(cid:13)(cid:13)(cid:0) Y Nu (cid:1) p (cid:13)(cid:13)(cid:13) σ i(cid:17) p (cid:27) du. (35)For every N ∈ N let z N : [0 , ∞ ) → R be a process that for all t ∈ [0 , ∞ ) satisfies z Nt = z N + Z t (cid:0) z Ns (cid:1) p − p n c + c (cid:0) z Ns (cid:1) p − c (cid:0) z Ns (cid:1) p o ds (36)with z N = E (cid:2)(cid:13)(cid:13)(cid:0) Y N (cid:1) p (cid:13)(cid:13) σ (cid:3) , where uniqueness follows from local Lipschitz continuity. Using classical comparisonresults from the theory of ODEs, the above computation shows that for all N ∈ N and all t ∈ [0 , ∞ ) we have E (cid:2)(cid:13)(cid:13)(cid:0) Y Nt (cid:1) p (cid:13)(cid:13) σ (cid:3) ≤ z Nt and for all N ∈ N we have sup t ∈ [0 , ∞ ) z Nt = max n E (cid:2)(cid:13)(cid:13)(cid:0) Y N (cid:1) p (cid:13)(cid:13) σ (cid:3) , (cid:16) c c + q ( c ) c + c c (cid:17) p o .We thereby conclude thatsup N ∈ N sup t ∈ [0 , ∞ ) E h(cid:13)(cid:13)(cid:13)(cid:0) ηH Nt + δP Nt (cid:1) p (cid:13)(cid:13)(cid:13) σ i ≤ sup N ∈ N sup t ∈ [0 , ∞ ) z Nt ≤ sup N ∈ N E h(cid:13)(cid:13)(cid:13)(cid:0) Y N (cid:1) p (cid:13)(cid:13)(cid:13) σ i + c c + r c c + c c ! p = sup N ∈ N E h(cid:13)(cid:13)(cid:13)(cid:0) Y N (cid:1) p (cid:13)(cid:13)(cid:13) σ i + c c (cid:16) q c c c (cid:17) p = sup N ∈ N E h(cid:13)(cid:13)(cid:13)(cid:0) ηH N + δP N (cid:1) p (cid:13)(cid:13)(cid:13) σ i + k k σ λ + ( − p + cp ) (¯ κ H +¯ κ P )2 min (cid:26) η λK , , δ γ (cid:27) p vuut (cid:26) η λK , , δ γ (cid:27)h η ¯ ι H + δ ¯ ι P +( p − (cid:16) η ¯ β H + 12 δ ¯ β P (cid:17)i ( λ + ( − p + cp ) (¯ κ H +¯ κ P ) ) p . (37)This finishes the proof of Lemma 2.5. 11 emma 2.6. Assume the setting of Section 2.1 and assume γ ≥ δ . Furthermore, assume that for all N ∈ N we have α N + κ NH ≤ λ , ι NP ≤ λ ( ν + λ )8 δ , and ι NH ≥ δκ NP ν + λ ) + β NH . Let ˆ D ⊆ D be a set. Then we have sup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i (cid:18) λ + ν P Nt ( i ) ( H Nt ( i ) ) + δ ( H Nt ( i ) ) (cid:19) ≤ sup N ∈ N E X i ∈ ˆ D σ i (cid:18) λ + ν P N ( H N ) + δ ( H N ) (cid:19) + κ P c λ ( λ + ν ) sup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i (cid:0) P Nt ( i ) (cid:1) + λ ( λ + ν ) (cid:16) η λ + λK (cid:17) sup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i P Nt ( i ) + K δ . (38) Proof.
If the right-hand side of (38) is infinite, then the claim trivially holds. For the remainder of the proofassume the right-hand side of (38) to be finite. Define D := ∅ and for every n ∈ N let D n ⊆ ˆ D be a set with |D n | = min (cid:8) n, | ˆ D| (cid:9) and D n ⊇ D n − . Define c := λ + ν and for all n ∈ N let c n := c ¯ κ P c sup N ∈ N sup t ∈ [0 , ∞ ) E " X i ∈D n σ i (cid:0) P Nt ( i ) (cid:1) + 2 c " η λ + λK sup N ∈ N sup t ∈ [0 , ∞ ) E " X i ∈D n σ i P Nt ( i ) + λK δ . (39)For N, n, l ∈ N define [0 , ∞ ]-valued stopping times τ N,nl := inf ( t ∈ [0 , ∞ ) : X i ∈D n σ i (cid:16) P Nt ( i ) + (cid:0) H Nt ( i ) (cid:1) − (cid:17) > l ) ∪ ∞ ! . (40)We infer from Lemma 2.2 that for all N, n ∈ N the map [0 , ∞ ) ∋ t P i ∈D n σ i (cid:16) P Nt ( i ) + (cid:0) H Nt ( i ) (cid:1) − (cid:17) ∈ R is P -a.s. continuous. Thereby, we have for all t ∈ [0 , ∞ ) and all N, n ∈ N that P (cid:20) lim l →∞ τ N,nl < t (cid:21) = 0 . (41)For all t ∈ [0 , ∞ ), N, n, l ∈ N applying Young’s inequality we get E " X i ∈D n σ i Z t ∧ τ N,nl (cid:18) c √ β NP P Nu ( i )( H Nu ( i )) (cid:19) du ≤ E " X i ∈D n σ i min k ∈D n { σ k } t sup u ∈ [0 ,t ] c ¯ β P (cid:18) (cid:16) P Nu ∧ τ N,nl ( i ) (cid:17) + (cid:16) H Nu ∧ τ N,nl ( i ) (cid:17) − (cid:19) ≤ t c ¯ β P min k ∈D n { σ k } E sup u ∈ [0 ,t ] X i ∈D n σ i (cid:18) P Nu ∧ τ N,nl ( i ) + (cid:16) H Nu ∧ τ N,nl ( i ) (cid:17) − (cid:19)! ≤ t c ¯ β P min k ∈D n { σ k } l < ∞ (42)and E " X i ∈D n σ i Z t ∧ τ N,nl (cid:18)(cid:0) c P Nu ( i ) + δ (cid:1) √ β NH H Nu ( i )( H Nu ( i )) (cid:19) du ≤ E " X i ∈D n σ i min k ∈D n { σ k } t sup u ∈ [0 ,t ] (4 c + δ ) ¯ β H (cid:18) (cid:16) P Nu ∧ τ N,nl ( i ) + 1 (cid:17) + (cid:16) H Nu ∧ τ N,nl ( i ) (cid:17) − (cid:19) ≤ t (4 c + δ ) ¯ β H min k ∈D n { σ k } E sup u ∈ [0 ,t ] X i ∈D n σ i (cid:18) P Nu ∧ τ N,nl ( i ) + 1 + (cid:16) H Nu ∧ τ N,nl ( i ) (cid:17) − (cid:19)! ≤ t (4 c + δ ) ¯ β H min k ∈D n { σ k } ( l + k k σ ) < ∞ . (43)12ence, we obtain for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E " X i ∈D n σ i Z t ∧ τ N,nl c √ β NP P Nu ( i )( H Nu ( i )) dW P,Nu ( i ) = 0 , E " X i ∈D n σ i Z t ∧ τ N,nl (cid:0) c P Nu ( i ) + δ (cid:1) √ β NH H Nu ( i )( H Nu ( i )) dW H,Nu ( i ) = 0 . (44)Define the function y : N × N × [0 , ∞ ) → [0 , ∞ ] by N × N × [0 , ∞ ) ∋ ( N, n, t ) y N,nt := E " X i ∈D n σ i (cid:18) c P Nt ( i ) ( H Nt ( i ) ) + δ ( H Nt ( i ) ) (cid:19) . (45)Recall from the beginning of the proof that we assume for all N, n ∈ N that y N,n < ∞ . Now, applying Itˆo’slemma and using (44), we obtain for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E " X i ∈D n σ i c P Nt ∧ τN,nl ( i )( H Nt ∧ τN,nl ( i )) + δ H Nt ∧ τN,nl ( i )) ! − y N,n = E " X i ∈D n σ i Z t ∧ τ N,nl c H Nu ( i )) κ NP X j ∈D m ( i, j ) P Nu ( j ) − ( κ NP + ν ) P Nu ( i ) − γ (cid:0) P Nu ( i ) (cid:1) + (cid:0) η − ρF Nu ( i ) (cid:1) P Nu ( i ) H Nu ( i ) + ι NP ! − (cid:16) c P Nu ( i )( H Nu ( i )) + δ H Nu ) (cid:17) κ NH X j ∈D m ( i, j ) H Nu ( j )+ ( − κ NH + λ − α N F Nu ( i )) H Nu ( i ) − λK (cid:0) H Nu ( i ) (cid:1) − δP Nu ( i ) H Nu ( i ) + ι NH ! + c P Nu ( i )( H Nu ( i )) β NH H Nu ( i ) +
12 12 δ H Nu ( i )) β NH H Nu ( i ) du . (46)Dropping some negative terms, we now get for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E " X i ∈D n σ i c P Nt ∧ τN,nl ( i )( H Nt ∧ τN,nl ( i )) + δ H Nt ∧ τN,nl ( i )) ! − y N,n ≤ E " X i ∈D n σ i Z t ∧ τ N,nl c ( H Nu ( i )) κ NP X j ∈D m ( i, j ) P Nu ( j ) − νP Nu ( i ) − γ (cid:0) P Nu ( i ) (cid:1) + ηP Nu ( i ) H Nu ( i ) + ι NP − (cid:16) c P Nu ( i )( H Nu ( i )) + δ H Nu ) (cid:17) (cid:16) ( − κ NH + λ − α N ) H Nu ( i ) − λK (cid:0) H Nu ( i ) (cid:1) − δP Nu ( i ) H Nu ( i ) + ι NH (cid:17) + 6 c P Nu ( i )( H Nu ( i )) β NH + δ H Nu ( i )) β NH du = E " X i ∈D n σ i Z t ∧ τ N,nl c κ NP H Nu ( i )) X j ∈D m ( i, j ) P Nu ( j ) − ν P Nu ( i )( H Nu ( i )) − γ ( P Nu ( i ) ) ( H Nu ( i )) + η P Nu ( i ) H Nu ( i ) + ι NP H Nu ( i )) − (cid:0) − κ NH + λ − α N (cid:1) P Nu ( i )( H Nu ( i )) + 2 λK P Nu ( i ) H Nu ( i ) + 2 δ ( P Nu ( i ) ) ( H Nu ( i )) − ι NH P Nu ( i )( H Nu ( i )) + 3 P Nu ( i )( H Nu ( i )) β NH ! + κ NH − λ + α N δ H Nu ) + λKδ H Nu + P Nu ( i )( H Nu ) − ι NH δ H Nu ) + β NH δ H Nu ( i )) du . (47)13sing Young’s inequality as well as Lemma 2.4 we get for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E " X i ∈D n σ i c P Nt ∧ τN,nl ( i )( H Nt ∧ τN,nl ( i )) + δ H Nt ∧ τN,nl ( i )) ! − y N,n ≤ E " X i ∈D n σ i Z t ∧ τ N,nl c κ NP H Nu ( i )) + κ NP c (cid:0) P Nu ( i ) (cid:1) − ν P Nu ( i )( H Nu ( i )) − γ ( P Nu ( i ) ) ( H Nu ( i )) + λ η η P Nu ( i )( H Nu ( i )) +
12 2 ηλ ηP Nu ( i ) + ι NP H Nu ( i )) − − κ NH + λ − α N ) P Nu ( i )( H Nu ( i )) + K λK P Nu ( i )( H Nu ( i )) +
12 4 K λK P Nu ( i ) + 2 δ ( P Nu ( i ) ) ( H Nu ( i )) − ι NH P Nu ( i )( H Nu ( i )) + 3 P Nu ( i )( H Nu ( i )) β NH ! + κ NH − λ + α N δ H Nu ) + K λKδ H Nu ) +
12 2
K λKδ + P Nu ( i )( H Nu ) − ι NH δ H Nu ) + β NH δ H Nu ( i )) du = E " X i ∈D n σ i Z t ∧ τ N,nl (cid:2) c κ NP − δ ι NH + δ β NH (cid:3) H Nu ( i )) + c κ NP c (cid:0) P Nu ( i ) (cid:1) + h c (cid:16) − ν + λ + 4( κ NH − λ + α N ) + λ (cid:17) + 1 i P Nu ( i )( H Nu ( i )) + 2 c [ − γ + 2 δ ] ( P Nu ( i ) ) ( H Nu ( i )) + 2 c h η λ + λK i P Nu ( i )+ h c ι NP + κ NH − λ + α N δ + λ δ i H Nu ( i )) + 2 c (cid:2) − ι NH + 3 β NH (cid:3) P Nu ( i )( H Nu ( i )) + λK δ du . (48)Recall ¯ κ P = sup N ∈ N κ NP and that for all N ∈ N we have α N + κ NH ≤ λ , ι NP ≤ λ ( ν + λ )8 δ , and ι NH ≥ δκ NP ν + λ ) + β NH .Furthermore, note that λ ≤ c . Together with the assumption that γ ≥ δ we see for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E " X i ∈D n σ i c P Nt ∧ τN,nl ( i )( H Nt ∧ τN,nl ( i )) + δ H Nt ∧ τN,nl ( i )) ! − y N,n ≤ E " X i ∈D n σ i Z t ∧ τ N,nl c ¯ κ P c (cid:0) P Nu ( i ) (cid:1) − P Nu ( i )( H Nu ( i )) + 2 c h η λ + λK i P Nu ( i ) − λ δ H Nu ( i )) + λK δ du ≤ Z t c n du − E " X i ∈D n σ i Z t ∧ τ N,nl λ (cid:16) c P Nu ( i )( H Nu ( i )) + δ H Nu ( i )) (cid:17) du . (49)Using Tonelli’s theorem, Fatou’s lemma, and (41) this implies for all t ∈ [0 , ∞ ) and all N, n ∈ N that y N,nt + Z t λ y N,nu du = y N,nt + E " X i ∈D n σ i Z t λ (cid:16) c P Nu ( i )( H Nu ( i )) + δ H Nu ( i )) (cid:17) du ≤ lim inf l →∞ E " X i ∈D n σ i c P Nt ∧ τN,nl ( i )( H Nt ∧ τN,nl ( i )) + δ H Nt ∧ τN,nl ( i )) ! + E " X i ∈D n σ i Z t ∧ τ N,nl λ (cid:16) c P Nu ( i )( H Nu ( i )) + δ H Nu ( i )) (cid:17) du ≤ y N,n + Z t c n du. (50)For every N, n ∈ N let z N,n : [0 , ∞ ) → R be a process that for all t ∈ [0 , ∞ ) satisfies z N,nt = z N,n + R t (cid:0) c n − λ z N,ns (cid:1) ds with z N,n = y N,n , where uniqueness follows from local Lipschitz continuity. Due to classical com-parison results of the theory of ODEs, the above computation yields for all t ∈ [0 , ∞ ) and all N, n ∈ N that14 N,nt ≤ z N,nt and for all
N, n ∈ N that sup t ∈ [0 , ∞ ) z N,nt = max n z N,n , c n λ o . We obtain for all n ∈ N thatsup N ∈ N sup t ∈ [0 , ∞ ) y N,nt ≤ sup N ∈ N sup t ∈ [0 , ∞ ) z N,nt = max (cid:26) sup N ∈ N z N,n , c n λ (cid:27) ≤ sup N ∈ N E " X i ∈D n σ i (cid:18) c P N ( H N ) + δ ( H N ) (cid:19) + c n λ . (51)Using monotone convergence we thereby concludesup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i (cid:18) λ + ν P Nt ( i ) ( H Nt ( i ) ) + δ ( H Nt ( i ) ) (cid:19) ≤ lim n →∞ sup N ∈ N sup t ∈ [0 , ∞ ) y N,nt ≤ lim n →∞ sup N ∈ N E " X i ∈D n σ i (cid:18) c P N ( H N ) + δ ( H N ) (cid:19) + c n λ ! ≤ sup N ∈ N E X i ∈ ˆ D σ i (cid:18) λ + ν P N ( H N ) + δ ( H N ) (cid:19) + κ P c λ ( λ + ν ) sup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i (cid:0) P Nt ( i ) (cid:1) + λ ( λ + ν ) (cid:16) η λ + λK (cid:17) sup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i P Nt ( i ) + K δ , (52)finishing the proof. Lemma 2.7.
