Stochastic eco-evolutionary model of a prey-predator community

Abstract

We are interested in the impact of natural selection in a prey-predator community. We introduce an individual-based model of the community that takes into account both prey and predator phenotypes. Our aim is to understand the phenotypic coevolution of prey and predators. The community evolves as a multi-type birth and death process with mutations. We first consider the infinite particle approximation of the process without mutation. In this limit, the process can be approximated by a system of differential equations. We prove the existence of a unique globally asymptotically stable equilibrium under specific conditions on the interaction among prey individuals. When mutations are rare, the community evolves on the mutational scale according to a Markovian jump process. This process describes the successive equilibria of the prey-predator community and extends the polymorphic evolutionary sequence to a coevolutionary framework. We then assume that mutations have a small impact on phenotypes and consider the evolution of monomorphic prey and predator populations. The limit of small mutation steps leads to a system of two differential equations which is a version of the canonical equation of adaptive dynamics for the prey-predator coevolution. We illustrate these different limits with an example of prey-predator community that takes into account different prey defense mechanisms. We observe through simulations how these various prey strategies impact the community.

Appendices

Appendix A: Construction of a trajectory of the prey-predator community process

We construct a trajectory of the prey-predator community process as solution of a system stochastic differential equations driven by Poisson point measures (see Fournier and Méléard 2004; Champagnat et al. 2006). We introduce two families of independent Poisson point measures on $(\mathbb {R}_+)^2$ with intensity $dsd\theta$: $(R_j)_{1\le j\le d+m}$ for the prey and predators reproduction events and $(M_j)_{1\le j\le d+m}$ for the death events. Then, $\forall 1\le i\le d$ and $\forall 1\le l\le m$

$$\begin{aligned} \begin{aligned} N^K_i(t)&=N^K_i(0) +\int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le b(x_i)N^K_i(s-)} R_i(ds,d\theta ) \\&\quad -\int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le \lambda (x_i,\mathbf {Z}(s-))N^K_i(s-) } M_i(ds, d\theta ) ,\\ H^K_l(t)&=H^K_l(0) + \int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le rH^K_l(s-)\bigl (\sum _{i=1}^d \frac{B(x_i,y_l)}{K}N^K_i(s-) \bigr )} R_{d+l}(ds,d\theta ) \\&\quad -\int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le D(y_l)H^K_l(s-) } M_{d+l}(ds, d\theta ). \end{aligned} \end{aligned}$$

(36)

Let us explain briefly these equations. We focus on the prey population $N^K_i$ with trait $x_i$. A trajectory is constructed using two Poisson point measures $R_i$ and $M_i$. The measure $R_i$ handles the reproduction events and $M_i$ the death events. A Poisson point measure $R$ on $(\mathbb {R}_+)^2$ with intensity $dsd\theta$ charges a countable set of points $\varOmega =\{(s_u,\theta _u),u\in \mathbb {N}\}$ (with mass $1$ on each point) (e.g. Watanabe and Ikeda 1981 Chapter I.8 for a complete definition). Then $\int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le b(x_i)N^K_i(s-)} R_i(ds,d\theta )$ only counts the points $(s^i_u,\theta ^i_u)_{u\in \mathbb {N}}$ such that $s_u^i\le t$ and $\theta ^i_u \le b(x_i)N^K_i(s^i_u-)$. Thus, we select the points of $R_i$ which correspond to birth events of the prey population. The other integrals have similar interpretations.

The existence of solutions of (36) is justified by Proposition 2.1(i). From this construction, we deduce the expression of the prey and the predator population sizes:

$$\begin{aligned} N^K(t)= & {} N^K(0) +\sum _{i=1}^d \left[ \int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le b(x_i)N^K_i(s-)} R_i(ds,d\theta )\right. \\&\quad -\left. \int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le \lambda (x_i,\mathbf {Z}(s-))N^K_i(s-) } M_i(ds, d\theta )\right] ,\\ H^K(t)= & {} H^K(0) +\sum _{l=1}^m \Bigl [ \int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le rH^K_l(s-)\left( \sum _{i=1}^d \frac{B(x_i,y_l)}{K}N^K_i(s-) \right) } R_{d+l}(ds,d\theta ) \\&\quad -\int _0^t\int _{\mathbb {R}_+} \mathbf {1}_{\theta \le D(y_l)H^K_l(s-) } M_{d+l}(ds, d\theta )\Bigr ]. \end{aligned}$$

Appendix B: Proof of Proposition 2.1

(i) For the first part, we compare the prey population with a population evolving in the absence of predators. Let us denote by $(\widetilde{N}^K_1,\dots ,\widetilde{N}^K_d)$ the sizes of the prey sub-populations evolving without predators and set $\widetilde{N}^K=\sum _{i=1}^d\widetilde{N}^K_i$. Using the description given in Appendix A, we construct the processes $N^K$ and $\widetilde{N}^K$ on the same probability space in such a way that $\forall t\ge 0$, $\widetilde{N}^K(t)\ge N^K(t)$ almost surely. Fournier and Méléard (2004, Theorem 5.3) and Champagnat et al. (2006, Lemma 1) established that

$$\begin{aligned} \sup _K\mathbb {E}\left( \sup _{t\in [0,T]} \left( \frac{\widetilde{N}^K(t)}{K} \right) ^3 \right) <\infty \quad \text {and }\quad \mathbb {E}\left( \sup _K \sup _{t\in [0,T]} \left( \frac{\widetilde{N}^K(t)}{K} \right) ^3 \right) <\infty . \end{aligned}$$

(37)