Assume the setting of Section 2.1 and assume λ > ν and η − ρ > λK . Furthermore, assume thatfor all N ∈ N we have ι NH ≥ β NH , κ NP + κ NH + α N ≤ λ − ν , ι NP ≥ β NP , and ι NH ≥ β NH . Let ˆ D ⊆ D be a set. Thenwe have sup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i (cid:16) ( η − ρ ) − λK κ P + ν ) 1 P Nt ( i ) + P Nt ( i ) H Nt ( i ) (cid:17) ≤ sup N ∈ N E X i ∈ ˆ D σ i (cid:16) ( η − ρ ) − λK κ P + ν ) 1 P N ( i ) + P N ( i ) H N ( i ) (cid:17) + { ¯ κ P + ν, λ − ν } γ ( η − ρ ) − λK κ P + ν ) + ( γ + δ ) sup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i H Nt ( i ) . (53) Proof.
If the right-hand side of (53) is infinite, then the claim trivially holds. For the remainder of the proofassume the right-hand side of (53) to be finite. Define D := ∅ and for every n ∈ N let D n ⊆ ˆ D be a set with |D n | = min n n, | ˆ D| o and D n ⊇ D n − . Define c := κ P + ν ) (cid:2) ( η − ρ ) − λK (cid:3) and for every n ∈ N let C n := γc + " γ + δ sup N ∈ N sup t ∈ [0 , ∞ ) E " X i ∈D n σ i H Nt ( i ) . (54)Note that due to the assumption η − ρ > λK we have c ∈ (0 , ∞ ). For all N, n, l ∈ N define [0 , ∞ ]-valuedstopping times τ N,nl := inf ( t ∈ [0 , ∞ ) : X i ∈D n σ i (cid:16)(cid:0) P Nt ( i ) (cid:1) − + (cid:0) H Nt ( i ) (cid:1) − (cid:17) > l ) ∪ ∞ ! . (55)We infer from Lemmas 2.2 and 2.3 that for all N, n ∈ N the map [0 , ∞ ) ∋ t P i ∈D n σ i (cid:16)(cid:0) P Nt ( i ) (cid:1) − + (cid:0) H Nt ( i ) (cid:1) − (cid:17) ∈ R is P -a.s. continuous which implies that we have for all t ∈ [0 , ∞ ) and all N, n ∈ N that P (cid:20) lim l →∞ τ N,nl < t (cid:21) = 0 . (56)15or all t ∈ [0 , ∞ ), N, n, l ∈ N applying Young’s inequality we see that E " X i ∈D n σ i Z t ∧ τ N,nl (cid:18) √ β NP P Nu ( i )( P Nu ( i )) (cid:16) c + H Nu ( i ) (cid:17)(cid:19) du ≤ ¯ β P E " t sup u ∈ [0 ,t ] X i ∈D n σ i min k ∈D n { σ k } (cid:16) P Nu ∧ τ N,nl ( i ) (cid:17) − + (cid:18) c + (cid:16) H Nu ∧ τ N,nl ( i ) (cid:17) − (cid:19) ! du ≤ ¯ β P min k ∈D n { σ k } E t sup u ∈ [0 ,t ] X i ∈D n σ i (cid:18)(cid:16) P Nu ∧ τ N,nl ( i ) (cid:17) − + (cid:16) H Nu ∧ τ N,nl ( i ) (cid:17) − + c (cid:19)! ≤ ¯ β P t ( l + c k k σ ) min k ∈D n { σ k } < ∞ (57)and E " X i ∈D n σ i Z t ∧ τ N,nl (cid:18) √ β NH H Nu ( i )( H Nu ( i )) P Nu ( i ) (cid:19) du ≤ ¯ β H E " t sup u ∈ [0 ,t ] X i ∈D n σ i min k ∈D n { σ k } (cid:18) (cid:16) H Nu ∧ τ N,nl ( i ) (cid:17) − + (cid:16) P Nu ∧ τ N,nl ( i ) (cid:17) − (cid:19) du ≤ t sup u ∈ [0 ,t ] ¯ β H min k ∈D n { σ k } E X i ∈D n σ i (cid:18)(cid:16) H Nu ∧ τ N,nl ( i ) (cid:17) − + (cid:16) P Nu ∧ τ N,nl ( i ) (cid:17) − (cid:19)! ≤ t ¯ β H l min k ∈D n { σ k } < ∞ . (58)Hence, we obtain for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E "Z t ∧ τ N,nl X i ∈D n σ i q β NP P Nt ( i ) ( P Nt ( i ) ) (cid:16) c + H Nt ( i ) (cid:17) dW P,Nu ( i ) = 0 , E "Z t ∧ τ N,nl X i ∈D n σ i √ β NH H Nt ( i ) ( H Nt ( i ) ) P Nt ( i ) dW H,Nu ( i ) = 0 . (59)Define the function y : N × N × [0 , ∞ ) → [0 , ∞ ] by N × N × [0 , ∞ ) ∋ ( N, n, t ) y N,nt := E " X i ∈D n σ i (cid:16) c P Nt ( i ) + P Nt ( i ) H Nt ( i ) (cid:17) . (60)Recall from the beginning of the proof that we assume for all N, n ∈ N that y N,n < ∞ . Applying Itˆo’s lemmaand using (59), we get for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E " X i ∈D n σ i c P Nt ∧ τN,nl ( i ) + P Nt ∧ τN,nl ( i ) H Nt ∧ τN,nl ( i ) ! − y N,n = E " X i ∈D n σ i Z t ∧ τ N,nl − (cid:16) c P Nu ( i )) + P Nu ( i )) H Nu ( i ) (cid:17) κ NP X j ∈D m ( i, j ) P Nu ( j ) − ( κ NP + ν ) P Nu ( i ) − γ (cid:0) P Nu ( i ) (cid:1) + (cid:0) η − ρF Nu ( i ) (cid:1) P Nu ( i ) H Nu ( i ) + ι NP ! + c P Nu ( i )) β NP P Nu ( i )+
12 2( P Nu ( i )) H Nu ( i ) β NP P Nu ( i ) − P Nu ( i )( H Nu ( i )) κ NH X j ∈D m ( i, j ) H Nu ( j ) + ( − κ NH + λ − α N F Nu ( i )) H Nu ( i ) − λK (cid:0) H Nu ( i ) (cid:1) − δP Nu ( i ) H Nu ( i ) + ι NH ! +
12 2 P Nu ( i )( H Nu ( i )) β NH H Nu ( i ) du . (61)16ropping some negative terms, we now get for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E " X i ∈D n σ i c P Nt ∧ τN,nl ( i ) + P Nt ∧ τN,nl ( i ) H Nt ∧ τN,nl ( i ) ! − y N,n ≤ E " X i ∈D n σ i Z t ∧ τ N,nl − (cid:16) c P Nu ( i )) + P Nu ( i )) H Nu ( i ) (cid:17) − ( κ NP + ν ) P Nu ( i ) − γ (cid:0) P Nu ( i ) (cid:1) + ( η − ρ ) P Nu ( i ) H Nu ( i ) + ι NP ! + c β NP P Nu ( i )) + β NP P Nu ( i )) H Nu ( i ) + β NH P Nu ( i )( H Nu ( i )) − P Nu ( i )( H Nu ( i )) (cid:16) ( − κ NH + λ − α N ) H Nu ( i ) − λK (cid:0) H Nu ( i ) (cid:1) − δP Nu ( i ) H Nu ( i ) + ι NH (cid:17) du = E " X i ∈D n σ i Z t ∧ τ N,nl (cid:0) κ NP + ν (cid:1) c P Nu ( i ) + γc − ( η − ρ ) c H Nu ( i ) P Nu ( i ) − ι NP c P Nu ( i )) + ( κ NP + ν ) P Nu ( i ) H Nu ( i ) + γ H Nu ( i ) − ( η − ρ ) P Nu ( i ) − ι NP P Nu ( i )) H Nu ( i ) + c β NP P Nu ( i )) + β NP P Nu ( i )) H Nu ( i ) + β NH P Nu ( i )( H Nu ( i )) − ( − κ NH + λ − α N ) P Nu ( i ) H Nu ( i ) + λK P Nu ( i ) + δ H Nu ( i ) − ι NH P Nu ( i )( H Nu ( i )) du = E " X i ∈D n σ i Z t ∧ τ N,nl (cid:2)(cid:0) κ NP + ν (cid:1) c − ( η − ρ ) + λK (cid:3) P Nu ( i ) + γc − ( η − ρ ) c H Nu ( i ) P Nu ( i ) + (cid:2) − ι NP c + c β NP (cid:3) P Nu ( i )) + (cid:2) ( κ NP + ν ) − ( − κ NH + λ − α N ) (cid:3) P Nu ( i ) H Nu ( i ) + [ γ + δ ] H Nu ( i ) + (cid:2) − ι NP + β NP (cid:3) P Nu ( i )) H Nu ( i ) + (cid:2) β NH − ι NH (cid:3) P Nu ( i )( H Nu ( i )) du . (62)Recall from Section 2.1 that ¯ κ P = sup N ∈ N κ NP , and from Assumption 1.1 that λ > ν , η − ρ > λK and that forall N ∈ N we have κ NP + κ NH + α N ≤ λ − ν , ι NP ≥ β NP , and ι NH ≥ β NH . Hence, we get for all t ∈ [0 , ∞ ) and all N, n, l ∈ N that E " X i ∈D n σ i c P Nt ∧ τN,nl ( i ) + P Nt ∧ τN,nl ( i ) H Nt ∧ τN,nl ( i ) ! − y N,n ≤ E " X i ∈D n σ i Z t ∧ τ N,nl − (¯ κ P + ν ) c P Nu ( i ) + γc − λ − ν P Nu ( i ) H Nu ( i ) + [ γ + δ ] H Nu ( i ) du ≤ Z t C n du − E "Z t ∧ τ N,nl min (cid:8) ¯ κ P + ν, λ − ν (cid:9) X i ∈D n σ i (cid:16) c P Nu ( i ) + P Nu ( i ) H Nu ( i ) (cid:17) du . (63)Applying Tonelli’s theorem, Fatou’s lemma, and (56) we obtain for all t ∈ [0 , ∞ ) and all N, n ∈ N that y N,nt + Z t min (cid:8) ¯ κ P + ν, λ − ν (cid:9) y N,nu du = y N,nt + E "Z t min (cid:8) ¯ κ P + ν, λ − ν (cid:9) X i ∈D n σ i (cid:16) c P Nu ( i ) + P Nu ( i ) H Nu ( i ) (cid:17) du ≤ lim inf l →∞ E " X i ∈D n σ i c P Nt ∧ τN,nl ( i ) + P Nt ∧ τN,nl ( i ) H Nt ∧ τN,nl ( i ) ! + E "Z t ∧ τ N,nl min (cid:8) ¯ κ P + ν, λ − ν (cid:9) X i ∈D n σ i (cid:16) c P Nu ( i ) + P Nu ( i ) H Nu ( i ) (cid:17) du ≤ y N,n + Z t C n du. (64)17or every N, n ∈ N , let z N,n : [0 , ∞ ) → R be a process that for all t ∈ [0 , ∞ ) satisfies z N,nt = z N,n + R t (cid:0) C n − min (cid:8) ¯ κ P + ν, λ − ν (cid:9) z N,ns (cid:1) ds , with z N,n = y N,n , where uniqueness follows from local Lipschitz continuity. Usingclassical comparison results from the theory of ODEs, the above computation yields for all t ∈ [0 , ∞ ) and all N, n ∈ N that y N,nt ≤ z N,nt and for all
N, n ∈ N that sup t ∈ [0 , ∞ ) z N,nt = max n z N,n , C n min { ¯ κ P + ν, λ − ν } o . Hence, weobtain for every n ∈ N thatsup N ∈ N sup t ∈ [0 , ∞ ) E " X i ∈D n σ i (cid:16) c P Nt ( i ) + P Nt ( i ) H Nt ( i ) (cid:17) = sup N ∈ N sup t ∈ [0 , ∞ ) y N,nt ≤ sup N ∈ N sup t ∈ [0 , ∞ ) z N,nt = max (cid:26) sup N ∈ N z N,n , C n min { ¯ κ P + ν, λ − ν } (cid:27) ≤ sup N ∈ N E " X i ∈D n σ i (cid:16) c P N ( i ) + P N ( i ) H N ( i ) (cid:17) + C n min { ¯ κ P + ν, λ − ν } . (65)Using monotone convergence, we thereby conclude thatsup N ∈ N sup t ∈ [0 , ∞ ) E X i ∈ ˆ D σ i (cid:16) c P Nt ( i ) + P Nt ( i ) H Nt ( i ) (cid:17) = lim n →∞ sup N ∈ N sup t ∈ [0 , ∞ ) E " X i ∈D n σ i (cid:16) c P Nt ( i ) + P Nt ( i ) H Nt ( i ) (cid:17) ≤ lim n →∞ sup N ∈ N E " X i ∈D n σ i (cid:16) c P N ( i ) + P N ( i ) H N ( i ) (cid:17) + C n min { ¯ κ P + ν, λ − ν } ! = sup N ∈ N E X i ∈ ˆ D σ i (cid:16) c P N ( i ) + P N ( i ) H N ( i ) (cid:17) + γc +( γ + δ ) sup N ∈ N sup t ∈ [0 , ∞ ) E (cid:20)P i ∈ ˆ D σ i H Nt ( i ) (cid:21) min { ¯ κ P + ν, λ − ν } , (66)finishing the proof. Theorem 2.8.
Assume the setting of Section 2.1 and let Assumption 1.1 hold. Then for all ( x, y, z ) ∈ (0 , ∞ ) × [0 , it holds that u ( x, y, z ) := ( η − ρz ) (cid:16) x − h ∞ ( z ) − h ∞ ( z ) ln (cid:16) xh ∞ ( z ) (cid:17)(cid:17) + δ (cid:16) y − p ∞ ( z ) − p ∞ ( z ) ln (cid:16) yp ∞ ( z ) (cid:17)(cid:17) ≥ . (67) Furthermore, there exists a constant c ∈ (0 , ∞ ) such that for every set ˆ D ⊆ D , for every N ∈ N , and every t ∈ [0 , ∞ ) it holds that E " X i ∈ ˆ D σ i u (cid:0) H Nt ( i ) , P Nt ( i ) , F Nt ( i ) (cid:1) + Z t ( η − ρ ) λK E X i ∈ ˆ D σ i (cid:0) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) + δγ E X i ∈ ˆ D σ i (cid:0) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) du ≤ E " X i ∈ ˆ D σ i u (cid:0) H N ( i ) , P N ( i ) , F N ( i ) (cid:1) + tc max (cid:8) κ NH , κ NP , α N , ι NH , ι NP , β NH , β NP (cid:9) . (68) Proof.
For the remainder of the proof fix a set ˆ
D ⊆ D . Define D := ∅ and for every n ∈ N let D n ⊆ ˆ D be a set with |D n | = min n n, | ˆ D| o and D n ⊇ D n − . We will first show that for all ( x, y, z ) ∈ (0 , ∞ ) × [0 , u ( x, y, z ) ≥
0. Define for all x ∈ (0 , ∞ ) the real-valued function (0 , ∞ ) ∋ y f x ( y ) := x − y − y ln (cid:16) xy (cid:17) . For all x ∈ (0 , ∞ ) the function f x has for all y ∈ (0 , ∞ ) first and second order derivatives df x dy ( y ) = ln( y ) − ln( x ) and d f x dy ( y ) = y >
0. Thus, for all x ∈ (0 , ∞ ) the function f x has its global minimum at x with f x ( x ) = 0. Consequently, for any ( x, y ) ∈ (0 , ∞ ) we have f x ( y ) ≥ f x ( x ) = 0. This shows that for all18 x, y, z ) ∈ (0 , ∞ ) × [0 ,
1] we have that u ( x, y, z ) ≥
0. In order to prove the second part of the claim, we willmake use of a Lyapunov function that is defined here analogously to the well-known Lyapunov function in thedeterministic setting. Define D V := (cid:16) l σ ∩ (0 , ∞ ) D (cid:17) × (cid:16) l σ ∩ (0 , ∞ ) D (cid:17) × E . For any subset ˆ D ′ ⊆ ˆ D define thefunction V ˆ D ′ : D V → [0 , ∞ ] for any ( h, p, f ) ∈ D V by V ˆ D ′ (( h, p, f )) := X i ∈ ˆ D ′ σ i u ( h i , p i , f i ) . (69)Due to the non-negativity of the mapping u , we obtain for any ˆ D ′ ⊆ ˆ D and any z ∈ D V that V ˆ D ′ ( z ) ∈ [0 , ∞ ] iswell-defined. From the fact that for all x ∈ (0 , ∞ ) we have − ln( x ) ≤ q x ≤ (cid:0) x + 1 (cid:1) as well as the assumptionsup N ∈ N E h(cid:13)(cid:13)(cid:13)(cid:0) H N + P N (cid:1) + H N ) + P N ( H N ) + P N + P N H N (cid:13)(cid:13)(cid:13) σ i < ∞ we obtainsup N ∈ N E (cid:2) V D (cid:0) H N , P N , F N (cid:1)(cid:3) < ∞ . (70)We now calculate the first and second order partial derivatives that we will need in the application of Itˆo’slemma below. For all n ∈ N , z = ( h, p, f ) ∈ D V , and i ∈ D n we get dV D n dh i ( z ) = σ i ( η − ρf i ) (cid:16) − h ∞ ( f i ) h i (cid:17) , d V D n dh i ( z ) = σ i ( η − ρf i ) h ∞ ( f i ) h i , dV D n dp i ( z ) = σ i δ (cid:16) − p ∞ ( f i ) p i (cid:17) , and d V D n dp i ( z ) = σ i δ p ∞ ( f i ) p i as well as dV D n df i ( z ) = σ i " − ρ (cid:16) h i − h ∞ ( f i ) − h ∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17)(cid:17) + ( η − ρf i ) − h ′∞ ( f i ) − h ′∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17) − ( h ∞ ( f i )) h i − h i ( h ∞ ( f i )) h ′∞ ( f i ) ! + δ (cid:16) − p ′∞ ( f i ) − p ′∞ ( f i ) ln (cid:16) p i p ∞ ( f i ) (cid:17) − ( p ∞ ( f i )) p i − p i ( p ∞ ( f i )) p ′∞ ( f i ) (cid:17) = σ i " − ρ (cid:16) h i − h ∞ ( f i ) − h ∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17)(cid:17) − ( η − ρf i ) h ′∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17) − δp ′∞ ( f i ) ln (cid:16) p i p ∞ ( f i ) (cid:17) (71)and d V D n df i ( z ) = σ i " ρ (cid:16) h ′∞ ( f i ) + h ′∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17) + ( h ∞ ( f i )) h i ( − h i ( h ∞ ( f i )) h ′∞ ( f i ) (cid:17) + ρh ′∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17) − ( η − ρf i ) (cid:16) h ′′∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17) + h ′∞ ( f i ) h ∞ ( f i ) h i − h i ( h ∞ ( f i )) h ′∞ ( f i ) (cid:17) − δ (cid:16) p ′′∞ ( f i ) ln (cid:16) p i p ∞ ( f i ) (cid:17) + p ′∞ ( f i ) p ∞ ( f i ) p i − p i ( p ∞ ( f i )) p ′∞ ( f i ) (cid:17) = σ i " ρh ′∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17) − ( η − ρf i ) (cid:16) h ′′∞ ( f i ) ln (cid:16) h i h ∞ ( f i ) (cid:17) − ( h ′∞ ( f i )) h ∞ ( f i ) (cid:17) − δ (cid:16) p ′′∞ ( f i ) ln (cid:16) p i p ∞ ( f i ) (cid:17) − ( p ′∞ ( f i )) p ∞ ( f i ) (cid:17) . (72)Recall that we have for all x ∈ [0 ,
1] that h ∞ ( x ) = b ( a − x ) and p ∞ ( x ) = λδ (cid:16) − Kb ( a − x ) (cid:17) and note that theassumption that η − ρ > νK implies for all x ∈ [0 ,
1] that p ∞ ( x ) >
0. Therefore, we get for all x ∈ [0 , h ′∞ ( x ) = b ( a − x ) > , h ′′∞ ( x ) = b ( a − x ) > ,p ′∞ ( x ) = − λδKb ( a − x ) < , p ′′∞ ( x ) = − λδKb ( a − x ) < . (73)So h ∞ , h ′∞ , and h ′′∞ are strictly monotonically increasing on [0 ,
1] while p ∞ , p ′∞ , and p ′′∞ are strictly mono-tonically decreasing on [0 , x ∈ [0 , δp ∞ ( x ) ≤ λ . Observe that for all x ∈ (0 , ∞ ) we have19 ln( x ) | ≤ √ x + √ x . Together with Young’s inequality as well as Lemmas 2.5, 2.6, and 2.7 we get for all t ∈ [0 , ∞ )and all N, n ∈ N that E " X i ∈D n σ i Z t (cid:18)q β NP P Nu ( i ) δ (cid:16) − p ∞ ( F Nu ( i )) P Nu ( i ) (cid:17)(cid:19) du ≤ ¯ β P δ E " X i ∈D n σ i Z t P Nu ( i ) (cid:16) ( p ∞ (0)) ( P Nu ( i )) (cid:17) du ≤ ¯ β P δ sup u ∈ [0 ,t ] E " X i ∈D n σ i t (cid:16) P Nu ( i ) + ( p ∞ (0)) P Nu ( i ) (cid:17) < ∞ (74)and E " X i ∈D n σ i Z t (cid:18)q β NH H Nu ( i )( η − ρF Nu ( i )) (cid:16) − h ∞ ( F Nu ( i )) H Nu ( i ) (cid:17)(cid:19) du ≤ ¯ β H η sup u ∈ [0 ,t ] E " X i ∈D n σ i t (cid:16) H Nu ( i ) + ( h ∞ (1)) H Nu ( i ) (cid:17) < ∞ (75)and E " X i ∈D n σ i Z t r β NH F Nu ( i ) ( − F Nu ( i ) ) H Nu ( i ) − ρ (cid:16) H Nu ( i ) − h ∞ ( F Nu ( i )) − h ∞ ( F Nu ( i )) ln (cid:16) H Nu ( i ) h ∞ ( F Nu ( i )) (cid:17)(cid:17) − ( η − ρF Nu ( i )) h ′∞ ( F Nu ( i )) ln (cid:16) H Nu ( i ) h ∞ ( F Nu ( i )) (cid:17) − δp ′∞ ( F Nu ( i )) ln (cid:16) P Nu ( i ) p ∞ ( F Nu ( i )) (cid:17) !! du ≤ ¯ β H E " X i ∈D n σ i Z t H Nu ( i ) ρH Nu ( i ) + ρh ∞ (1) + ρh ∞ (1) (cid:18) √ H Nu ( i ) √ h ∞ (0) + √ h ∞ (1) √ H Nu ( i ) (cid:19) + ηh ′∞ (1) (cid:18) √ H Nu ( i ) √ h ∞ (0) + √ h ∞ (1) √ H Nu ( i ) (cid:19) + δ | p ′∞ (1) | (cid:18) √ P Nu ( i ) √ p ∞ (1) + √ p ∞ (0) √ P Nu ( i ) (cid:19) ! du ≤ ¯ β H sup u ∈ [0 ,t ] E " X i ∈D n σ i ρ H Nu ( i ) + ρ ( h ∞ (1)) (cid:16) H Nu ( i ) + h ∞ (0) + h ∞ (1)( H Nu ( i )) (cid:17) + η ( h ′∞ (1)) (cid:16) h ∞ (0) + h ∞ (1)( H Nu ( i )) (cid:17) + δ ( p ′∞ (1)) (cid:16) p ∞ (1) P Nu ( i ) H Nu ( i ) + p ∞ (0) P Nu ( i ) H Nu ( i ) (cid:17) ! < ∞ . (76)Hence, we obtain for all t ∈ [0 , ∞ ) and all N, n ∈ N that E "Z t X i ∈D n σ i q β NP P Nu ( i ) δ (cid:16) − p ∞ ( F Nu ( i )) P Nu ( i ) (cid:17) dW P,Nu ( i ) = 0 , E "Z t X i ∈D n σ i q β NH H Nu ( i )( η − ρF Nu ( i )) (cid:16) − h ∞ ( F Nu ( i )) H Nu ( i ) (cid:17) dW H,Nu ( i ) = 0 , E "Z t X i ∈D n σ i r β NH F Nu ( i ) ( − F Nu ( i ) ) H Nu ( i ) " − ρ (cid:16) H Nu ( i ) − h ∞ ( F Nu ( i )) − h ∞ ( F Nu ( i )) ln (cid:16) H Nu ( i ) h ∞ ( F Nu ( i )) (cid:17)(cid:17) − ( η − ρF Nu ( i )) h ′∞ ( F Nu ( i )) ln (cid:16) H Nu ( i ) h ∞ ( F Nu ( i )) (cid:17) − δp ′∞ ( F Nu ( i )) ln (cid:16) P Nu ( i ) p ∞ ( F Nu ( i )) (cid:17) dW F,Nu ( i ) = 0 . (77)20or all t ∈ [0 , ∞ ), all N ∈ N , and all i ∈ D define R Nt ( i ) := max ( max n ηc, ρc, ρ, cη h ′∞ (1) h ∞ (0) , η h ′∞ (1) h ∞ (0) o H Nt ( i ) , ηh ∞ (1) , η, max n η h ∞ (1) , ρ ( h ∞ (1)) , η ( h ′∞ (1) ) h ∞ (0) , δ ( p ′∞ (1) ) p ∞ (1) o H Nt ( i ) , δcP Nt ( i ) , δp ∞ (0) , δ, δ p ∞ (0) P Nt ( i ) , max n ρ ( h ∞ (1)) , ρ ( h ∞ (1)) , (cid:16) ηh ′∞ (1) p h ∞ (1) (cid:17) , , η h ′′∞ (1) h ∞ (1) o ( H Nt ( i ) ) , c (cid:0) H Nt ( i ) (cid:1) , c (cid:0) H Nt ( i ) (cid:1) , (cid:18) δ | p ′∞ (1) |√ p ∞ (1) (cid:19) P Nt ( i ) ( H Nt ( i ) ) , δ | p ′∞ (1) | p ∞ (0) P Nt ( i ) H Nt ( i ) ,δ | p ′∞ (1) | q p ∞ (0) P Nt ( i ) , ρ h ′∞ (1) h ∞ (0) , δ | p ′′∞ (1) | p ∞ (1) P Nt ( i ) H Nt ( i ) , ) ,b N := max (cid:8) κ NH , κ NP , α N , ι NH , ι NP , β NH , β NP (cid:9) . (78)Note that lim N →∞ b N = 0. Define c := 32 sup M ∈ N sup u ∈ [0 , ∞ ) E (cid:2)(cid:13)(cid:13) R Mu (cid:13)(cid:13) σ (cid:3) . Observe that due to Lemmas 2.5, 2.6, and 2.7we have c ∈ (0 , ∞ ). For all t ∈ [0 , ∞ ), all N ∈ N , and all a ∈ n η, ρ, η h ′∞ (1) h ∞ (0) o we have that X i ∈D σ i a X j ∈D m ( i, j ) H Nt ( j ) ≤ X i ∈D σ i caH Nt ( i ) ≤ X i ∈D σ i R Nt ( i ) . (79)Furthermore, we have for all t ∈ [0 , ∞ ) and all N ∈ N that X i ∈D σ i δ X j ∈D m ( i, j ) P Nt ( j ) ≤ X i ∈D σ i δcP Nt ( i ) ≤ X i ∈D σ i R Nt ( i ) . (80)Using Young’s inequality and Lemma 2.4 we get for all t ∈ [0 , ∞ ) and all N ∈ N that X i ∈D σ i ρ h ∞ ( F Nt ( i ) ) H Nt ( i ) X j ∈D m ( i, j ) H Nt ( j ) ≤ X i ∈D σ i (cid:16) ρ h ∞ ( F Nt ( i ) ) H Nt ( i ) (cid:17) + (cid:16) X j ∈D m ( i, j ) H Nt ( j ) (cid:17) ! ≤ X i ∈D σ i R Nt ( i ) + c (cid:0) H Nt ( i ) (cid:1) ! ≤ X i ∈D σ i R Nt ( i ) , (81)and X i ∈D σ i − δp ′∞ ( F Nt ( i ) ) q p ∞ ( F Nt ( i ) ) √ P Nt ( i ) H Nt ( i ) X j ∈D m ( i, j ) H Nt ( j ) ≤ X i ∈D σ i (cid:16) δ | p ′∞ (1) |√ p ∞ (1) √ P Nt ( i ) H Nt ( i ) (cid:17) + (cid:16) X j ∈D m ( i, j ) H Nt ( j ) (cid:17) ! ≤ X i ∈D σ i R Nt ( i ) + c (cid:0) H Nt ( i ) (cid:1) ! ≤ X i ∈D σ i R Nt ( i ) , (82)and X i ∈D σ i ( − δp ′∞ (cid:0) F Nt ( i ) (cid:1) q p ∞ ( F Nt ( i ) ) √ P Nt ( i ) H Nt ( i ) X j ∈D m ( i, j ) H Nt ( j ) ≤ X i ∈D σ i δ | p ′∞ (1) | p ∞ (0) P Nt ( i ) H Nt ( i ) + (cid:16) X j ∈D m ( i, j ) H Nt ( j ) (cid:17) H Nt ( i ) ! ≤ X i ∈D σ i R Nt ( i ) + (cid:16) X j ∈D m ( i, j ) H Nt ( j ) (cid:17) +