The process $N^K$ then satisfies the same moment properties. To study the number of predators, we define $\tau _n=\inf \{t\ge 0,H^K(t)\ge n\}$. By neglecting the death events, we obtain that

$$\begin{aligned}&\mathbb {E}\left( \sup _{t\in [0,T\wedge \tau _n]}\left( \frac{H^{K}(t)}{K}\right) ^3 \right) \le \mathbb {E}\left( \left( \frac{H^{K}(0)}{K}\right) ^3\right) \\&\quad +\,4\mathbb {E}\left( \int _0^{T\wedge \tau _n} \left( 1+\left( \frac{H^{K}(s)}{K}\right) ^{2}\right) \frac{H^{K}(s)}{K} \frac{N^K(s)}{K} r\bar{B}ds \right) , \end{aligned}$$

where we used that $(1+x)^3-x^3\le 4(1+x^2)$, $\forall x\ge 0$. Since the process $\widetilde{N}^K$ is independent of the number $H^K$ of predators we get that

$$\begin{aligned} \begin{aligned}&\mathbb {E}\left( \sup _{t\in [0,T\wedge \tau _n]}\left( \frac{H^{K}(t)}{K}\right) ^3\right) \\&\quad \le \phi (T) +2r\bar{B}\int _0^{T} \mathbb {E}\left( \sup _{t\in [0,s\wedge \tau _n]}\left( \frac{H^{K}(t)}{K}\right) ^{3}\right) \mathbb {E}\left( \sup _{t\in [0,s]} \frac{\widetilde{N}^K(t)}{K}\right) ds,\\ \end{aligned} \end{aligned}$$

where $\phi (T)=\mathbb {E}\left( \left( \frac{H^{K}(0)}{K}\right) ^3\right) +2r\bar{B}T\mathbb {E}\left( \sup _{t\in [0,T]}\frac{\widetilde{N}^K(t)}{K}\right)$. By Gronwall’s Lemma and (37), we obtain that

$$\begin{aligned} \mathbb {E}\left( \sup _{t\in [0,T\wedge \tau _n]}\left( \frac{H^{K}(t)}{K}\right) ^3\right) \le C(T) \end{aligned}$$

(38)

which concludes point (i) and proves the existence of $\mathbf {Z}^K$ for all times.

(ii) The second part is much more difficult since using such a coupling is not possible: the constant $C(T)$ obtained in (38) goes to $\infty$ as $T\rightarrow \infty$. In the sequel we study the behavior of the time derivative of $\mathbb {E}\Bigl (\bigl ( \frac{N^K(t)+H^K(t)}{K} \bigr )^2\Bigr )$. We gather together the terms related to predation and bound the other terms using Assumption A to obtain

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\mathbb {E}&\left( \left( \frac{N^K(t)}{K}+\frac{H^K(t)}{K} \right) ^2\right) \le \mathbb {E}\bigl (K\Psi (\mathbf {Z}^K(t))\bigr ), \end{aligned} \end{aligned}$$

(39)

where

$$\begin{aligned} \begin{aligned} \Psi (\mathbf {Z}^K)&\,= \sum _{i=1}^d\sum _{l=1}^m \frac{H^K_l}{K}\frac{N^K_i}{K}B(x_i,y_l)\\&\quad \times \left[ \left( \frac{N^K+H^K-1}{K}\right) ^2-\left( \frac{N^K+H^K}{K}\right) ^2\right. \\&\left. \qquad +\,r\left( \frac{N^K+H^K+1}{K}\right) ^2-r\left( \frac{N^K+H^K}{K}\right) ^2 \right] \\&\quad +\,\frac{N^K}{K}\bar{b} \left[ \left( \frac{N^K+H^K+1}{K}\right) ^2-\left( \frac{N^K+H^K}{K}\right) ^2\right] \\&\quad +\underline{c}\frac{ (N^K)^2 }{K^2} \left[ \left( \frac{N^K+H^K-1}{K}\right) ^2-\left( \frac{N^K+H^K}{K}\right) ^2\right] \\&\quad +\frac{H^K}{K} \underline{D}\left[ \left( \frac{N^K+H^K-1}{K} \right) ^2-\left( \frac{N^K+H^K}{K}\right) ^2\right] . \end{aligned} \end{aligned}$$

(40)

The function $\Psi$ is the sum of three terms that we handle separately. The first term gathers together all the predation effects. The second term (sum of the second and third terms) only depends on the prey population. The last term is related to the death of predators. We start with the first term. To remove the dependence on the traits, we search for conditions on the term between square brackets to be non positive. This is equivalent to consider the sign of $(1-\frac{1}{n+h})^2-1+r(1+\frac{1}{n+h})^2-r,$ for $(n,h)\in \mathbb {N}^2{\setminus }\{(0,0)\}$. It is non positive as soon as $n+h\ge \frac{(1+r)}{2(1-r)}=n_1$. Thus if $N^K>n_1$,

$$\begin{aligned} \begin{aligned}&\sum _{i=1}^d\sum _{l=1}^m \frac{H^K_l}{K}\frac{N^K_i}{K}B(x_i,y_l)\left( \frac{N^K+H^K}{K}\right) ^2\\&\quad \times \left[ \left( 1-\frac{1}{N^K+H^K }\right) ^2-1+r(1+\frac{1}{N^K+H^K})^2 -r \right] \\&\le \frac{N^KH^K}{K^2}\left( \frac{N^K+H^K}{K}\right) ^2\underline{B}\left[ \left( 1-\frac{1}{N^K+H^K }\right) ^2\right. \\&\left. \qquad -1+r\left( 1+\frac{1}{N^K+H^K}\right) ^2 -r \right] , \end{aligned} \end{aligned}$$

(41)

which is non positive.

For the second term, let us remark that if $N^K>K \frac{2\bar{b}}{\underline{c}}=Kn_2$, then

$$\begin{aligned} \begin{aligned}&N^K\bar{b}\left( \left( 1+\frac{1}{N^K+H^K}\right) ^2-1\right) +\frac{\underline{c}}{K}(N^K)^2\left( \left( 1-\frac{1}{N^K+H^K}\right) ^2-1\right) \\&\quad \le N^K\bar{b}\left[ \left( \left( 1+\frac{1}{N^K+H^K}\right) ^2-1\right) +2\left( \left( 1-\frac{1}{N^K+H^K}\right) ^2-1\right) \right] , \end{aligned} \end{aligned}$$

(42)

We set $n_0=\max (n_1,n_2)$. If $N^K\ge Kn_0$, we obtain by combining (41) and (42) that:

$$\begin{aligned} \begin{aligned} \Psi (\mathbf {Z}^K)&\le \frac{1}{K}\left( \frac{N^K+H^K}{K}\right) ^2 \left[ N^K\bar{b}\left[ \left( \left( 1+\frac{1}{N^K+H^K}\right) ^2-1\right) \right. \right. \\&\left. \left. \qquad +\,2\left( \left( 1-\frac{1}{N^K+H^K}\right) ^2-1\right) \right] \right. \\&\quad +\left. H^K\underline{D}\left[ \left( 1 -\frac{1}{N^K+H^K}\right) ^2-1\right] \right] . \end{aligned} \end{aligned}$$

(43)

Finally the term between square brackets in (43) is smaller than $-\min (\bar{b},\underline{D})$, as soon as $N^K\ge Kn_0$ for $n_0$ large enough. Thus $\forall t\ge 0$,