14 1 ( H Nt ( i ) ) ! ≤ X i ∈D σ i R Nt ( i ) + c (cid:0) H Nt ( i ) (cid:1) + R Nt ( i ) ! ≤ X i ∈D σ i R Nt ( i ) . (83)21gain using Young’s inequality and Lemma 2.4 we get for all a ∈ n ρ (cid:0) h ∞ (1) (cid:1) , ηh ′∞ (1) p h ∞ (1) o , all t ∈ [0 , ∞ ),and all N ∈ N that X i ∈D σ i a (cid:16) H Nt ( i ) (cid:17) X j ∈D m ( i, j ) H Nt ( j ) ≤ X i ∈D σ i a (cid:16) H Nt ( i ) (cid:17) + (cid:16) X j ∈D m ( i, j ) H Nt ( j ) (cid:17) ! ≤ X i ∈D σ i R Nt ( i ) + c (cid:0) H Nt ( i ) (cid:1) ! ≤ X i ∈D σ i R Nt ( i ) . (84)Due to Lemma 2.1 we have that W H,N ( i ), W F,N ( i ), N ∈ N , i ∈ D , are independent Brownian motions and dueto Lemmas 2.5, 2.6, and 2.7 we have for all t ∈ [0 , ∞ ) and all N ∈ N that P -a.s. (cid:0) H Nt , P Nt , F Nt (cid:1) ∈ D V . Thus,applying Itˆo’s lemma and using (77) we obtain for all t ∈ [0 , ∞ ) and all N, n ∈ N that E (cid:2) V D n (cid:0)(cid:0) H Nt , P Nt , F Nt (cid:1)(cid:1)(cid:3) − E (cid:2) V D n (cid:0)(cid:0) H N , P N , F N (cid:1)(cid:1)(cid:3) = E " Z t X i ∈D n σ i (cid:0) η − ρF Nu ( i ) (cid:1) (cid:18) − h ∞ ( F Nu ( i ) ) H Nu ( i ) (cid:19) n κ NH X j ∈D m ( i, j ) (cid:0) H Nu ( j ) − H Nu ( i ) (cid:1) + H Nu ( i ) h λ (cid:16) − H Nu ( i ) K (cid:17) − δP Nu ( i ) − α N F Nu ( i ) i + ι NH o + ( η − ρF Nu ( i ) ) h ∞ ( F Nu ( i ) ) ( H Nu ( i )) β NH H Nu ( i )+ δ (cid:18) − p ∞ ( F Nu ( i ) ) P Nu ( i ) (cid:19) n κ NP X j ∈D m ( i, j ) (cid:0) P Nu ( j ) − P Nu ( i ) (cid:1) + P Nu ( i ) (cid:2) − ν − γP Nu ( i ) + (cid:0) η − ρF Nu ( i ) (cid:1) H Nu ( i ) (cid:3) + ι NP o + δ p ∞ ( F Nu ( i ))( P Nu ( i )) β NP P Nu ( i ) + " − ρ (cid:16) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1) − h ∞ (cid:0) F Nu ( i ) (cid:1) ln (cid:16) H Nu ( i ) h ∞ ( F Nu ( i )) (cid:17) (cid:17) − (cid:0) η − ρF Nu ( i ) (cid:1) h ′∞ (cid:0) F Nu ( i ) (cid:1) ln (cid:16) H Nu ( i ) h ∞ ( F Nu ( i )) (cid:17) − δp ′∞ (cid:0) F Nu ( i ) (cid:1) ln (cid:16) P Nu ( i ) p ∞ ( F Nu ( i )) (cid:17) κ NH X j ∈D m ( i, j ) (cid:0) F Nu ( j ) − F Nu ( i ) (cid:1) H Nu ( j ) H Nu ( i ) − α N F Nu ( i ) (cid:0) − F Nu ( i ) (cid:1) o + n ρh ′∞ (cid:0) F Nu ( i ) (cid:1) ln (cid:16) H Nu ( i ) h ∞ ( F Nu ( i )) (cid:17) − η − ρF Nu ( i )2 (cid:16) h ′′∞ (cid:0) F Nu ( i ) (cid:1) ln (cid:16) H Nu ( i ) h ∞ ( F Nu ( i )) (cid:17) − ( h ′∞ ( F Nu ( i ) )) h ∞ ( F Nu ( i )) (cid:17) − δ (cid:16) p ′′∞ ( F Nu ( i )) ln (cid:16) P Nu ( i ) p ∞ ( F Nu ( i )) (cid:17) − ( p ′∞ ( F Nu ( i )) ) p ∞ ( F Nu ( i )) (cid:17)o β NH F Nu ( i ) ( − F Nu ( i ) ) H Nu ( i ) ! du . (85)Note that for all x ∈ [0 ,
1] it holds that 0 < η − ρx ≤ η . Together with the fact that for all x ∈ (0 , ∞ ) we haveln( x ) ≤ √ x , ln( x ) ≤ x , (cid:12)(cid:12)(cid:12) ln( x ) (cid:12)(cid:12)(cid:12) ≤ √ x + q x , and (cid:12)(cid:12)(cid:12) ln( x ) (cid:12)(cid:12)(cid:12) ≤ x + q x and dropping negative terms, this implies22or all t ∈ [0 , ∞ ) and all N, n ∈ N that E (cid:2) V D n (cid:0)(cid:0) H Nt , P Nt , F Nt (cid:1)(cid:1)(cid:3) − E (cid:2) V D n (cid:0)(cid:0) H N , P N , F N (cid:1)(cid:1)(cid:3) ≤ E " Z t X i ∈D n σ i ηκ NH X j ∈D m ( i, j ) H Nu ( j ) + ηh ∞ (cid:0) F Nu ( i ) (cid:1) κ NH + (cid:0) η − ρF Nu ( i ) (cid:1) (cid:0) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) h λ (cid:16) − H Nu ( i ) K (cid:17) − δP Nu ( i ) i + ηh ∞ (cid:0) F Nu ( i ) (cid:1) α N + ηι NH + η h ∞ (cid:0) F Nu ( i ) (cid:1) β NH H Nu ( i ) + δκ NP X j ∈D m ( i, j ) P Nu ( j ) + δp ∞ (cid:0) F Nu ( i ) (cid:1) κ NP + δ (cid:0) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) (cid:2) − ν − γP Nu ( i ) + (cid:0) η − ρF Nu ( i ) (cid:1) H Nu ( i ) (cid:3) + δι NP + δ p ∞ ( F Nu ( i ) ) P Nu ( i ) β NP + ρκ NH X j ∈D m ( i, j ) H Nu ( j ) + ρH Nu ( i ) α N + ρ h ∞ ( F Nu ( i ) ) H Nu ( i ) H Nu ( i ) h ∞ ( F Nu ( i )) + r h ∞ ( F Nu ( i ) ) H Nu ( i ) ! κ NH X j ∈D m ( i, j ) H Nu ( j ) + ρh ∞ (cid:0) F Nu ( i ) (cid:1) h ∞ ( F Nu ( i ) ) H Nu ( i ) α N + " ηh ′∞ (cid:0) F Nu ( i ) (cid:1) r h ∞ ( F Nu ( i ) ) H Nu ( i ) + H Nu ( i ) h ∞ ( F Nu ( i )) ! − δp ′∞ (cid:0) F Nu ( i ) (cid:1) r P Nu ( i ) p ∞ ( F Nu ( i )) + r p ∞ ( F Nu ( i ) ) P Nu ( i ) ! κ NH X j ∈D m ( i, j ) H Nu ( j ) H Nu ( i ) + " ηh ′∞ (cid:0) F Nu ( i ) (cid:1) H Nu ( i ) h ∞ ( F Nu ( i )) − δp ′∞ (cid:0) F Nu ( i ) (cid:1) r p ∞ ( F Nu ( i ) ) P Nu ( i ) α N + ρh ′∞ (cid:0) F Nu ( i ) (cid:1) H Nu ( i ) h ∞ ( F Nu ( i )) β NH H Nu ( i ) + η (cid:16) h ′′∞ (cid:0) F Nu ( i ) (cid:1) h ∞ ( F Nu ( i ) ) H Nu ( i ) + ( h ′∞ ( F Nu ( i ) )) h ∞ ( F Nu ( i )) (cid:17) β NH H Nu ( i ) + δ (cid:18) − p ′′∞ ( F Nu ( i )) P Nu ( i ) p ∞ ( F Nu ( i )) + ( p ′∞ ( F Nu ( i )) ) p ∞ ( F Nu ( i )) (cid:19) β NH H Nu ( i ) ! du . (86)Using (79), (80), (81), (82), (83), and (84) we get for all t ∈ [0 , ∞ ) and all N, n ∈ N that E (cid:2) V D n (cid:0)(cid:0) H Nt , P Nt , F Nt (cid:1)(cid:1)(cid:3) − E (cid:2) V D n (cid:0)(cid:0) H N , P N , F N (cid:1)(cid:1)(cid:3) ≤ E " Z t X i ∈D σ i b N R Nu ( i ) + (cid:0) η − ρF Nu ( i ) (cid:1) (cid:0) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) h λ (cid:16) − H Nu ( i ) K (cid:17) − δP Nu ( i ) i + δ (cid:0) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) h − ν − γP Nu ( i ) + (cid:0) η − ρF Nu ( i ) (cid:1) H Nu ( i ) i! du . (87)Note that for all x ∈ [0 ,
1] we have δp ∞ ( x ) + λK h ∞ ( x ) − λ = δλK ( η − ρx ) − δλν + λδν + λ γλγ + δK ( η − ρx ) − λ = δK ( η − ρx )+ λγλγ + δK ( η − ρx ) λ − λ = 0 ,ν − ( η − ρx ) h ∞ ( x ) + γp ∞ ( x ) = ν − ( η − ρx ) Kδν +( η − ρx ) Kγλ − γλK ( η − ρx )+ γλνλγ + δK ( η − ρx ) = ν − ( η − ρx ) Kδ + γλλγ + δK ( η − ρx ) ν = 0 . (88)From (88) we see that for all t ∈ [0 , ∞ ) and all N, n ∈ N it holds that E (cid:2) V D n (cid:0)(cid:0) H Nt , P Nt , F Nt (cid:1)(cid:1)(cid:3) − E (cid:2) V D n (cid:0)(cid:0) H N , P N , F N (cid:1)(cid:1)(cid:3) ≤ E " Z t X i ∈D σ i b N R Nu ( i ) + (cid:0) η − ρF Nu ( i ) (cid:1) (cid:0) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) h λ − λK (cid:16) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1) (cid:17) − δ (cid:16) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1) (cid:17) − λ i + δ (cid:0) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) h − ν − γ (cid:16) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1) (cid:17) + (cid:0) η − ρF Nu ( i ) (cid:1) (cid:16) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1) (cid:17) + ν i! du . (89)23ence, we obtain for every N, n ∈ N and every t ∈ [0 , ∞ ) that E (cid:2) V D n (cid:0) H Nt , P Nt , F Nt (cid:1)(cid:3) + Z t ( η − ρ ) λK E " X i ∈D n σ i (cid:0) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) + δγ E " X i ∈D n σ i (cid:0) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) du ≤ E (cid:2) V ˆ D (cid:0) H N , P N , F N (cid:1)(cid:3) + tb N
32 sup M ∈ N sup u ∈ [0 , ∞ ) E (cid:2)(cid:13)(cid:13) R Mu (cid:13)(cid:13) σ (cid:3) . (90)Applying monotone convergence we now see that for every N ∈ N and every t ∈ [0 , ∞ ) we have E (cid:2) V ˆ D (cid:0) H Nt , P Nt , F Nt (cid:1)(cid:3) + Z t ( η − ρ ) λK E X i ∈ ˆ D σ i (cid:0) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) + δγ E X i ∈ ˆ D σ i (cid:0) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) du = lim n →∞ E (cid:2) V D n (cid:0) H Nt , P Nt , F Nt (cid:1)(cid:3) + Z t ( η − ρ ) λK E " X i ∈D n σ i (cid:0) H Nu ( i ) − h ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) + δγ E " X i ∈D n σ i (cid:0) P Nu ( i ) − p ∞ (cid:0) F Nu ( i ) (cid:1)(cid:1) du ! ≤ E (cid:2) V ˆ D (cid:0) H N , P N , F N (cid:1)(cid:3) + tb N c . (91)The set ˆ D ⊆ D was arbitrarily chosen and thus, this finishes the proof of Theorem 2.8.
For convenience of the reader, we restate Lemma 3.3 of Klenke and Mytnik [20].
Lemma 2.9.
Let D be a countable set, let σ ∈ (0 , ∞ ) D such that P i ∈D σ i < ∞ , and let l σ := { z ∈ R D : k z k σ := P i ∈D σ i z i < ∞} . A subset K ⊆ l σ is relatively compact if and only if(i) sup x ∈ K k x k σ < ∞ (ii) for every ε ∈ (0 , ∞ ) there exists a finite subset E ⊆ D such that sup x ∈ K k x D\E k σ < ε . Lemma 2.10.
Let (Ω , F , P ) be a probability space, let D be a countable set, let σ ∈ (0 , ∞ ) D such that P i ∈D σ i < ∞ , let l σ := { z ∈ R D : k z k σ := P i ∈D σ i z i < ∞} , let E := l σ ∩ [0 , ∞ ) D , let I be a set, and let Z i : Ω → E , i ∈ I ,be a family of random variables. Assume that sup i ∈ I E [ k Z i k σ ] < ∞ and inf S⊆D , |S| < ∞ sup i ∈ I P k ∈D\S σ k E [ Z ik ] =0 . Then the family { Z i : i ∈ I } is relatively compact in E .Proof. Fix ε ∈ (0 , ∞ ). For each m ∈ N by assumption there exists a set S m,ε ⊆ D such thatsup i ∈ I X k ∈D\S m,ε σ k E [ Z ik ] < ε m ( m +1) . (92)Define the set K ε ⊆ E by K ε := ( x ∈ E : k x k σ ≤ i ∈ I E [ k Z i k σ ] ε , sup m ∈ N n m X k ∈D\S m,ε σ k | x k | o ≤ ) . (93)24ue to the Heine-Borel theorem we can apply Lemma 2.9 to obtain relative compactness of K ε . By Markov’sinequality we getsup i ∈ I P h Z i / ∈ K ε i ≤ sup i ∈ I P h Z i / ∈ K ε i ≤ sup i ∈ I P h k Z i k σ > j ∈ I E [ k Z j k σ ] ε i + sup i ∈ I ∞ X m =1 P h X k ∈D\S m,ε σ k Z ik > m i ≤ ε j ∈ I E [ k Z j k σ ] sup i ∈ I E h k Z i k σ i + ∞ X m =1 m sup i ∈ I X k ∈D\S m,ε σ k E h Z ik i ≤ ε + ∞ X m =1 m ε m ( m +1) = ε. (94)Since ε was arbitrarily chosen it follows that { Z i : i ∈ I } is tight in E . Due to Prohorov’s theorem (e.g.,Theorem 3.2.2 in Ethier and Kurtz [7]) the claim follows. Assume the setting of Section 2.1 and assume that for all N ∈ N we have P i ∈D σ i E h H N ( i ) i < ∞ . For all n ∈ N denote by m n the n -fold matrix product of m . Then we get for all t ∈ [0 , ∞ ) , all i ∈ D , andall N ∈ N that E (cid:2) H Nt ( i ) (cid:3) ≤ E X j ∈D ∞ X n =0 e − tκ NH ( tκ NH ) n n ! m n ( i, j ) H N ( j ) + K (cid:16) q ι H Kλ (cid:17) . (95) Proof.
We have for every n ∈ N and every i, j ∈ D that m n ( i, j ) ∈ [0 , T ∈ [0 , ∞ ) andall i, j ∈ D that ∞ X n =0 sup t ∈ [0 ,T ] e − t t n n ! m n ( i, j ) < ∞ . (96)Thereby, for all t ∈ [0 , ∞ ) and all i, j ∈ D we can define m t ( i, j ) := ∞ X n =0 e − t t n n ! m n ( i, j ) . (97)By (96) and using dominated convergence, we can compute for all t ∈ [0 , ∞ ) and all i, j ∈ D that ddt m t ( i, j ) = − m t ( i, j ) + ∞ X n =1 e − t t n − ( n − m n ( i, j ) = − m t ( i, j ) + ∞ X n =0 e − t t n n ! m n +1 ( i, j )= − m t ( i, j ) + ∞ X n =0 e − t t n n ! X k ∈D m n ( i, k ) m ( k, j ) = X k ∈D m t ( i, k )( m ( k, j ) − j = k ) . (98)Furthermore, note that for all t ∈ [0 , ∞ ) and all i ∈ D we have X j ∈D m t ( i, j ) = X j ∈D ∞ X n =0 e − t t n n ! m n ( i, j ) = ∞ X n =0 e − t t n n ! = 1 . (99)For all t ∈ [0 , ∞ ), s ∈ [0 , t ], i ∈ D , N ∈ N define Y N,ts ( i ) := X j ∈D m ( t − s ) κ NH ( i, j ) H Ns ( j ) . (100)Observe that since for all i, j ∈ D it holds that m ( i, j ) = i = j we have for all t ∈ [0 , ∞ ), all i ∈ D , and all N ∈ N that Y N,tt ( i ) = H Nt ( i ) . (101)25urthermore, using (3) we have for all t ∈ [0 , ∞ ) and all N ∈ N that X i ∈D σ i E h Y N,t ( i ) i = X i ∈D σ i E X j ∈D m tκ NH ( i, j ) H N ( j ) = X i ∈D σ i X j ∈D ∞ X n =0 e − tκ NH ( tκ NH ) n n ! m n ( i, j ) E (cid:2) H N ( j ) (cid:3) = X j ∈D ∞ X n =0 e − tκ NH ( tκ NH ) n n ! E (cid:2) H N ( j ) (cid:3) X i ∈D σ i m n ( i, j ) ≤ X j ∈D ∞ X n =0 e − tκ NH ( tκ NH ) n n ! E (cid:2) H N ( j ) (cid:3) c n σ j = X j ∈D ∞ X n =0 e − tκ NH ( tκ NH c ) n n ! E (cid:2) H N ( j ) (cid:3) σ j = e tκ NH ( c − E (cid:2)(cid:13)(cid:13) H N (cid:13)(cid:13) σ (cid:3) . (102)For all t ∈ [0 , ∞ ), s ∈ [0 , t ], i, j ∈ D , N ∈ N we see from (98) that we have dds m ( t − s ) κ NH ( i, j ) = − κ NH X k ∈D m ( t − s ) κ NH ( i, k )( m ( k, j ) − j = k ) . (103)For t ∈ [0 , ∞ ), N, l ∈ N , i ∈ D define τ N,tl ( i ) := inf (cid:0)(cid:8) u ∈ [0 , t ] : Y N,tu ( i ) > l (cid:9) ∪ ∞ (cid:1) . (104)Using the fact that for all t ∈ [0 , ∞ ), all u ∈ [0 , t ], all N ∈ N , and all i, j ∈ D we have m ( t − u ) κ NH ( i, j ) ∈ [0 ,
1] weget for all t ∈ [0 , ∞ ), all s ∈ [0 , t ], all N, l ∈ N , and all i ∈ D that Z s ∧ τ N,tl ( i )0 X j ∈D (cid:18) m ( t − u ) κ NH ( i, j ) q β NH H Nu ( j ) (cid:19) du ≤ Z s ∧ τ N,tl ( i )0 X j ∈D m ( t − u ) κ NH ( i, j ) β NH H Nu ( j ) du = Z s ∧ τ N,tl ( i )0 β NH Y N,tu ( i ) du ≤ Z s β NH Y N,tu ∧ τ N,tl ( i ) ( i ) du ≤ tβ NH l. (105)For all t ∈ [0 , ∞ ), s ∈ [0 , t ], i ∈ D , N ∈ N using Itˆo’s lemma with (99) and (103) we get P -a.s. Y N,ts ( i ) − Y N,t ( i ) = Z s X j ∈D m ( t − u ) κ NH ( i, j ) κ NH X k ∈D m ( j, k ) H Nu ( k ) + (cid:0) λ − κ NH − α N F Nt ( j ) (cid:1) H Nu ( j ) − λK (cid:0) H Nu ( j ) (cid:1) − δH Nu ( j ) P Nu ( j ) + ι NH ! − X j ∈D κ NH X k ∈D m ( t − u ) κ NH ( i, k )( m ( k, j ) − j = k ) H Nu ( j ) du + X j ∈D Z s m ( t − u ) κ NH ( i, j ) q β NH H Nu ( j ) dW N,Hu ( j )= Z s X j ∈D m ( t − u ) κ NH ( i, j ) (cid:0) λ − α N F Nt ( j ) (cid:1) H Nu ( j ) − λK (cid:0) H Nu ( j ) (cid:1) − δH Nu ( j ) P Nu ( j ) ! + ι NH du + X j ∈D Z s m ( t − u ) κ NH ( i, j ) q β NH H Nu ( j ) dW N,Hu ( j ) ≤ Z s X j ∈D m ( t − u ) κ NH ( i, j ) λH Nu ( j ) − λK (cid:0) H Nu ( j ) (cid:1) ! + ι NH du + X j ∈D Z s m ( t − u ) κ NH ( i, j ) q β NH H Nu ( j ) dW N,Hu ( j ) . (106)26hus, using (105) and (106) we get for all t ∈ [0 , ∞ ), all s ∈ [0 , t ], all i ∈ D , and all N, l ∈ N that E h Y N,ts ∧ τ N,tl ( i ) ( i ) i − E h Y N,t ( i ) i ≤ E Z s ∧ τ N,tl ( i )0 X j ∈D m ( t − u ) κ NH ( i, j ) λH Nu ( j ) + ι NH du ≤ E Z s X j ∈D m ( t − u ∧ τ N,tl ( i )) κ NH ( i, j ) λH Nu ∧ τ N,tl ( i ) ( j ) + ι NH du = Z s λ E h Y N,tu ∧ τ N,tl ( i ) ( i ) i + ι NH du ≤ tι NH + λ Z s E h Y N,tu ∧ τ N,tl ( i ) ( i ) i du. (107)Now, using Gronwall’s lemma (e.g., Klenke [19]), we get for all t ∈ [0 , ∞ ), all s ∈ [0 , t ], all i ∈ D , and all N, l ∈ N that E h Y N,ts ∧ τ N,tl ( i ) ( i ) i ≤ (cid:16) E h Y N,t ( i ) i + tι NH (cid:17) e λs ≤ (cid:16) E h Y N,t ( i ) i + tι NH (cid:17) e λt . (108)For all t ∈ [0 , ∞ ), N ∈ N , i ∈ D the P -a.s. continuous paths of (cid:0) Y N,tu ( i ) (cid:1) u ∈ [0 ,t ] imply P h sup u ∈ [0 ,t ] Y N,tu ( i ) < ∞ i = 1. Hence, we get for all t ∈ [0 , ∞ ), all N ∈ N , and all i ∈ D that P (cid:20) lim l →∞ τ N,tl ( i ) = ∞ (cid:21) = 1 . (109)Using the assumption that for all N ∈ N we have P i ∈D σ i E h H N ( i ) i < ∞ together with (101), (102), (108),and (109) with Fatou’s lemma we obtain for all t ∈ [0 , ∞ ) and all N ∈ N that X i ∈D σ i E (cid:2) H Nt ( i ) (cid:3) = X i ∈D σ i E h Y N,tt ( i ) i = X i ∈D σ i E (cid:20) lim l →∞ Y N,tt ∧ τ N,tl ( i ) ( i ) (cid:21) ≤ X i ∈D σ i lim inf l →∞ E h Y N,tt ∧ τ N,tl ( i ) ( i ) i ≤ X i ∈D σ i lim inf l →∞ (cid:16) E h Y N,t ( i ) i + tι NH (cid:17) e λt = X i ∈D σ i (cid:16) E h Y N,t ( i ) i + tι NH (cid:17) e λt ≤ e tκ NH ( c − E (cid:2)(cid:13)(cid:13) H N (cid:13)(cid:13) σ (cid:3) + X i ∈D σ i tι NH ! e λt < ∞ . (110)Using the fact that for all t ∈ [0 , ∞ ), all u ∈ [0 , t ], all N ∈ N , and all i, j ∈ D we have m ( t − u ) κ NH ( i, j ) ∈ [0 , t ∈ [0 , ∞ ), all s ∈ [0 , t ], all N ∈ N , and all i ∈ D , that E X j ∈D Z s (cid:18) m ( t − u ) κ NH ( i, j ) q β NH H Nu ( j ) (cid:19) du = Z s E X j ∈D (cid:18) m ( t − u ) κ NH ( i, j ) q β NH H Nu ( j ) (cid:19) du ≤ β NH Z s E X j ∈D m ( t − u ) κ NH ( i, j ) H Nu ( j ) du = β NH Z s E (cid:2) Y N,tu ( i ) (cid:3) du < ∞ . (111)Thus, taking expectations in (106) gives for all t ∈ [0 , ∞ ), all s ∈ [0 , t ], all i ∈ D , and all N ∈ N using Jensen’sinequality E (cid:2) Y N,ts ( i ) (cid:3) − E h Y N,t ( i ) i ≤ Z s λ E (cid:2) Y N,tu ( i ) (cid:3) − λK E X j ∈D m ( t − u ) κ NH ( i, j ) (cid:0) H Nu ( j ) (cid:1) ! + ι NH du ≤ Z s λ E (cid:2) Y N,tu ( i ) (cid:3) − λK E h(cid:0) Y N,tu ( i ) (cid:1) i ! + ι NH du ≤ Z s λ E (cid:2) Y N,tu ( i ) (cid:3) − λK (cid:0) E (cid:2) Y N,tu ( i ) (cid:3)(cid:1) ! + ¯ ι H du. (112)27or t ∈ [0 , ∞ ), i ∈ D , N ∈ N let z N,t ( i ) : [0 , ∞ ) → R be a process that for all s ∈ [0 , ∞ ) satisfies z N,ts ( i ) = z N,t ( i ) + Z s (cid:16) λz N,tu ( i ) − λK (cid:0) z N,tu ( i ) (cid:1) + ¯ ι H (cid:17) du (113)with z N,t ( i ) = E h Y N,t ( i ) i where uniqueness follows from local Lipschitz continuity. Define c := K + q K + K ¯ ι H λ ∈ (0 , ∞ ). Using classical comparison results from the theory of ODEs, the above computationshows that for all N ∈ N , all i ∈ D , and all t ∈ [0 , ∞ ) we have E (cid:2) H Nt ( i ) (cid:3) = E h Y N,tt ( i ) i ≤ z N,tt ( i ) ≤ max (cid:26) E h Y N,t ( i ) i , lim sup s →∞ z N,ts ( i ) (cid:27) = max n E h Y N,t ( i ) i , c o ≤ E h Y N,t ( i ) i + c = E X j ∈D m tκ NH ( i, j ) H N ( j ) + c (114)This finishes the proof of Lemma 2.11. Proof of Theorem 1.3.
We will use stochastic averaging (see Theorem 2.1 in Kurtz [21]) to prove the result.So we first check that all conditions of the aforementioned theorem are fulfilled. Note that E = [0 , D and E = l σ ∩ [0 , ∞ ) D are complete separable metric spaces. Tychonoff’s theorem implies that E is compact.Since for all N ∈ N and all t ∈ [0 , ∞ ) the random variable F NtN takes values in the compact space E , thecompact containment condition holds for (cid:8) (cid:0) F NtN (cid:1) t ∈ [0 , ∞ ) : N ∈ N (cid:9) . We will now use Lemma 2.10 to show foreach T ∈ [0 , ∞ ) that the family (cid:8) H NtN : t ∈ [0 , T ] , N ∈ N (cid:9) is relatively compact in E . From Lemma 2.5 andthe assumption sup N ∈ N E h(cid:13)(cid:13)(cid:13)(cid:16) H N + P N (cid:17) i < ∞ we see thatsup N ∈ N sup t ∈ [0 , ∞ ) E (cid:2)(cid:13)(cid:13) H NtN (cid:13)(cid:13) σ (cid:3) < ∞ . (115)Define D := ∅ and for all n ∈ N let D n ⊆ D be a set with |D n | = min { n, |D|} and D n ⊇ D n − . Define c := K (cid:16) q ι H Kλ (cid:17) . From Lemma 2.11 with the assumption that P i ∈D sup N ∈ N σ i E h H N ( i ) i < ∞ we getfor all T ∈ [0 , ∞ ) that X i ∈D σ i sup N ∈ N sup t ∈ [0 ,T ] E (cid:2) H NtN ( i ) (cid:3) ≤ X i ∈D σ i sup N ∈ N sup t ∈ [0 ,T ] X j ∈D ∞ X n =0 e − tNκ NH ( tNκ NH ) n n ! m n ( i, j ) E (cid:2) H N ( j ) (cid:3) + c ≤ X j ∈D ∞ X n =0 X i ∈D σ i m n ( i, j ) ! sup N ∈ N sup t ∈ [0 ,T Nκ NH ] e − t t n n ! E (cid:2) H N ( j ) (cid:3) + c X i ∈D σ i ≤ X j ∈D ∞ X n =0 c n σ j sup N ∈ N ( T Nκ NH ) n n ! E (cid:2) H N ( j ) (cid:3) + c k k σ ≤ e cT sup M ∈ N Mκ MH X j ∈D σ j sup N ∈ N E (cid:2) H N ( j ) (cid:3) + c k k σ < ∞ . (116)Now we can use the dominated convergence theorem to obtain for all T ∈ [0 , ∞ ) thatlim n →∞ sup N ∈ N sup t ∈ [0 ,T ] X k ∈D\D n σ k E h H NtN ( k ) i ≤ lim n →∞ X k ∈D\D n sup N ∈ N sup t ∈ [0 ,T ] σ k E h H NtN ( k ) i = 0 . (117)Hence, for all T ∈ [0 , ∞ ) we can apply Lemma 2.10 to the family (cid:8) H NtN : t ∈ [0 , T ] , N ∈ N (cid:9) and conclude thatit is relatively compact in E . Denote by C b ( E , R ) the set of bounded, continuous real-valued functions on E and by C b ( E , R ) the set of all real-valued functions on E that are twice continuously differentiable andbounded, with bounded first and second order partial derivatives. For f ∈ C b ( E , R ) let c f ∈ (0 , ∞ ) be suchthat for all x ∈ E and all i ∈ D we have (cid:12)(cid:12)(cid:12) dfdx i ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) d fdx i ( x ) (cid:12)(cid:12)(cid:12) ≤ c f . DefineDom( A ) := (cid:8) f ∈ C b ( E , R ) : f depends only on finitely many coordinates (cid:9) (118)28nd for any f ∈ Dom( A ) denote by D f the finite set of coordinates that f depends on. Due to the Stone-Weierstrass theorem (e.g., Theorem 15.2 in Klenke [19]) we see that Dom( A ) is dense in C b ( E , R ) in thetopology of uniform convergence. Denote by C ( E × E , R ) the set of real-valued continuous functions on E × E and define the operator A : Dom( A ) → C ( E × E , R ) for all f ∈ Dom( A ), all x ∈ E , and all y ∈ E by( A f ) ( x, y ) := X i ∈D y i > h κ H X j ∈D (cid:16) m ( i, j ) y j y i ( x j − x i ) (cid:17) − αx i (1 − x i ) i dfdx i ( x ) + β H x i (1 − x i ) y i d fdx i ( x ) ! . (119)For all f ∈ Dom( A ), all N ∈ N , and all t ∈ [0 , ∞ ) define ε Nf ( t ) := N Z t ( A f ) (cid:0) F Nu , H Nu (cid:1) du − X i ∈D Z t dfdx i ( F Nu ) " κ NH X j ∈D m ( i, j ) (cid:0) F Nu ( j ) − F Nu ( i ) (cid:1) H Nu ( j ) H Nu ( i ) − α N F Nu ( i ) (cid:0) − F Nu ( i ) (cid:1) + d fdx i ( F Nu ) β H F Nu ( i ) ( − F Nu ( i ) ) NH Nu ( i ) du. (120)From Itˆo’s lemma and Lemma 2.4 we get for all f ∈ Dom( A ), all N ∈ N , and all t ∈ [0 , ∞ ) that P -a.s. f (cid:0) F Nt (cid:1) − f (cid:0) F N (cid:1) = X i ∈D Z t dfdx i ( F Nu ) dF Nu ( i ) + X i,j ∈D Z t (cid:16) d fdx i dx j ( F Nu ) (cid:17) d (cid:10) F N ( i ) , F N ( j ) (cid:11) u = X i ∈D Z t dfdx i ( F Nu ) " κ NH X j ∈D m ( i, j ) (cid:0) F Nu ( j ) − F Nu ( i ) (cid:1) H Nu ( j ) H Nu ( i ) − α N F Nu ( i ) (cid:0) − F Nu ( i ) (cid:1) + d fdx i ( F Nu ) β NH F Nu ( i ) ( − F Nu ( i ) ) H Nu ( i ) du + X i ∈D Z t dfdx i ( F Nu ) r β NH F Nu ( i ) ( − F Nu ( i ) ) H Nu ( i ) dW F,Nu ( i ) . (121)Hence, we get for all f ∈ Dom( A ), all N ∈ N , and all t ∈ [0 , ∞ ) that P -a.s. f (cid:0) F NtN (cid:1) − Z t ( A f ) (cid:0) F NuN , H