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\mathbb {E}&\left( \left( \frac{N^K(t)+H^K(t)}{K}\right) ^2\right) \\&\le \mathbb {E}\left( -\min (\bar{b},\underline{D})\left( \frac{N^K(t)+H^K(t)}{K}\right) ^2\mathbf {1}_{\{N^K(t)>Kn_0\}} +K\Psi (\mathbf {Z}^K(t))\mathbf {1}_{\{N^K\le Kn_0\}} \right) . \end{aligned} \end{aligned}$$

We now consider the event $\{N^K\le Kn_0\}$. On this event we aim at bounding from above the function $\Psi$ with

$$\begin{aligned} \Psi (\mathbf {Z}^K)\le \frac{1}{K}\left( \frac{N^K+H^K}{K}\right) ^2\varPhi ^K\left( \frac{N^K}{K},\frac{H^K}{K}\right) . \end{aligned}$$

(44)

Since for $(n,h)\in \mathbb {N}^2{\setminus }\{(0,0)\}$,

$$\begin{aligned} \left( 1-\frac{1}{n+h}\right) ^2-1 +r\left( 1+\frac{1}{n+h}\right) ^2-r = -2(1-r)\frac{1}{n+h}+(1+r)\frac{1}{(n+h)^2}, \end{aligned}$$

and Assumption 7, we set for every $(u,v)\in (\mathbb {R}_+)^2{\setminus }\{(0,0)\}$,

$$\begin{aligned} \varPhi ^K(u,v)= & {} \frac{2u}{u+v}(\bar{b}-\underline{c}u)-\frac{2v}{u+v}((1-r)\underline{B}u+\underline{D})\nonumber \\&\quad +\,\frac{u}{K(u+v)^2}(\bar{b}+\underline{c}u+(1+r)\bar{B}v)+\underline{D}\frac{v}{K(u+v)^2}. \end{aligned}$$

(45)

We seek a condition on $v$ to obtain that $\varPhi ^K(u,v)\le -D$, $\forall K\ge 0$, $\forall 0\le u\le n_0$. This inequality can be written as a polynomial

$$\begin{aligned} v^2 \alpha (u)+ v\beta (u,K)+\gamma (u,K)\le 0, \end{aligned}$$

(46)

where the coefficients are given by

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \alpha (u)=-2(1-r)\underline{B}u-\underline{D},\\ \beta (u,K)=2u(\bar{b}-\underline{c}u)-2u^2(1-r)\underline{B}+\frac{u}{K}(1+r)\bar{B}+\frac{\underline{D}}{K},\\ \displaystyle \gamma (u,K)=\frac{u}{K}(\bar{b}+\underline{c}u +\underline{D}\displaystyle u^2+2u^2(\bar{b}-\underline{c}u)).\\ \end{array}\right. \end{aligned}$$

As $\alpha (u)<0$, this polynomial remains negative for every $v$ greater than its largest real root. If the polynomial (46) has real roots, then we can bound from above the largest one with

$$\begin{aligned} \frac{|\beta (u,K)|+\sqrt{\beta (u,K)^2-4\alpha (u)\gamma (u,K)}}{-2\alpha (u)}. \end{aligned}$$

The coefficient $\beta (u,K)$ decreases with $K$, thus for every $K\ge 1$ and $0\le u\le n_0$,

$$\begin{aligned} \begin{aligned}&2u(\bar{b}-\underline{c}u)-2u^2(1-r)\underline{B}\le \beta (u,K)\le 2u(\bar{b}-\underline{c}u)\\&\qquad -2u^2(1-r)\underline{B}+u(1+r)\bar{B}+\underline{D}\\&\qquad -2(1+\underline{c}) n_0^2\underline{B}\le \beta (u,K)\le 2n_0 \bar{b}+n_0(1+r)\underline{B}+\underline{D}. \end{aligned} \end{aligned}$$

In the case where $\gamma (u,K)<0$, the discriminant $\Delta (u,K)=\beta (u,K)^2-4\alpha (u)\gamma (u,K)$ is bounded by $|\beta (u,K)|$. Otherwise $\Delta \le \beta (u,K)^2+8((1-r)\underline{B}u+\underline{D})u (\bar{b}+\underline{c}u +\underline{D}u^2+2u^2(u-\underline{c}u))$ which can be bounded uniformly for $u\in [0,n_0]$. Thus there exists $h_0$ independent on $K$ such that

$$\begin{aligned} \forall n\le Kn_0,\quad \forall \, h>Kh_0,\quad \varPhi ^K\left( \frac{n}{K},\frac{h}{K}\right) \le -D. \end{aligned}$$

Finally

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\mathbb {E}\left( \left( \frac{N^K(t)+H^K(t)}{K} \right) ^2\right) \le \mathbb {E}\left( K\Psi (\mathbf {Z}^K(t))\mathbf {1}_{\{N^K(t)\le Kn_0,H^K(t)\le Kh_0\}} \right) \\&\quad +\mathbb {E}\left( -C\left( \frac{N^K(t)+H^K(t)}{K}\right) ^2\left( \mathbf {1}_{\{N^K(t)>Kn_0\}}+\mathbf {1}_{\{N^K(t)\le Kn_0, H^K(t)>Kh_0\}}\right) \right) , \end{aligned} \end{aligned}$$

with $C>0$. To conclude it remains to bound the expectation of $\Psi$ on the event $\{N^K\le Kn_0 \text { and }H^K\le Kh_0\}$. Keeping only the positive terms we obtain that

$$\begin{aligned} \begin{aligned}&\mathbb {E}\left( K\Psi (\mathbf {Z}^K(t))=\mathbf {1}_{\{N^K(t)\le Kn_0,H^K(t)\le Kh_0\}} \right) \,\le \mathbb {E}\left( \mathbf {1}_{\{N^K(t)\le Kn_0,H^K(t)\le Kh_0\}}\right. \\&\quad \left. \times \left( \bar{b}N^K(t)+r\bar{B} N^K(t)\frac{H^K(t)}{K}\right) \left( \left( \frac{N^K(t)+H^K(t)}{K}+\frac{1}{K} \right) ^2-\left( \frac{N^K(t)+H^K(t)}{K} \right) ^2\right) \right) \\&\quad \,\le \sum _{n=0}^{Kn_0}\sum _{h=0}^{Kh_0}\left( \frac{n+h}{K}\right) ^2\left( \bar{b}n+r\bar{B} n\frac{h}{K}\right) \left( \left( 1+\frac{1}{n+h}\right) ^2-1\right) \\&\quad \,\le \sum _{n=0}^{Kn_0}\sum _{h=0}^{Kh_0}\left( \frac{n+h}{K}\right) ^2 3\left( \bar{b}+r\bar{B} \frac{h}{K}\right) , \end{aligned} \end{aligned}$$

where the last inequality derives from $(1+u)^2-1\le 3u$, for all $u\in [0,1]$.