NuN (cid:1) du + ε Nf ( tN )= f (cid:0) F N (cid:1) + X i ∈D Z t dfdx i ( F NuN ) r β H F NuN ( i ) ( − F NuN ( i ) ) H NuN ( i ) dW F,NuN ( i ) . (122)From Tonelli’s theorem and Lemma 2.6 we obtain for all f ∈ Dom( A ), all N ∈ N , and all t ∈ [0 , ∞ ) that E " Z t X i ∈D dfdx i ( F Nu ) r β NH F Nu ( i ) ( − F Nu ( i ) ) H Nu ( i ) ! du ≤ t |D f | c f ¯ β H max i ∈D f sup M ∈ N sup u ∈ [0 , ∞ ) E h H Mu ( i ) i ≤ t |D f | c f ¯ β H max i ∈D f σ i sup M ∈ N sup u ∈ [0 , ∞ ) E h(cid:13)(cid:13)(cid:13) H Mu (cid:13)(cid:13)(cid:13) σ i < ∞ . (123)Thus for all f ∈ Dom( A ), all N ∈ N , and all t ∈ [0 , ∞ ) the left-hand side of (122) is a martingale. Next, for all29 ∈ Dom( A ) and all T ∈ [0 , ∞ ) it holds thatsup N ∈ N E "Z T (cid:12)(cid:12) ( A f ) (cid:0) F NtN , H
NtN (cid:1)(cid:12)(cid:12) dt = sup N ∈ N E " Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈D f κ H X j ∈D (cid:16) m ( i, j ) H NtN ( j ) H NtN ( i ) (cid:0) F NtN ( j ) − F NtN ( i ) (cid:1)(cid:17) − αF NtN ( i ) (cid:0) − F NtN ( i ) (cid:1) ! dfdx i (cid:0) F NtN (cid:1) + X i ∈D f β H F
NtN ( i ) ( − F NtN ( i ) ) H NtN ( i ) d fdx i (cid:0) F NtN (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ sup N ∈ N E " Z T X i ∈D f (cid:12)(cid:12)(cid:12) κ H X j ∈D (cid:16) m ( i, j ) H NtN ( j ) H NtN ( i ) (cid:17) c f (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) αc f (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) β H H NtN ( i ) c f (cid:12)(cid:12)(cid:12)!! dt . (124)Using Young’s inequality and Jensen’s inequality we get for all f ∈ Dom( A ) and all T ∈ [0 , ∞ ) thatsup N ∈ N E "Z T (cid:12)(cid:12) ( A f ) (cid:0) F NtN , H
NtN (cid:1)(cid:12)(cid:12) dt ≤ sup N ∈ N E " Z T X i ∈D f (cid:16) κ H c f H NtN ( i ) (cid:17) + (cid:16) X j ∈D m ( i, j ) H NtN ( j ) (cid:17) + αc f + β H H NtN ( i ) c f !! dt ≤ sup N ∈ N E " Z T |D f | ) min k ∈D f { σ k } X i ∈D f σ i (cid:16) (cid:17) (cid:16) κ H c f H NtN ( i ) (cid:17) + (cid:16) (cid:17) (cid:16) X j ∈D m ( i, j ) H NtN ( j ) (cid:17) + (cid:16) αc f (cid:17) + (cid:16) β H H NtN ( i ) c f (cid:17) ! dt . (125)Using Lemma 2.4, Tonelli’s theorem, and Lemmas 2.5 and 2.6 we obtain for all f ∈ Dom( A ) and all T ∈ [0 , ∞ )that sup N ∈ N E "Z T (cid:12)(cid:12) ( A f ) (cid:0) F NtN , H
NtN (cid:1)(cid:12)(cid:12) dt ≤ sup N ∈ N (4 |D f | ) min k ∈D f { σ k } Z T (cid:16) (cid:17) ( κ H c f ) E " (cid:13)(cid:13)(cid:13)(cid:13) ( H NtN ) (cid:13)(cid:13)(cid:13)(cid:13) σ + (cid:16) (cid:17) c E " (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) H NtN (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) σ + ( αc f ) k k σ + (cid:16) (cid:17) ( β H c f ) E " (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) H NtN (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) σ dt < ∞ . (126)Furthermore, for all f ∈ Dom( A ), all N ∈ N , and all T ∈ [0 , ∞ ) we have that E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) ε Nf ( tN ) (cid:12)(cid:12)(cid:12) = E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈D f Z t dfdx i ( F NuN ) h (cid:0) κ H − N κ NH (cid:1) X j ∈D m ( i, j ) (cid:0) F NuN ( j ) − F NuN ( i ) (cid:1) H NuN ( j ) H NuN ( i ) + (cid:0) α − N α N (cid:1) F NuN ( i ) (cid:0) − F NuN ( i ) (cid:1) i + d fdx i ( F NuN ) (cid:0) β H − N β NH (cid:1) F NuN ( i ) ( − F NuN ( i ) ) H NuN ( i ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E " Z T X i ∈D f c f (cid:16) (cid:12)(cid:12) κ H − N κ NH (cid:12)(cid:12) X j ∈D m ( i, j ) H NuN ( j ) H NuN ( i ) + (cid:12)(cid:12) α − N α N (cid:12)(cid:12) + (cid:12)(cid:12) β H − N β NH (cid:12)(cid:12) H NuN ( i ) (cid:17) du . (127)30sing Young’s inequality, Lemma 2.4, and Tonelli’s theorem we get for all f ∈ Dom( A ), all N ∈ N , and all T ∈ [0 , ∞ ) that E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) ε Nf ( tN ) (cid:12)(cid:12)(cid:12) ≤ E " Z T X i ∈D f σ i c f min k ∈D f { σ k } | κ H − Nκ NH | (cid:16) H NuN ( i ) (cid:17) + (cid:16) X j ∈D m ( i, j ) H NuN ( j ) (cid:17) ! + (cid:12)(cid:12) α − N α N (cid:12)(cid:12) + | β H − Nβ NH | H NuN ( i ) ! du ≤ c f min k ∈D f { σ k } Z T | κ H − Nκ NH | E (cid:20)(cid:13)(cid:13)(cid:13)(cid:13) ( H NuN ) (cid:13)(cid:13)(cid:13)(cid:13) σ (cid:21) + c E h(cid:13)(cid:13)(cid:13)(cid:0) H NuN (cid:1) (cid:13)(cid:13)(cid:13) σ i ! + (cid:12)(cid:12) α − N α N (cid:12)(cid:12) k k σ + | β H − Nβ NH | E h(cid:13)(cid:13)(cid:13) H NuN (cid:13)(cid:13)(cid:13) σ i du. (128)Hence, from Lemmas 2.5 and 2.6 we see for all f ∈ Dom( A ) and all T ∈ [0 , ∞ ) that0 ≤ lim N →∞ E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) ε Nf ( tN ) (cid:12)(cid:12)(cid:12) ≤ lim N →∞ T c f min k ∈D f { σ k } | κ H − Nκ NH | sup M ∈ N sup t ∈ [0 , ∞ ) (cid:16) E (cid:20)(cid:13)(cid:13)(cid:13)(cid:13) ( H Mt ) (cid:13)(cid:13)(cid:13)(cid:13) σ (cid:21) + c E h(cid:13)(cid:13)(cid:13)(cid:0) H Mt (cid:1) (cid:13)(cid:13)(cid:13) σ i (cid:17) + (cid:12)(cid:12) α − N α N (cid:12)(cid:12) k k σ + | β H − Nβ NH | sup M ∈ N sup t ∈ [0 , ∞ ) E h(cid:13)(cid:13)(cid:13) H Mt (cid:13)(cid:13)(cid:13) σ i ! = 0 . (129)Define the set R := n × i ∈D B i : ( B i ) i ∈D ⊆ B ([0 , ∞ ) D ) , B i = [0 , ∞ ) for all but finitely many i ∈ D o . For all N ∈ N , all t ∈ [0 , ∞ ), and all B ∈ R define the measure-valued random variablesΛ N ([0 , t ] × B ) := Z t B (cid:0) H NuN (cid:1) du = Z t Y i ∈D B i (cid:0) H NuN ( i ) (cid:1) du, (130)Due to Carath´eodory’s theorem (see e.g., Theorem 1.41 in Klenke [19]) there is a unique extension of thispre-measure to a measure on [0 , t ] × E , which we will denote by the same name. Define the space ℓ ( E ) := { µ : µ is a measure on [0 , ∞ ) × E such that for all t ∈ [0 , ∞ ) it holds that µ ([0 , t ] × E ) = t } and the space D ([0 , ∞ )) := { f : [0 , ∞ ) → E | f is c`adl`ag } . Having checked all assumptions, we can now apply Theorem2.1 from Kurtz [21] and conclude that the sequence (cid:8)(cid:0)(cid:0) F NtN (cid:1) t ∈ [0 , ∞ ) , Λ N (cid:1) : N ∈ N (cid:9) is relatively compactin D ([0 , ∞ )) × ℓ ( E ). Let ( F, Λ) be a D ([0 , ∞ )) × ℓ ( E )-valued random variable and let ( N k ) k ∈ N ⊆ N be anincreasing sequence such that lim k →∞ (cid:0)(cid:0) F N k tN k (cid:1) t ∈ [0 , ∞ ) , Λ N k (cid:1) = ( F, Λ). Due to Skorohod’s representation theorem(see Theorem 3.1.8 of Ethier and Kurtz [7]) we can assume without loss of generality and for ease of notationthat ( F, Λ) acts on the probability space (Ω , F , P ). Using H¨older’s inequality and Theorem 1.2 we see for all t ∈ [0 , ∞ ) that0 ≤ lim N →∞ Z t E h(cid:13)(cid:13)(cid:13) H NuN − (cid:0) h ∞ (cid:0) F NuN ( i ) (cid:1)(cid:1) i ∈D (cid:13)(cid:13)(cid:13) σ i du = lim N →∞ Z t E "X i ∈D σ i (cid:12)(cid:12) H NuN ( i ) − h ∞ (cid:0) F NuN ( i ) (cid:1)(cid:12)(cid:12) du ≤ lim N →∞ vuutZ t E "X i ∈D σ i (cid:0) H NuN ( i ) − h ∞ (cid:0) F NuN ( i ) (cid:1)(cid:1) du s t X k ∈D σ k = 0 . (131)For any bounded Lipschitz continuous function f : l σ → R , with Lipschitz constant ¯ c f , and all t ∈ [0 , ∞ ),applying (131), we then have0 ≤ E h(cid:12)(cid:12)(cid:12) Z t Z E f ( y ) Λ( du × dy ) − Z t f ( h ∞ ( F u )) du (cid:12)(cid:12)(cid:12)i = lim k →∞ E h(cid:12)(cid:12)(cid:12) Z t f (cid:0) H N k uN k (cid:1) du − Z t f (cid:0) h ∞ (cid:0) F N k uN k (cid:1)(cid:1) du (cid:12)(cid:12)(cid:12)i ≤ ¯ c f lim k →∞ E h Z t (cid:13)(cid:13) H N k uN k − h ∞ (cid:0) F N k uN k (cid:1)(cid:13)(cid:13) du i = 0 . (132)31efine the operator A : Dom( A ) → C ( E , R ) for all f ∈ Dom( A ) and all x ∈ E by( A f ) ( x ) := X i ∈D κ H X j ∈D (cid:16) m ( i, j ) a − x i a − x j ( x j − x i ) (cid:17) − αx i (1 − x i ) dfdx i ( x )+ X i ∈D β H b ( a − x i ) x i (1 − x i ) d fdx i ( x ) . (133)For all t ∈ [0 , ∞ ), all f ∈ Dom( A ), and all x ∈ E we have P -a.s. Z t Z E ( A f ) ( F s , y )Λ( ds × dy )= Z t Z E " X i ∈D κ H X j ∈D (cid:16) m ( i, j ) y j y i ( F s ( j ) − F s ( i )) (cid:17) − αF s ( i )(1 − F s ( i )) dfdx i ( F s )+ X i ∈D β H F s ( i )(1 − F s ( i )) y i d fdx i ( F s ) i ∈D y i ( h ∞ ( F s ( i ))) dy ds = Z t ( A f ) ( F s ) ds. (134)Applying Theorem 2.1 of Kurtz [21] together with (134), we see for each f ∈ Dom( A ) that (cid:0) f ( F t ) − Z t ( A f ) ( F s ) ds (cid:1) t ∈ [0 , ∞ ) (135)is a martingale. Hence, F is a solution to (5). Note that for all z , z ∈ [0 ,
1] we have that a − z a − z ( z − z ) =( a − z )( a − z a − z − S ⊆ D and any x, y ∈ E that X i ∈S σ i x i ≥ y i (cid:16) κ H X j ∈D m ( i, j ) (cid:0) ( a − x i ) a − x j − ( a − x i ) − ( a − y i ) a − y j + ( a − y i ) (cid:1) − α ( x i (1 − x i ) − y i (1 − y i )) (cid:17) = X i ∈S σ i x i ≥ y i (cid:16) κ H X j ∈D m ( i, j ) (cid:0) ( x i − y i ) + (( a − x i ) − ( a − y i ) ) a − x j − ( a − y i ) (cid:0) a − y j − a − x j (cid:1)(cid:1) + α ( − ( x i − y i ) + x i − y i ) (cid:17) ≤ X i ∈S σ i (cid:16) κ H X j ∈D m ( i, j ) x j ≥ y j ( a − y i ) (cid:0) a − x j − a − y j (cid:1)(cid:17) + X i ∈S σ i ( κ H + 2 α ) x i ≥ y i ( x i − y i ) ≤ X i ∈S σ i (cid:16) κ H X j ∈D m ( i, j ) x j ≥ y j a ( a − ( x j − y j ) (cid:17) + X i ∈S σ i ( κ H + 2 α ) x i ≥ y i ( x i − y i ) ≤ X i ∈S σ i cκ H x i ≥ y i a ( x i − y i )( a − + X i ∈S σ i ( κ H + 2 α ) x i ≥ y i ( x i − y i ) = X i ∈S σ i (cid:0) cκ H a ( a − + κ H + 2 α (cid:1) ( x i − y i ) + . (136)This implies that equation (26) of Hutzenthaler and Wakolbinger [17] is fulfilled. Together with the assumptionson m in Assumption 1.1 we now infer, analogous to Proposition 2.1 of Hutzenthaler and Wakolbinger [17], thatthe system (5) has a unique strong solution with a.s. continuous paths. We conclude that any limit point of (cid:8) (cid:0) F NtN (cid:1) t ∈ [0 , ∞ ) : N ∈ N (cid:9) solves (5). Combining this with the fact that (cid:8) (cid:0) F NtN (cid:1) t ∈ [0 , ∞ ) : N ∈ N (cid:9) is relativelycompact we obtain (cid:0) F NtN (cid:1) t ∈ [0 , ∞ ) = ⇒ ( X t ) t ∈ [0 , ∞ ) , as N → ∞ . This finishes the proof of Theorem 1.3. In this section we investigate convergence of a sequence of exchangeable systems of stochastic differential equa-tions. 32 .1 Setting
Let (Ω , F , P ) be a probability space, let I ⊂ [0 , ∞ ) be an interval of length | I | ∈ (0 , ∞ ] which is either ofthe form [0 , | I | ] if | I | < ∞ or of the form [0 , ∞ ) if | I | = ∞ , let A ⊆ R be a convex set, and let ψ : I → A , ξ : A × I → R , and σ : I → [0 , ∞ ) be functions. The function σ : I → [0 , ∞ ) is locally Lipschitz continuousin I and satisfies σ (0) = 0 and if | I | < ∞ , then σ ( | I | ) = 0. Furthermore, the function σ is strictly positiveon (0 , | I | ). There exists a constant L ∈ (0 , ∞ ) such that σ satisfies the growth condition that for all y ∈ I wehave σ ( y ) ≤ L ( y + y ) and such that ξ satisfies for all ( u, x ) , ( v, y ) ∈ A × I that x ≥ y (cid:0) ξ ( u, x ) − ξ ( v, y ) (cid:1) ≤ L ( u − v ) + + L ( x − y ) + . (137)The function ψ : I → [0 , ∞ ) satisfies for all x, y ∈ I that | ψ ( x ) − ψ ( y ) | ≤ L | x − y | . Let W ( i ) : [0 , ∞ ) × Ω → R , i ∈ N , be independent Brownian motions with continuous sample paths. For all D ∈ N let X D : [0 , ∞ ) ×{ , . . . , D } × Ω → I be an adapted stochastic process with continuous sample paths that for all t ∈ [0 , ∞ ) andall i ∈ { , . . . , D } P -a.s. satisfies X Dt ( i ) = X D ( i )+ Z t ξ (cid:16) D X j ∈{ ,...,D } ψ (cid:0) X Ds ( j ) (cid:1) , X Ds ( i ) (cid:17) ds + Z t q σ ( X Ds ( i )) dW s ( i ) . (138)Let M : [0 , ∞ ) × Ω → I be an adapted stochastic process with continuous sample paths that for all t ∈ [0 , ∞ ) P -a.s. satisfies M t = M + Z t ξ ( E [ ψ ( M s )] , M s ) ds + Z t p σ ( M s ) dW s (1) . (139) The following proposition, Proposition 3.1, partly generalizes Proposition 4.29 in Hutzenthaler [16] where ξ depends linearly on its first argument. Proposition 3.1.
Assume the setting of Section 3.1, let M be an I -valued random variable, for every D ∈ N let (cid:0) X D ( j ) (cid:1) j ∈{ ,...,D } be exchangeable and integrable random variables with values in I . Then, there exists aunique solution M of (139) and for all D ∈ N and all t ∈ [0 , ∞ ) we have that √ D E h(cid:12)(cid:12) X Dt (1) − M t (cid:12)(cid:12)i ≤ e ( L + L + L µ ) t (cid:18) √ D E h(cid:12)(cid:12) X D (1) − M (cid:12)(cid:12)i + L Z t (cid:16) Var (cid:0) ψ (cid:0) M s (cid:1)(cid:1)(cid:17) ds (cid:19) . (140) Proof.
Existence of a weak solution is straightforward using a tightness argument. Next we show pathwiseuniqueness for the SDE (139). Let M, ¯ M : [0 , ∞ ) × Ω → I be two solutions of the SDE (139). Then ourassumptions and a standard Yamada-Watanabe argument (cf., e.g., Theorem 1 in Yamada and Watanabe [42])shows for all t ∈ [0 , ∞ ) that P -a.s. | M t − ¯ M t | = | M − ¯ M | + Z t sgn( M s − ¯ M s ) d ( M s − ¯ M s ) . (141)Let ( τ l ) l ∈ N be a localizing sequence for the local martingale (cid:0) R t sgn( M s − ¯ M s )( σ ( M s ) − σ ( ¯ M s )) dW s (cid:1) t ∈ [0 , ∞ ) .Then Fatou’s Lemma and our assumptions imply for all t ∈ [0 , ∞ ) that E [ | M t − ¯ M t | ] ≤ lim l →∞ E [ | M t ∧ τ l − ¯ M t ∧ τ l | ] ≤ E [ | M − ¯ M | ] + E h Z t sgn( M s − ¯ M s ) (cid:0) ξ ( E [ ψ ( M s )] , M s ) − ξ ( E [ ψ ( ¯ M s )] , ¯ M s ) (cid:1) ds i ≤ E [ | M − ¯ M | ] + L Z t (cid:12)(cid:12) E [ ψ ( M s )] − E [ ψ ( ¯ M s )] (cid:12)(cid:12) + E (cid:2) | M s − ¯ M s | (cid:3) ds ≤ E [ | M − ¯ M | ] + ( L + 1) Z t E (cid:2) | M s − ¯ M s | (cid:3) ds. (142)This together with Gronwall’s lemma implies pathwise uniqueness for the SDE (139). Therefore, the theoremof Yamada and Watanabe (see Yamada and Watanabe [42]) implies that the SDE (139) is exact. The rest ofthe proof is analogous to the proof of Proposition 4.29 in Hutzenthaler [16] and we omit it here.33 .3 Application to altruistic defense in structured populations In this section we verify the applicability of Proposition 3.1 to the case of altruistic defense in structuredpopulations.
Lemma 3.2.
Let α, β, κ ∈ (0 , ∞ ) and a ∈ (1 , ∞ ) , let I = [0 , and define the function σ : I → [0 , ∞ ) by I ∋ x σ ( x ) := β ( a − x ) x (1 − x ) , the function ψ : I → [0 , ∞ ) by I ∋ x ψ ( x ) := a − x , and the function ξ : [0 , ∞ ) × I → R by [0 , ∞ ) × I ∋ ( u, x ) ξ ( u, x ) := κ ( a − x ) (cid:0) ( a − x ) u − (cid:1) − αx (1 − x ) . Then the interval I and the functions σ , ψ , and ξ satisfy the setting of Section 3.1 with L = max (cid:8) βa, κa , κ + α, a − (cid:9) .Proof. For all ( u, x ) , ( v, y ) ∈ [0 , ∞ ) × [0 ,
1] it holds that x ≥ y (cid:0) ξ ( u, x ) − ξ ( v, y ) (cid:1) = x ≥ y (cid:0) κ ( a − x )(( a − x ) u − − κ ( a − y )(( a − y ) v − − αx (1 − x ) + αy (1 − y ) (cid:1) = x ≥ y (cid:0) κ [( a − x ) u − ( a − x ) − ( a − y ) v + ( a − y )] − α (1 − ( x + y ))( x − y ) (cid:1) = x ≥ y (cid:0) κ [( x − y ) + (( a − x ) − ( a − y ) ) u − ( a − y ) ( v − u )] − α (1 − ( x + y ))( x − y ) (cid:1) ≤ ( κ + α )( x − y ) + + κa ( u − v ) + ≤ L ( x − y ) + + L ( u − v ) + . (143)Moreover, for all x, y ∈ I it holds that σ ( x ) = β ( a − x ) x (1 − x ) ≤ βax ≤ L ( x + x ) and that | ψ ( x ) − ψ ( y ) | = (cid:12)(cid:12) a − x − a − y (cid:12)(cid:12) = (cid:12)(cid:12) Z xy a − z ) dz (cid:12)(cid:12) ≤ a − | x − y | ≤ L | x − y | . (144)This finishes the proof of Lemma 3.2. Let (Ω , F , P ) be a probability space, let κ, α, β ∈ (0 , ∞ ), a ∈ (1 , ∞ ), c ∈ (0 , W : [0 , ∞ ) × Ω → R be aBrownian motion with continuous sample paths, let Z : [0 , ∞ ) × Ω → [0 ,
1] be an adapted process with continuoussample paths that for all t ∈ [0 , ∞ ) satisfies P -a.s. Z t = Z + Z t (cid:0) κ ( a − Z s ) (cid:0) ( a − Z s ) E (cid:2) a − Z s (cid:3) − (cid:1) − αZ s (1 − Z s ) (cid:1) ds + Z t p β ( a − Z s ) Z s (1 − Z s ) dW s . (145)Moreover, for all θ ∈ (cid:0) a , a − (cid:1) let Z θ : [0 , ∞ ) × Ω → [0 ,
1] be an adapted process with continuous sample pathsthat for all t ∈ [0 , ∞ ) satisfies P -a.s. Z θt = Z θ + Z t (cid:0) κ (cid:0) a − Z θs (cid:1) (cid:0)(cid:0) a − Z θs (cid:1) θ − (cid:1) − αZ θs (cid:0) − Z θs (cid:1)(cid:1) ds + Z t q β ( a − Z θs ) Z θs (1 − Z θs ) dW s . (146)For all θ ∈ (cid:0) a , a − (cid:1) and all z ∈ [0 ,
1] define m θ ( z ) := βc κβ ( aθ − (1 − c ) κβ (1 − θ ( a − ( a − c ) αβ β ( a − z ) z (1 − z ) exp (cid:18)Z zc κ ( a − y )(( a − y ) θ − − αy (1 − y ) β ( a − y ) y (1 − y ) dy (cid:19) = z κβ ( aθ − − (1 − z ) κβ (1 − θ ( a − − ( a − z ) αβ − . (147)Note that this defines the speed density (see p. 95 in Karlin and Taylor [18]) for (146). Furthermore, note thatfor all θ ∈ (cid:0) a , a − (cid:1) it holds that Z m θ ( z ) dz < ∞ . (148)34or all θ ∈ (cid:0) a , a − (cid:1) define c θ := R m θ ( z ) dz , for all x ∈ { , } denote by δ x the Dirac measure on [0 , θ ∈ (cid:2) a , a − (cid:3) define the mapping Ψ θ : B ([0 , → [0 ,
1] by B ([0 , ∋ A Ψ θ ( A ) := δ ( A ) , if θ = a ,δ ( A ) , if θ = a − , R A c θ m θ ( z ) dz, if θ ∈ (cid:0) a , a − (cid:1) . (149) Assume the setting of Section 4.1. Existence and uniqueness of the solution of (145) follow from Proposition3.1. When θ ∈ (cid:0) a , a − (cid:1) we have that Ψ θ defines a probability distribution by (148), and we can apply TheoremV.54.5 of Rogers and Williams [34] to conclude that it is the unique equilibrium distribution for (146). Theproof of the following lemma, Lemma 4.1, is clear and therefore omitted. Lemma 4.1.
Assume the setting of Section 4.1. A probability measure
Φ : B ([0 , → [0 , is an equilibriumdistribution of the dynamics (145) if and only if there exists a θ ∈ (cid:2) a , a − (cid:3) such that Φ = Ψ θ . Lemma 4.2.
Assume the setting of Section 4.1 and let θ ∈ (cid:0) a , a − (cid:1) . Then we have Z
10 1 a − z Ψ θ ( dz ) < θ, if α > β, = θ, if α = β,> θ, if α < β. (150) Proof.
Define u := κβ ( aθ −
1) and v := κβ (1 − θ ( a − u, v ∈ (0 , ∞ ). Let Γ : (0 , ∞ ) → (0 , ∞ ) bethe Gamma function, i.e., for all x ∈ (0 , ∞ ) let Γ( x ) := R ∞ z x − e − z dz . It is well-known that for all x ∈ (0 , ∞ )the Gamma function satisfies Γ( x + 1) = x Γ( x ) and that for all x, y ∈ (0 , ∞ ) it holds that R z x − (1 − z ) y − dz = Γ( x )Γ( y )Γ( x + y ) . Thus, we obtain Z z u − (1 − z ) v − ( a − z ) (cid:16) a − z − θ (cid:17) dz = Z z u − (1 − z ) v − dz − aθ Z z u − (1 − z ) v − dz + θ Z z u (1 − z ) v − dz = Γ( u )Γ( v )Γ( u + v ) − aθ Γ( u )Γ( v )Γ( u + v ) + θ Γ( u +1)Γ( v )Γ( u + v +1) = (cid:16) (1 − aθ ) ( u + v )Γ( u )Γ( v )Γ( u + v +1) + θ u Γ( u )Γ( v )Γ( u + v +1) (cid:17) = ( u (1 − aθ + θ ) + v (1 − aθ )) Γ( u )Γ( v )Γ( u + v +1) = κβ (( aθ − − θ ( a − − θ ( a − − aθ )) Γ( u )Γ( v )Γ( u + v +1) = (cid:16) κβ (1 − θ ( a − aθ − − aθ ) (cid:17) Γ( u )Γ( v )Γ( u + v +1) = 0 . (151)First, consider the case α = β . Using (151) we see that Z
10 1 a − z Ψ θ ( dz ) − θ = Z c θ z κβ ( aθ − − (1 − z ) κβ (1 − θ ( a − − ( a − z ) αβ − (cid:16) a − z − θ (cid:17) dz = c θ Z z u − (1 − z ) v − ( a − z ) (cid:16) a − z − θ (cid:17) dz = 0 . (152)Now, consider the case α > β . Let ˆ δ := α − β , δ := δβ , and z ∗ := sup { z ∈ (0 ,
1) : a − z − θ < } . Note thatˆ δ, δ > z ∗ = a − θ ∈ (0 , z ∈ (0 , z ∗ ) we have a − z − θ < a − z ) δ > ( a − z ∗ ) δ .Furthermore, for all z ∈ ( z ∗ ,
1) we have a − z − θ > a − z ) δ < ( a − z ∗ ) δ . Together with (151) we thereby35btain Z
10 1 a − z Ψ θ ( dz ) − θ = Z (cid:16) a − z − θ (cid:17) Ψ θ ( dz ) = Z c θ z u − (1 − z ) v − ( a − z ) αβ − (cid:16) a − z − θ (cid:17) dz = Z z ∗ c θ z u − (1 − z ) v − ( a − z ) δ (cid:16) a − z − θ (cid:17) dz + Z z ∗ c θ z u − (1 − z ) v − ( a − z ) δ (cid:16) a − z − θ (cid:17) dz< c θ ( a − z ∗ ) δ (cid:18) Z z ∗ z u − (1 − z ) v − ( a − z ) (cid:16) a − z − θ (cid:17) dz + Z z ∗ z u − (1 − z ) v − ( a − z ) (cid:16) a − z − θ (cid:17) dz (cid:19) = c θ ( a − z ∗ ) δ Z z u − (1 − z ) v − ( a − z ) (cid:16) a − z − θ (cid:17) dz = 0 . (153)The case α < β can be proved analogously and thereby, we omit it here. This finishes the proof. Proof of Theorem 1.4.