Combining all these results

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\mathbb {E}\left( \left( \frac{N^K(t)+H^K(t)}{K}\right) ^2\right)&\le \mathbb {E}\left( -C\left( \frac{N^K(t)+H^K(t)}{K})^2\right) \right) \\&+ \int _0^{n_0}\int _{0}^{h_0}\left( 3( n+h)^2(C+\bar{b})+ (n+h)^3r\underline{B}\right) dh dn \\&\le C^{\prime }-C\mathbb {E}\left( \left( \frac{N^K(t)+H^K(t)}{K}\right) ^2\right) , \end{aligned} \end{aligned}$$

with $C^{\prime }>0$. We solve this inequality to get that

$$\begin{aligned} \begin{aligned} \mathbb {E}\left( \left( \frac{N^K(t)+H^K(t)}{K}\right) ^2\right)&\le C^{\prime } +\left( \mathbb {E}\left( \left( \frac{N^K(0)+H^K(0)}{K}\right) ^2\right) -C^{\prime }\right) e^{-Ct}. \end{aligned} \end{aligned}$$

which gives the uniform bound.

Appendix C: Proof of Theorem 3.2

The proof relies on the expression of Linear Complementarity Problems as variational inequality problems.

Definition 2

The variational inequality problem associated with a function $f:\mathbb {R}^u\rightarrow \mathbb {R}^u$ and a subset $E\subset \mathbb {R}^u$ seeks a vector $z\in E$ such that

$$\begin{aligned} \forall a\in E,\quad (a-z)^T f(z)\ge 0. \end{aligned}$$

(47)

The existence of solutions is not true in a general setting but we are interested in a specific framework where the subset $E$ is compact and convex.

Theorem C1

Let $E$ be a non empty compact convex of $\mathbb {R}^u$ and $f$ continuous function, then the variational inequality problem associated to $(f,E)$ admits a solution.

The proof of Theorem C1 is rather classical and requires to express a solution as a fix point of a projection of the subset $E$ (see Cottle et al. 1992, Theorem 3.7.1). With this result we can prove the Theorem 3.2.

Proof

(Proof of Theorem 3.2 ) Let us recall that a solution to the Linear complementarity problem associated to the couple $(\tilde{M},\tilde{q})$ defined in (16) is a vector $\mathbf {z}=(\mathbf {n},\mathbf {h})\in \mathbb {R}^d\times \mathbb {R}^m$ such that: for every $1\le i\le d$ and $1\le l\le m$,

$$\begin{aligned} n_i\ge 0,\quad (q+M\mathbf {n}+B\mathbf {h})_i\ge 0,\quad (\mathbf {n})^T(q+M\mathbf {n}+B\mathbf {h})=0 \end{aligned}$$

(48)

and

$$\begin{aligned} h_l\ge 0,\quad (D-B^T\mathbf {n})_l\ge 0, \quad (\mathbf {h})^T(D -B^T\mathbf {n})=0 \end{aligned}$$

(49)

These conditions (48) entail that the vector $\mathbf {n}$ is a solution to $LCP(M,q+B\mathbf {h})$.

Note that if $\mathbf {n}\in \mathbb {R}^d$ is solution to the restricted problem $LCP(M,q)$ satisfying moreover $(-B^T\mathbf {n}+D)_l\ge 0$ for all $1\le l\le m$, then the vector $(\mathbf {n},0)$ is solution to $LCP(\tilde{M},\tilde{q})$. Similarly we seek a suitable vector $\mathbf {n}$ and adjust it thanks to the vector $\mathbf {h}$.

We consider the variational inequality problem associated to the set

$$\begin{aligned} E=\{\mathbf {n}\in (\mathbb {R}_+)^d,\quad \forall 1\le l\le m \quad (D-B^T\mathbf {n})_l\ge 0 \}, \end{aligned}$$

and the continuous function $f(\mathbf {n})=q+M\mathbf {n}$.

Since $D$ is non negative, the set $E$ is not empty. Moreover $E$ is convex, closed and bounded thus compact. Theorem C1 ensures the existence of a solution $\mathbf {n}^*$ to this problem. Note that (47) can be written as

$$\begin{aligned} \forall a\in E, \quad a^Tf(\mathbf {n}^*)\ge (\mathbf {n}^*)^Tf(\mathbf {n}^*). \end{aligned}$$

Thus $\mathbf {n}^*$ minimizes the function $a\rightarrow a^Tf(\mathbf {n}^*)$ on $E$. Therefore

• either $\mathbf {n}^*$ is in the interior of $E$ and is therefore a global minimizer of the function $a\rightarrow a^Tf(\mathbf {n}^*)$ on $\mathbb {R}^d$ and $(\mathbf {n}^*,0)$ is a solution to $LCP(\widetilde{M},\widetilde{q})$.

• otherwise we can define the Lagrange multipliers for this problem. There exist $d+m$ non negative real $h_1,\ldots ,h_{d+m}$ such that $\forall 1\le i\le d$, $\forall 1\le l\le m$,

  $$\begin{aligned} (q+M\mathbf {n}^*)_i=h_i-\sum _{l=1}^mB_{il}h_{d+l},\quad h_in_i^*=0,\text { and }\quad h_{d+l}(-B^T \mathbf {n}^*+D)_k=0. \end{aligned}$$

  The first condition entails that $h_i=(q+M\mathbf {n}^*)_i+\sum _{l=1}^mB_{il}h_{d+l}$ and therefore the vector $(\mathbf {n}^*,h_{d+1},\dots ,h_{d+m})$ is a solution to $LCP(\widetilde{M},\widetilde{q})$.

Appendix D: Proof of Theorem 4.2

A perturbation $\fancyscript{Z}^K=(\fancyscript{N}^K_1,\ldots ,\fancyscript{N}^K_d,\fancyscript{H}^K_1,\ldots ,\fancyscript{H}^K_m)$ of the prey-predator community process is defined by $2$ families of $d+m$ real-valued random processes $(u^K_i)_{1\le i \le d+m}$ and $(v^K_i)_{1\le i \le d+m}$ which are predictable with respect to the filtration $\fancyscript{F}_t$ generated by the processes $\mathbf {Z}^K$. Both families are uniformly bounded by a parameter $\kappa >0$.

The perturbation $\fancyscript{Z}^K$ is solution of the following system of stochastic differential equations driven by the Poisson point measures $R_i$ and $M_i$ introduced in Appendix A.

$$\begin{aligned} \begin{aligned} \fancyscript{Z}^K(t)&\,=\fancyscript{Z}^K(0) +\sum _{i=1}^d \left[ \int _0^t\int _{\mathbb {R}_+} \frac{e_i}{K} \mathbf {1}_{\theta \le b(x_i)\fancyscript{N}^K_i(s-)\, +\,u^K_i(s)} R_i(ds,d\theta ) \right. \\&\quad \left. -\int _0^t\int _{\mathbb {R}_+}\frac{e_i}{K} \mathbf {1}_{\theta \le \fancyscript{N}^K_i(s-))\lambda (x_,\fancyscript{Z}^K(s-)) \,+\,v^K_i(s)} M_i(ds, d\theta )\right] \\&\quad +\sum _{l=1}^m \left[ \int _0^t\int _{\mathbb {R}_+} \frac{e_{d+l}}{K} \mathbf {1}_{\theta \le r\fancyscript{H}^K_l(s-)\left( \sum _{i=1}^d \frac{B(x_i,y_l)}{K}\fancyscript{N}^K_i(s-) \right) \,+\,u^K_{d+l}(s)} R_{d+l}(ds,d\theta ) \right. \\&\quad \left. -\int _0^t\int _{\mathbb {R}_+}\frac{e_{d+l}}{K} \mathbf {1}_{\theta \le D(y_l)\fancyscript{H}^K_l(s-) \,+\,v^K_{d+l}(s)} M_{d+l}(ds, d\theta )\right] . \end{aligned} \end{aligned}$$

(50)

where $(e_1,\dots ,e_d,e_{d+1},\dots ,e_{d+m})$ is the canonical basis of $\mathbb {R}^{d+m}$.

The proof relies on the study of the stochastic process $L(\fancyscript{Z}^K)$ where $L$ is the Lyapunov function for the system $LVP(\mathbf {x},\mathbf {y})$ introduced in (11) with an appropriate choice of $\gamma$. The function $L$ is the sum of two functions $V$ and $W$. $V$ defined in (8) is linear in the coordinate $n_i$, $i\in P$ and $h_l$, $l\in Q$ and strictly convex in the other coordinates. Moreover, its Hessian matrix at $\mathbf {z}^*$ is diagonal. $W$ defined (12) is a quadratic form in $(\mathbf {z}-\mathbf {z}^*)$. This justifies the inequality (20):

$$\begin{aligned} \begin{aligned} ||\mathbf {z}-\mathbf {z}^*||^2&\le \sum _{i\notin P}|n_i-n_i^*|^2 +\sum _{i\in P}|n_i|+\sum _{l\notin Q}|h_l-h_l^*| +\sum _{l\in Q}|h_l|\\&\le C\left( L(\mathbf {z})-L(\mathbf {z}^*)\right) \le CC^{\prime }\nonumber \\&\quad \,\times \left( \sum _{i\notin P}|n_i-n_i^*|^2 +\sum _{i\in P}|n_i|+\sum _{l\notin Q}|h_l-h_l^*| +\sum _{l\in Q}|h_l|\right) , \end{aligned} \end{aligned}$$

where $P$ and $Q$ have been defined in (6). We set in the following

$$\begin{aligned} ||\mathbf {z}-\mathbf {z}^*||_{PQ}=\sum _{i\notin P}|n_i-n_i^*|^2 +\sum _{i\in P}|n_i|+\sum _{l\notin Q}|h_l-h_l^*| +\sum _{l\in Q}|h_l| \end{aligned}$$

The derivative of $L(\mathbf {z}(t))$ given in (13) can be bounded from above in the neighbourhood of $\mathbf {z}^*$ by

$$\begin{aligned} \frac{d}{dt}L(\mathbf {z}(t))\le & {} -C_1||\mathbf {n}(t)-\mathbf {n}^*||^2 -C_1\left( \sum _{i\in P} n_i(t) +\sum _{l\in Q} h_l(t) \right) \\&-\,C_1 \sum _{i\notin P}\left( \sum _{l\notin Q} B_{il} (h_l(t)-h^*_l)\right) ^2, \end{aligned}$$

for a positive real number $C_1$. If we set

$$\begin{aligned} C_2=\inf \left\{ \sum _{i\notin P}\left( \sum _{l\notin Q} B_{il} (h_l-h^*_l)\right) ^2, \mathbf {h}\in (\mathbb {R}_+)^m, ||\mathbf {h}-\mathbf {h}^*||=1\right\} >0, \end{aligned}$$

then

$$\begin{aligned} \begin{aligned} \frac{d}{dt}L(\mathbf {z}(t))&\le -C_1||\mathbf {n}(t)-\mathbf {n}^*||^2 -C_1\left( \sum _{i\in P} n_i(t) +\sum _{l\in Q} h_l(t) \right) \\&\quad -C_1C_2 \sum _{l\notin Q}(h_l(t)-h^*_l)^2. \end{aligned} \end{aligned}$$

We then obtain (21):

$$\begin{aligned} \frac{d}{dt}L(\mathbf {z}(t)) \le -C^{\prime \prime } ||\mathbf {z}-\mathbf {z}^*||^2. \end{aligned}$$

We introduce $\tau _{\varepsilon }^K=\inf \{t\ge 0, \fancyscript{Z}^K(t)\notin B_{\varepsilon }\}$. In the sequel we prove that there exist $\varepsilon ^{\prime \prime }<\varepsilon$ and $V>0$ such that if $\fancyscript{Z}^K(0)\in \fancyscript{B}{_{\varepsilon ^{\prime \prime }}}$, then

$$\begin{aligned} \lim _{K\rightarrow \infty } \mathbb {P}\Bigl ( \tau ^K_{\varepsilon }>e^{KV}\Bigr ) =1. \end{aligned}$$

(51)