Applying Itˆo’s lemma, we get for all t ∈ [0 , ∞ ) that a − Z t − a − Z = Z t a − Z s ) (cid:16) κ ( a − Z s ) (cid:16) ( a − Z s ) E h a − Z s i − (cid:17) − αZ s (1 − Z s ) (cid:17) +
12 2( a − Z s )( a − Z s ) β ( a − Z s ) Z s (1 − Z s ) ds + Z t a − Z s ) p β ( a − Z s ) Z s (1 − Z s ) dW s = Z t κ (cid:16) E h a − Z s i − a − Z s (cid:17) − αZ s (1 − Z s )( a − Z s ) + βZ s (1 − Z s )( a − Z s ) ds + Z t a − Z s ) p β ( a − Z s ) Z s (1 − Z s ) dW s . (154)After taking expectations we can apply Fubini’s theorem to obtain for all t ∈ [0 , ∞ ) that E (cid:2) a − Z t (cid:3) − E h a − Z i = Z t κ (cid:16) E h a − Z s i − E h a − Z s i(cid:17) − α E h Z s (1 − Z s )( a − Z s ) i + β E h Z s (1 − Z s )( a − Z s ) i ds = ( β − α ) Z t E h Z s (1 − Z s )( a − Z s ) i ds. (155)Since for all s ∈ [0 , ∞ ) it holds that E (cid:2) Z s (1 − Z s )( a − Z s ) (cid:3) ≥ , ∞ ) ∋ t E (cid:2) a − Z t (cid:3) ∈ (cid:2) a , a − (cid:3) converges monotonically non-increasing as t → ∞ if α > β , monotonically non-decreasing if α < β , oris constant if α = β .First, assume α > β . From (145) we see that δ is an invariant measure for Z . So if P [ Z = 1] = 1, thenfor all t ∈ [0 , ∞ ) it holds that P [ Z t = 1] = 1. Now let P [ Z = 1] <
1, implying E h a − Z i ∈ (cid:2) a , a − (cid:1) . Define θ := lim t →∞ E (cid:2) a − Z t (cid:3) and fix it for the rest of the paragraph. Note that due to the monotonicity stated above wehave θ ∈ (cid:2) a , a − (cid:1) . Aiming at a contradiction, we assume that θ ∈ (cid:0) a , a − (cid:1) . Choose any ε ∈ (cid:0) , a − − θ (cid:1) andfix it for the rest of the proof. By definition of θ there exists an s ε ∈ (0 , ∞ ), such that for all t ∈ [ s ε , ∞ ) itholds that E (cid:2) a − Z t (cid:3) < θ + ε . Let ˜ W : [0 , ∞ ) × Ω → R be a Brownian motion with continuous sample paths, let˜ Z : [0 , ∞ ) × Ω → [0 ,
1] and ˜ Z θ + ε : [0 , ∞ ) × Ω → [0 ,
1] be adapted processes with continuous sample paths thatsatisfy for all t ∈ [0 , ∞ ) P -a.s.˜ Z t = ˜ Z + Z t (cid:16) κ ( a − ˜ Z s ) (cid:16) ( a − ˜ Z s ) E h a − ˜ Z s i − (cid:17) − α ˜ Z s (1 − ˜ Z s ) (cid:17) ds + Z t q β ( a − ˜ Z s ) ˜ Z s (1 − ˜ Z s ) d ˜ W s , ˜ Z θ + εt = ˜ Z θ + ε + Z t (cid:16) κ ( a − ˜ Z θ + εs ) (cid:16) ( a − ˜ Z θ + εs )( θ + ε ) − (cid:17) − α ˜ Z θ + εs (1 − ˜ Z θ + εs ) (cid:17) ds + Z t q β ( a − ˜ Z θ + εs ) ˜ Z θ + εs (1 − ˜ Z θ + εs ) d ˜ W s , (156)36uch that ˜ Z θ + ε = ˜ Z and such that ˜ Z and Z s ε are equal in distribution. Then for each t ∈ [ s ε , ∞ ) we have that Z t and ˜ Z t − s ε are equal in distribution and the drift term of ˜ Z t − s ε is lower than that of ˜ Z θ + εt − s ε . Together withthe fact that the mapping [0 , ∋ z a − z is strictly monotonically increasing this implies for all t ∈ [ s ε , ∞ )that E (cid:2) a − Z t (cid:3) = E h a − ˜ Z t − sε i ≤ E h a − ˜ Z θ + εt − sε i . (157)Recall from Section 4.2 that for any η ∈ (cid:0) a , a − (cid:1) we have that Ψ η is the unique equilibrium distribution of ˜ Z η .Combining this with (157) we obtain (see, e.g., Theorem V.54.5 in Rogers and Williams [34]) θ = lim t →∞ E (cid:2) a − Z t (cid:3) ≤ lim t →∞ E h a − ˜ Z θ + εt − sε i = Z
10 1 a − z Ψ θ + ε ( dz ) . (158)The dominated convergence theorem yields that the mapping (cid:0) a , a − (cid:1) ∋ η Ψ η is continuous with respect tothe weak topology. Applying this, (158) together with the fact that ε ∈ (cid:0) , a − − θ (cid:1) was arbitrarily chosen, andLemma 4.2, we obtain the contradiction θ ≤ lim δ → Z
10 1 a − z Ψ θ + δ ( dz ) = Z
10 1 a − z Ψ θ ( dz ) < θ. (159)Hence, we have θ = a , implying0 ≤ lim t →∞ E [ Z t ] ≤ lim t →∞ a E h Z t a ( a − Z t ) i = lim t →∞ a E (cid:2) a − Z t (cid:3) − a a = 0 . (160)The case α < β can be proved analogously and we omit it here.Finally, assume α = β , define θ := E [ a − Z ], and fix it for the rest of the proof. We see from (155) that E [ a − Z t ] is constant in t ∈ [0 , ∞ ). Thus, assuming that Z and Z θ are equal in distribution we see from (145)and (146) that for all t ∈ [0 , ∞ ) it holds that Z t and Z θt are equal in distribution. Recall from Section 4.2 thatΨ θ is the unique equilibrium distribution of Z θ . Consequently, Ψ θ is the unique equilibrium distribution of Z .This finishes the proof of Theorem 1.4. Let (Ω , F , P ) be a probability space, let κ, α, β ∈ (0 , ∞ ), a ∈ (1 , ∞ ), and let W ( i ) : [0 , ∞ ) × Ω → R , i ∈ N , beindependent Brownian motions with continuous sample paths. For all D ∈ N let X D : [0 , ∞ ) × { , . . . , D } × Ω → [0 ,
1] be an adapted process with continuous sample paths that for all t ∈ [0 , ∞ ) and all i ∈ { , . . . , D } P -a.s. satisfies X Dt ( i ) = X D ( i ) + Z t ( a − X Ds ( i )) (cid:18) ( a − X Ds ( i )) D D X j =1 1 a − X Ds ( j ) − (cid:19) − αX Ds ( i )(1 − X Ds ( i )) ds + Z t q β ( a − X Ds ( i )) X Ds ( i )(1 − X Ds ( i )) dW s ( i ) . (161)Let ˜ a : [0 , ∞ ) → [0 , ∞ ) be a function defined by[0 , ∞ ) ∋ x ˜ a ( x ) := κa min { x, } a − min { x, } + ( x − + . (162)Then, assuming there is positive mass only in deme 1, the dynamics in deme 1 follows asymptotically thefollowing process Y . Let Y : [0 , ∞ ) × Ω → [0 ,
1] be an adapted process with continuous sample paths such thatfor all t ∈ [0 , ∞ ) it P -a.s. holds that Y t = Y − Z t κa Y s ( a − Y s ) + αY s (1 − Y s ) ds + Z t p β ( a − Y s ) Y s (1 − Y s ) dW s (1) . (163)37n addition, let Q Y be the excursion measure which satisfies Q Y = lim <ε → ε P [ Y ∈ ·| Y = ε ] in a suitablesense; see Pitman and Yor [29] and Hutzenthaler [15] for details. Asymptotically in the many-demes limit, everydeme with population path χ ∈ C ([0 , ∞ ) , [0 , a ( χ t ) dt × Q Y ( dψ ). Now let ( V t ) t ∈ [0 , ∞ ) be the totalmass process of the associated tree of excursions with initial island measure that equals the distribution of Y in (163) and excursion measure Q Y . Proposition 5.1.
Assume the setting of Section 5.1. Let x ∈ (0 , and assume Y = x = V . Then the totalmass process dies out (i.e., converges in probability to zero as t → ∞ ) if and only if α ≥ β. (164) Proof.
Define the functions s : [0 , → [0 , ∞ ) and S : [0 , → [0 , ∞ ) by [0 , ∋ z s ( z ) := exp (cid:16) − R z − κa x ( a − x ) − αx (1 − x ) β ( a − x ) x (1 − x ) dx (cid:17) and [0 , ∋ y S ( y ) := R y s ( z ) dz . Note that for all z ∈ [0 ,
1] it holds that s ( z ) = exp (cid:16) Z z κaβ − x + αβ a − x dx (cid:17) = (1 − z ) − κaβ (cid:16) a − za (cid:17) − αβ (165)and S ( z ) = Z z s ( x ) dx ≤ zs ( z ) . (166)We will apply Theorem 5 from Hutzenthaler [15] to show the result. First, we verify that the assumptions ofthe aforementioned theorem are satisfied. Using (166), we see that Z S ( y ) β ( a − y ) y (1 − y ) s ( y ) dy ≤ Z β ( a − y )(1 − y ) dy ≤
12 2 β ( a − )(1 − ) < ∞ . (167)Furthermore, we getlim ε → Z ε − κa ( a − y ) y − αy (1 − y ) β ( a − y ) y (1 − y ) dy = lim ε → Z ε − κaβ (1 − y ) − αβ ( a − y ) dy = lim ε → (cid:16) κaβ (ln(1 − ) − ln(1 − ε )) + αβ (ln( a − ) − ln( a − ε )) (cid:17) = κaβ ln(1 − ) + αβ (ln( a − ) − ln( a )) ∈ ( −∞ , ∞ ) . (168)From (165) as well as the fact that κaβ > Z ˜ a ( y ) β ( a − y ) y (1 − y ) s ( y ) dy = Z κa ya − y β ( a − y ) y (1 − y ) (1 − y ) κaβ (cid:16) a − ya (cid:17) αβ dy = 2 κaβ a − αβ Z (1 − y ) κaβ − ( a − y ) αβ − dy ≤ κaβ a − αβ (cid:16) ( a − ) αβ − + ( a − αβ − (cid:17) Z (1 − y ) κaβ − dy < ∞ . (169)We obtain from (167), (168), and (169) together with a straightforward adaptation of Lemmas 9.6, 9.9, and9.10 in Hutzenthaler [15] to the state space [0 ,
1] that the assumptions of Theorem 5 in Hutzenthaler [15] aresatisfied. Applying the aforementioned theorem shows that the total mass process dies out if and only if
Z Z ∞ ˜ a ( χ t ) dt Q Y ( dχ ) ≤ . (170)38oreover, a straight forward adaptation of Lemma 9.8 in Hutzenthaler [15] to the state space [0 ,
1] togetherwith (165) shows that
Z Z ∞ ˜ a ( χ t ) dt Q Y ( dχ ) = Z κa ya − y β ( a − y ) y (1 − y ) (1 − y ) κaβ (cid:16) a − ya (cid:17) αβ dy. (171)Observe that we have κaβ R (1 − y ) κaβ − dy = 1. Combining this with (170) and (171) we see that the total massprocess dies out if and only if0 ≥ Z κa ya − y β ( a − y ) y (1 − y ) (1 − y ) κaβ (cid:16) a − ya (cid:17) αβ dy − κaβ Z (1 − y ) κaβ − (cid:16) a − ya (cid:17) αβ − dy − κaβ Z (1 − y ) κaβ − (cid:18)(cid:16) a − ya (cid:17) αβ − − (cid:19) dy. (172)Consequently, the total mass process dies out if and only if α ≥ β . This finishes the proof of Proposition 5.1. References [1]
Abbot, P., Abe, J., Alcock, J., Alizon, S., Alpedrinha, J. A., Andersson, M., Andre, J.-B.,van Baalen, M., Balloux, F., Balshine, S., et al.
Inclusive fitness theory and eusociality.
Nature471 , 7339 (2011), E1–E4.[2]
Best, A., Webb, S., White, A., and Boots, M.
Host resistance and coevolution in spatially structuredpopulations.
Proc. R. Soc. B 278 , 1715 (2011), 2216–2222.[3]
Bshary, R., and Grutter, A. S.
Image scoring and cooperation in a cleaner fish mutualism.
Nature441 , 7096 (2006), 975–978.[4]
Comins, H., Hassell, M., and May, R.
The spatial dynamics of host–parasitoid systems.
J. Anim.Ecol. (1992), 735–748.[5]
Darwin, C. R.
The origin of species . J. Murray, London, 1859.[6]
Dobrinevski, A., and Frey, E.
Extinction in neutrally stable stochastic Lotka-Volterra models.
Phys.Rev. E 85 , 5 (2012), 051903.[7]
Ethier, S. N., and Kurtz, T. G.
Markov processes: Characterization and convergence . Wiley Seriesin Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & SonsInc., New York, 1986.[8]
Fukuyo, M., Sasaki, A., and Kobayashi, I.
Success of a suicidal defense strategy against infection ina structured habitat.
Sci. rep. 2 (2012).[9]
Gardner, A.
The genetical theory of multilevel selection.
J. Evol. Biol. (2015).[10]
Goodnight, C. J., Rauch, E. M., Sayama, H., De Aguiar, M. A., Baranger, M., and Bar-yam,Y.
Evolution in spatial predator–prey models and the “prudent predator”: The inadequacy of steady-stateorganism fitness and the concept of individual and group selection.
Complexity 13 , 5 (2008), 23–44.[11]
Goodnight, C. J., and Stevens, L.
Experimental studies of group selection: what do they tell us aboutgroup selection in nature?
Am. Nat. 150 , S1 (1997), s59–s79.[12]
Goodnight, C. J., and Wade, M. J.
The ongoing synthesis: a reply to Coyne, Barton, and Turelli.
Evolution 54 , 1 (2000), 317–324.[13]
Hamilton, W.
The genetical evolution of social behaviour. I.
J. Theor. Biol. 7 , 1 (1964), 1–16.3914]
Haraguchi, Y., and Sasaki, A.
The evolution of parasite virulence and transmission rate in a spatiallystructured population.
J. Theor. Biol. 203 , 2 (2000), 85–96.[15]
Hutzenthaler, M.
The virgin island model.
Electron. J. Probab. 14 (2009), no. 39, 1117–1161 (elec-tronic).[16]
Hutzenthaler, M.
Interacting diffusions and trees of excursions: convergence and comparison.
Electron.J. Probab. 17 (2012), no. 71, 1–49.[17]
Hutzenthaler, M., and Wakolbinger, A.
Ergodic behavior of locally regulated branching populations.
Ann. Appl. Probab. 17 , 2 (2007), 474–501.[18]
Karlin, S., and Taylor, H. M.
A second course in stochastic processes . Academic Press Inc. [HarcourtBrace Jovanovich Publishers], New York, 1981.[19]
Klenke, A.
Probability theory . Universitext. Springer-Verlag London Ltd., London, 2008. A comprehensivecourse, Translated from the 2006 German original.[20]
Klenke, A., and Mytnik, L.
Infinite rate mutually catalytic branching in infinitely many colonies:construction, characterization and convergence.
Probab. Theory Related Fields 154 , 3-4 (2012), 533–584.[21]
Kurtz, T. G.
Averaging for martingale problems and stochastic approximation. In
Applied stochasticanalysis (New Brunswick, NJ, 1991) , vol. 177 of
Lecture Notes in Control and Inform. Sci.
Springer, Berlin,1992, pp. 186–209.[22]
Lotka, A. J.
Undamped oscillations derived from the law of mass action.
J. Am. Chem. Soc. 42 , 8 (1920),1595–1599.[23]
Maynard Smith, J.
Group selection and kin selection.
Nature 201 (1964), 1145–1147.[24]
Maynard Smith, J.
Group selection.
Q. Rev. Biol. 51 , 2 (1976), pp. 277–283.[25]
McGregor, P. K. , Ed.
Animal communication networks . Cambridge University Press, 2005.[26]
Nowak, M. A.
Five rules for the evolution of cooperation.
Science 314 , 5805 (2006), 1560–1563.[27]
Nowak, M. A., Tarnita, C. E., and Wilson, E. O.
The evolution of eusociality.
Nature 466 , 7310(2010), 1057–1062.[28]
Pamminger, T., Foitzik, S., Metzler, D., and Pennings, P. S.
Oh sister, where art thou? spatialpopulation structure and the evolution of an altruistic defence trait.
J. Evol. Biol. 27 , 11 (2014), 2443–2456.[29]
Pitman, J., and Yor, M.
A decomposition of Bessel bridges.
Z. Wahrsch. Verw. Gebiete 59 , 4 (1982),425–457.[30]
Queller, D. C.
Quantitative genetics, inclusive fitness, and group selection.
Am. Nat. (1992), 540–558.[31]
Rand, D., Keeling, M., and Wilson, H.
Invasion, stability and evolution to criticality in spatiallyextended, artificial host-pathogen ecologies.
Proc. R. Soc. B 259 , 1354 (1995), 55–63.[32]
Rauch, E. M., Sayama, H., and Bar-Yam, Y.
Relationship between measures of fitness and time scalein evolution.
Phys. Rev. Lett. 88 , 22 (2002), 228101.[33]
Rauch, E. M., Sayama, H., and Bar-Yam, Y.
Dynamics and genealogy of strains in spatially extendedhost–pathogen models.
J. Theor. Biol. 221 , 4 (2003), 655–664.[34]
Rogers, L. C. G., and Williams, D.
Diffusions, Markov processes and martingales. Vol. 2 . CambridgeMathematical Library. Cambridge University Press, Cambridge, 2000. Itˆo calculus, Reprint of the second(1994) edition. 4035]
Shorter, J., and Rueppell, O.
A review on self-destructive defense behaviors in social insects.
InsectesSoc. 59 , 1 (2012), 1–10.[36]
Traulsen, A.
Mathematics of kin- and group-selection: formally equivalent?
Evolution 64 , 2 (2010),316–323.[37]
Traulsen, A., and Nowak, M. A.
Evolution of cooperation by multilevel selection.
Proc. Natl. Acad.Sci. USA 103 , 29 (2006), 10952–10955.[38]
Van Valen, L.
Group selection and the evolution of dispersal.
Evolution (1971), 591–598.[39]
Volterra, V.
Fluctuations in the abundance of a species considered mathematically.
Nature 118 (1926),558–560.[40]
Wade, M. J.
A critical review of the models of group selection.
Q. Rev. Biol. (1978), 101–114.[41]
West, S. A., Griffin, A. S., and Gardner, A.
Evolutionary explanations for cooperation.
Curr. Biol.17 , 16 (2007), R661–R672.[42]
Yamada, T., and Watanabe, S.
On the uniqueness of solutions of stochastic differential equations.