For every $t\le \tau _{\varepsilon }^K$,

$$\begin{aligned} \begin{aligned} L(\fancyscript{Z}^K(t))&=L(\fancyscript{Z}^K(0)) +M^K(t)\\&\quad +\int _0^t\sum _{i=1}^d \left( L\left( \fancyscript{Z}^K(s)+\frac{e_i}{K}\right) -L\left( \fancyscript{Z}^K(s)\right) \right) \left( \fancyscript{N}^K_i(s)b(x_i)+u^K_i(s)\right) ds \\&\quad +\int _0^t\sum _{i=1}^d\left( L\left( \fancyscript{Z}^K(s)-\frac{e_i}{K}\right) -L\left( \fancyscript{Z}^K(s)\right) \right) \left( \fancyscript{N}^K_i(s)\lambda (x_i,\fancyscript{Z}^K(s))+v^K_i(s)\right) ds\\&\quad + \int _0^t\sum _{l=1}^m\left( L\left( \fancyscript{Z}^K(s)+\frac{e_{d+l}}{K}\right) -L\left( \fancyscript{Z}^K(s)\right) \right) \\&\qquad \times \left( \fancyscript{H}^K_l(s)\left( r\sum _{i=1}^dB(x_i,y_m)\frac{\fancyscript{N}^K_i(s)}{K}\right) +u^K_{d+l}\right) ds \\&\quad + \int _0^t\sum _{l=1}^m\left( L\left( \fancyscript{Z}^K(s)-\frac{e_{d+l}}{K}\right) -L\left( \fancyscript{Z}^K(s)\right) \right) \left( \fancyscript{H}^K_l(s)D(y_l)+v^K_{d+l}\right) ds . \end{aligned} \end{aligned}$$

where $M^K_t$ is a local martingale which can be expressed with respect to the compensated Poisson point measures $(\widetilde{R}_i)_{1\le i\le d+m}$ and $(\widetilde{M}_{i})_{1\le i\le d+ m}$:

(52)

For every $t\le \tau _{\varepsilon }^K$ and $1\le i\le d$ we give the second order expansion of the terms

$$\begin{aligned}&L\left( \fancyscript{Z}^K(t)+\frac{e_i}{K}\right) -L(\fancyscript{Z}^K(t))=\frac{1}{K}\frac{\partial }{\partial e_i} L(\fancyscript{Z}^K(t))\\&\quad +\frac{1}{2} \int _{0}^{\frac{1}{K}}\left( \frac{\fancyscript{N}^K_i(t)}{K}+\frac{1}{K}-u\right) \frac{\partial ^2}{\partial e_i^2} L\left( \fancyscript{Z}^K(t)-\left( \frac{\fancyscript{N}^K_i(t)}{K}-u\right) e_i\right) du. \end{aligned}$$

We obtain a similar equality for the derivative with respect to $e_{d+l}$ for $1\le l\le m$.

Let us remark that $\sup \{\frac{\partial ^2}{\partial e_j^2} L(u,v) ,(u,v)\in \fancyscript{B}_{\varepsilon }\}<\infty$ for $\varepsilon$ small enough, for all $1\le j \le d+m$. Therefore the integrated term is of order $1/K^2$ for large $K$. The impact of the perturbed terms can be bounded similarly using the first derivative. Thus

$$\begin{aligned} \begin{aligned} L(\fancyscript{Z}^K(t))&= L(\fancyscript{Z}^K(0)) +M^K(t)\\&\quad +\int _0^t \sum _{i=1}^d \frac{\partial L(\fancyscript{Z}^K(s))}{\partial e_i} \frac{\fancyscript{N}^K_i(s)}{K}\\&\quad \times \left[ b(x_i)-d(x_i)-\sum _{j=1}^dc(x_i,x_j)\frac{\fancyscript{N}^K_j(s)}{K}-\sum _{l=1}^m\frac{\fancyscript{H}^K_l(s)}{K}B(x_i,y_l) \right] ds\\&\quad + \int _0^t \sum _{l=1}^m \frac{\partial L(\fancyscript{Z}^K(s))}{\partial e_{d+l}} \frac{\fancyscript{H}^K_l(s)}{K} \left[ r\sum _{i=1}^dB(x_i)\frac{\fancyscript{N}^K_i(s)}{K}-D(y_l)\right] ds \\&\quad + \fancyscript{O}\Bigl (\frac{t}{K}\Bigr )+\fancyscript{O}\bigl (\kappa t\bigr ). \end{aligned} \end{aligned}$$

Note that if $\mathbf {z}(t)$ is a solution of $LVP(\mathbf {x},\mathbf {y})$ then:

$$\begin{aligned} \begin{aligned} \frac{\partial L(\mathbf {z}(t))}{\partial t}&=\sum _{i=1}^d \frac{\partial }{\partial e_i} L(\mathbf {z}(t))n_i(t)\\&\quad \times \, \left[ b(x_i)-d(x_i)-\sum _{j=1}^dc(x_i,x_j)n_j(t)-\sum _{k=1}^mB(x_i,y_k)h_k(t) \right] \\&\quad + \sum _{l=1}^m \frac{\partial }{\partial e_{d+l}}L(\mathbf {z}(t)) h_l(t)\left[ r\sum _{i=1}^dB(x_i,y_l)n_i(t)-D(y_l)\right] . \end{aligned} \end{aligned}$$

We denote by $\frac{\partial L(\fancyscript{Z}^K(t))}{\partial t}$ the derivative along the solution $\mathbf {z}$ such that $\mathbf {z}(t)=\fancyscript{Z}^K(t)$. Then for $\kappa \ge 1{/}K$:

$$\begin{aligned} L(\fancyscript{Z}^K(t))=&L(\fancyscript{Z}^K(0)) +M^K(t)+ \int _0^t \frac{\partial L(\fancyscript{Z}^K(s))}{\partial t}ds +\fancyscript{O}\bigl (\kappa t\bigr ). \end{aligned}$$

Using inequalities (20) and (21) we obtain that there exists $C^{\prime \prime \prime }>0$, such that if $t\le T\wedge \tau _{\varepsilon }^K$ then

$$\begin{aligned} \begin{aligned} ||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2&\le C\left[ C^{\prime }\left( ||\fancyscript{Z}^K(0)-\mathbf {z}^*||_{PQ}\right) + \sup _{t\in [0,T]}|M^K(t)|\right. \\&\quad \left. -\,C^{\prime \prime }\int _0^t\left( ||\fancyscript{Z}^K(s)-\mathbf {z}^*||^2-C^{\prime \prime \prime }\kappa \right) ds\right] .\\ \end{aligned} \end{aligned}$$

(53)

This inequality is the main tool of the proof. It connects the time spent by the process above a given threshold with the values it takes during this time interval.

We define $S_{\kappa }=\inf \{t\ge 0,||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2 \le 2C^{\prime \prime \prime }\kappa \}$. Then for every $t\le S_{\kappa }\wedge T \wedge \tau _{\varepsilon }^K$:

$$\begin{aligned} ||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2\le C\left[ C^{\prime }\left( ||\fancyscript{Z}^K(0)-\mathbf {z}^*||_{PQ}\right) + \sup _{[0,T]}|M^K(t)|-C^{\prime \prime }C^{\prime \prime \prime }\kappa t \bigr ) \right] . \end{aligned}$$

As the l.h.s. is nonnegative we define

$$\begin{aligned} T_{\kappa }=\frac{C^{\prime }(||\fancyscript{Z}^K(0)-\mathbf {z}^*||_{PQ}+ \sup _{[0,T]}|M^K(t)|}{C^{\prime \prime }C^{\prime \prime \prime }\kappa }\ge 0, \end{aligned}$$

(54)

which can be seen as the maximal time spent by the process $||\mathbf {Z}^K(t)-\mathbf {z}^*||^2$ above $2C^{\prime \prime \prime }\kappa$ before the time $T\wedge \tau _{\varepsilon }^K$. Therefore for every $t\le S_{\kappa }\wedge T \wedge \tau _{\varepsilon }^K$:

$$\begin{aligned} ||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2\le CC^{\prime \prime }C^{\prime \prime \prime }\kappa T_{\kappa }. \end{aligned}$$

To control the norm $||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2$ it remains to control $T_{\kappa }$ and thus the martingale $M^K$. To obtain the uniform bound, we use the exponential bound given by Lemma 1. On the event

$$\begin{aligned} \Bigl \{ T_{\kappa }\le T\wedge \frac{\varepsilon ^2}{2CC^{\prime \prime }C^{\prime \prime \prime }\kappa } \Bigr \}, \end{aligned}$$

(55)

then $\sup _{[0,S_{\kappa }]}(||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2) \le \frac{\varepsilon ^2}{2},$ and in particular $S_{\kappa }\le \tau _{\varepsilon }^K\wedge T_{\kappa }.$

Moreover applying (53) on the same event we get

$$\begin{aligned} \sup _{[0,T\wedge \tau _{\varepsilon }^K]}(||\mathbf {Z}^K(t)-\mathbf {z}^*||^2)\le CC^{\prime \prime }C^{\prime \prime \prime }\kappa (T+T_{\kappa })\le \frac{\varepsilon ^2}{2}+CC^{\prime \prime }C^{\prime \prime \prime }\kappa T. \end{aligned}$$

(56)

Thus if furthermore $\kappa <\varepsilon ^2{/}(2CC^{\prime \prime }C^{\prime \prime \prime }T)$ then $\tau _{\varepsilon }^K> T$.

These results lead to the Theorem. Let $\varepsilon ^{\prime }>0$ such that $\varepsilon ^{\prime \prime }<\varepsilon ^{\prime }{/}2<\varepsilon ^{\prime }<\varepsilon$.

We introduce a sequence of stopping times that describes the back and forth of the process $\fancyscript{Z}^K$ between the balls $\fancyscript{B}_{\varepsilon ^{\prime \prime }}$ and $\fancyscript{B}_{\varepsilon ^{\prime }/2}$ (see Fig. 6). Set $\tau _0=0$ and for every $k\ge 1$ such that $\tau _k<\tau _{\varepsilon }^K$:

$$\begin{aligned} \begin{aligned} \tau ^{\prime }_k=&\inf \bigl \{t\ge \tau _{k-1}: \fancyscript{Z}^K(t)\notin B_{\varepsilon ^{\prime }/2} \bigr \},\\ \tau _k=&\inf \bigl \{t\ge \tau ^{\prime }_{k}: \fancyscript{Z}^K(t)\in B_{\varepsilon ^{\prime \prime }} \text { ou } \fancyscript{Z}^K(t)\notin B_{\varepsilon } \bigr \}. \end{aligned} \end{aligned}$$

(57)

We denote by $k_{\varepsilon }$ the number of back and forths before the exit:

$$\begin{aligned} k_{\varepsilon }=\inf \{k\in \mathbb {N}, \tau _k=\tau _{\varepsilon }^K\}. \end{aligned}$$

In the sequel we bound $k_{\varepsilon }$ from below.

Fig. 6

A trajectory of $\fancyscript{Z}^K$ in the neighbourhood of $\mathbf {z}^*$ for $d=m=1$

We consider an initial condition $\fancyscript{Z}^K(0)\in \fancyscript{B}_{\varepsilon ^{\prime }}$. We set $\kappa =(\varepsilon ^{\prime \prime })^2/2C^{\prime \prime \prime }$ and apply the previous results. The time $\tau _1$ corresponds to the first return in $\fancyscript{B}_{\varepsilon ^{\prime \prime }}$ therefore it is equal to the time $S_{\kappa }$ introduced before. We deduce from the previous computations that on the event (55)

$$\begin{aligned} \mathbb {P}\bigl ( \tau _{1}<\tau _{\varepsilon }^K\bigr )&= \mathbb {P}\bigl ( \sup _{[0,\tau _{1}]} ||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2 <\varepsilon ^2 \bigr )\ge \mathbb {P}\bigl ( T_{\kappa }\le T\wedge \frac{\varepsilon ^2}{2CC^{\prime \prime }C^{\prime \prime \prime }\kappa } \bigr ). \end{aligned}$$

We replace $T_{\kappa }$ by its value (54) to get that

$$\begin{aligned} \begin{aligned}&\mathbb {P}\bigl ( T_{\kappa }>T\wedge \frac{\varepsilon ^2}{2CC^{\prime \prime }C^{\prime \prime \prime }\kappa } \bigr ) \\&\quad =\mathbb {P}\Bigl ( \sup _{[0,T]}|M^K(t)| >\bigl (C^{\prime \prime }C^{\prime \prime \prime }\kappa T\wedge \frac{\varepsilon ^2}{2C} \bigr )-C^{\prime }(||\fancyscript{Z}^K(0)-\mathbf {z}^*||_{PQ})\Bigr )\\&\quad \le \mathbb {P}\Bigl ( \sup _{[0,T]}|M^K(t)| >\bigl (C^{\prime \prime }C^{\prime \prime \prime }\kappa T\wedge \frac{\varepsilon ^2}{2C} \bigr )-C^{\prime }\varepsilon ^{\prime }\Bigr ), \end{aligned} \end{aligned}$$

where we used that $\fancyscript{Z}^K(0)\in \fancyscript{B}_{\varepsilon ^{\prime }}$ to obtain the last inequality.

If we choose $T=2C^{\prime }\varepsilon ^{\prime }/C^{\prime \prime }C^{\prime \prime \prime }\kappa$ and $\varepsilon ^{\prime }$ such that $2C^{\prime }\varepsilon ^{\prime }<\frac{\epsilon ^2}{2C}$ then the inequality becomes

$$\begin{aligned} \mathbb {P}\bigl ( T_{\kappa }>T\wedge \frac{\varepsilon ^2}{2CC^{\prime \prime }C^{\prime \prime \prime }\kappa } \bigr )&\le \mathbb {P}\bigl ( \sup _{[0,T]}|M^K(t)| >C^{\prime }\varepsilon ^{\prime }\bigr ). \end{aligned}$$

We finally use Lemma 1 to obtain

$$\begin{aligned} \mathbb {P}\bigl ( T_{\kappa }>T\wedge \frac{\varepsilon ^2}{2CC^{\prime \prime }C^{\prime \prime \prime }\kappa } \bigr )&\le \exp (-KV), \end{aligned}$$

where $V>0$ only depends on $\varepsilon ^{\prime }$ and $\varepsilon ^{\prime \prime }$.

Since this inequality remains true as long as the initial condition is in $B_{\varepsilon ^{\prime }}$ we deduce that

$$\begin{aligned} \sup _{\fancyscript{Z}^K(0)\in B_{\varepsilon ^{\prime }}}\mathbb {P}\Bigl ( \tau _1<\tau _{\varepsilon }^K\Bigr )\ge 1-\exp (-KV). \end{aligned}$$

(58)

Applying the strong Markov property at the stopping time $\tau _k$ for $k\ge 1$

$$\begin{aligned} \sup _{\fancyscript{Z}^K(0)\in B_{\varepsilon ^{\prime }}}\mathbb {P}\Bigl ( \tau _k<\tau _{\varepsilon }^K|\tau _{k-1}<\tau _{\varepsilon }^K\Bigr )\ge 1-\exp (-KV). \end{aligned}$$

therefore we can bound $k_{\varepsilon }$ from below by a random variable distributed according to a geometric law of parameter $\exp (-KV)$. Then

$$\begin{aligned} \lim _{K\rightarrow \infty } \mathbb {P}(k_{\varepsilon }>\exp (\textit{KV}{/}2))=1. \end{aligned}$$

(59)

It remains to prove that these back and forths do not happen too fast. We establish that the time intervals $\tau _k-\tau _{k-1}$ are of order $1$ for $k\ge 2$. To this aim we search for $T^{\prime }$ such that for every $k\ge 2$, $\mathbb {P}(\tau ^{\prime }_k-\tau _{k-1}>T^{\prime })>0$. Using the strong Markov property again, it is sufficient to prove that $\inf _{\fancyscript{Z}^K(0)\in B_{\varepsilon ^{\prime \prime }}}\mathbb {P}( \tau ^{\prime }_1>T^{\prime })>0$:

$$\begin{aligned} \inf _{\fancyscript{Z}^K(0)\in B_{\varepsilon ^{\prime \prime }}}\mathbb {P}( \tau ^{\prime }_1>T^{\prime })=\inf _{\fancyscript{Z}^K(0)\in B_{\varepsilon ^{\prime \prime }}}\mathbb {P}\left( \sup _{[0,T^{\prime }\wedge \tau ^{\prime }_1]}||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2<\frac{\varepsilon ^{\prime 2}}{4}\right) . \end{aligned}$$

(60)

We deduce from (56) with $\varepsilon =\varepsilon ^{\prime }/2$ that on the event $\{T_{\kappa }\le T^{\prime }\wedge \frac{\varepsilon ^{\prime 2}}{8CC^{\prime \prime }C^{\prime \prime \prime }\kappa }\}$:

$$\begin{aligned} \sup _{[0,T^{\prime }\wedge \tau ^{\prime }_1]}\left( ||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2\right) ^2\le CC^{\prime \prime }C^{\prime \prime \prime }\kappa (T^{\prime }+T_{\kappa })\le \frac{\varepsilon ^{\prime 2}}{8}+CC^{\prime \prime }C^{\prime \prime \prime }\kappa T^{\prime }. \end{aligned}$$

Setting $T^{\prime }=2C^{\prime }\varepsilon ^{\prime \prime }/C^{\prime \prime }C^{\prime \prime \prime }\kappa$ and $\varepsilon ^{\prime \prime }$ such that $2C^{\prime }\varepsilon ^{\prime \prime }<\varepsilon ^{\prime 2}/4C$, we get that

$$\begin{aligned} \sup _{[0,T^{\prime }\wedge \tau ^{\prime }_1]}||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2< \frac{\varepsilon ^{\prime 2}}{4}, \end{aligned}$$

and thus $\tau ^{\prime }_1>T^{\prime }$.

Lemma 1 ensures again that for any initial condition in $\fancyscript{B}_{\varepsilon ^{\prime \prime }}$:

$$\begin{aligned} \begin{aligned} \mathbb {P}\left( T_{\kappa }> T^{\prime }\wedge \frac{\varepsilon ^{\prime 2}}{8CC^{\prime \prime }C^{\prime \prime \prime }\kappa } \right) \le \mathbb {P}\left( \sup _{[0,T^{\prime }]}|M^K(t)| > C^{\prime }\varepsilon ^{\prime \prime } \right) \underset{{K\rightarrow \infty }}{\longrightarrow }0. \end{aligned} \end{aligned}$$

and thus

$$\begin{aligned} \inf _{\fancyscript{Z}^K(0)\in B_{\varepsilon ^{\prime \prime }}}\mathbb {P}\left( \sup _{[0,T^{\prime }\wedge \tau ^{\prime }_1]}||\fancyscript{Z}^K(t)-\mathbf {z}^*||^2<\frac{\varepsilon ^{\prime 2}}{4}\right) \underset{{K\rightarrow \infty }}{\longrightarrow } 1. \end{aligned}$$

Finally (51) is deduced from (59).