A general theory of coexistence and extinction for stochastic ecological communities

Abstract

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393–431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893–1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka–Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka–Volterra models.

Appendices

Appendix A. Persistence proofs

Lemma A.1

For any ${\mathbf {z}}\in {\mathcal {S}}, t\in {\mathbb {N}}$ we have the bounds

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}(V({\mathbf {Z}}(t))\le \rho ^t V({\mathbf {z}})+C\sum _{s=0}^t\rho ^s\le \rho ^t V({\mathbf {z}})+\dfrac{C}{1-\rho }, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}h({\mathbf {Z}}(t), \xi (t)) \le&\, {\mathbb {E}}_{\mathbf {z}}\left[ V({\mathbf {Z}}(t)) h({\mathbf {Z}}(t), \xi (t))\right] \le&\, \rho {\mathbb {E}}_{\mathbf {z}}V({\mathbf {Z}}(t))+C\\ \le&\,\rho ^{t+1} V({\mathbf {z}})+\dfrac{C}{1-\rho }. \end{aligned} \end{aligned}$$

If a function $\psi$ satisfies

$$\begin{aligned} \lim _{{\mathbf {z}}\rightarrow \infty } \frac{\psi ({\mathbf {z}})}{{\mathbb {E}}V({\mathbf {z}},\xi (t))h({\mathbf {z}},\xi (t))}=0 \end{aligned}$$

then $\psi$ is $\mu$-integrable for any invariant probability measure $\mu$. Moreover, if $\mu ({\mathcal {S}}_+)=1$ then $r_i(\mu )=0$ for any $i\in I$.

Proof

By Assumption A3) we have $PV({\mathbf {z}})\le \rho V({\mathbf {z}})+C$. Using this and the Markov property yields ${\mathbb {E}}_{\mathbf {z}}(V({\mathbf {Z}}(t+1))\le \rho {\mathbb {E}}_{\mathbf {z}}V({\mathbf {Z}}(t))+C$. As a result

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}(V({\mathbf {Z}}(t))\le \rho ^t V({\mathbf {z}})+C\sum _{s=0}^t\rho ^s\le \rho ^t V({\mathbf {z}})+\dfrac{C}{1-\rho }. \end{aligned}$$

On the other hand Assumption A3) and the above imply that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}\left[ V({\mathbf {Z}}(t+1)) h({\mathbf {Z}}(t), \xi (t))\right] \le&\, \rho {\mathbb {E}}_{\mathbf {z}}V({\mathbf {Z}}(t))+C\\ \le&\,\rho ^{t} V({\mathbf {z}})+\dfrac{C}{1-\rho }. \end{aligned} \end{aligned}$$

(A.1)

Since by A3) part i) $V({\mathbf {z}})\ge 1$ we also get that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}h({\mathbf {Z}}(t), \xi (t)) \le&\rho ^{t+1} V({\mathbf {z}})+\dfrac{C}{1-\rho }. \end{aligned} \end{aligned}$$

This implies that if $\mu$ is an ergodic invariant probability measure then

$$\begin{aligned} {\mathbb {E}}_\mu h({\mathbf {Z}}(t), \xi (t)) = \int {\mathbb {E}}h({\mathbf {z}}, \xi (1)) \mu (d{\mathbf {z}})\le \frac{C}{1-\rho }. \end{aligned}$$

Since

$$\begin{aligned} |\log F_i({\mathbf {z}},\xi )|^k\le c_k \max \left\{ F_i({\mathbf {z}},\xi ),\frac{1}{F_i({\mathbf {z}},\xi )}\right\} ^{\gamma _3}\le c_k h({\mathbf {z}},\xi ) \end{aligned}$$

for some $c_k>0$, we see that for any $k\in {\mathbb {N}}$ there exists $C_k>0$ such that

$$\begin{aligned} \int {\mathbb {E}}|\log F_i({\mathbf {z}},\xi (1))|^k\mu (d{\mathbf {z}})\le C_k. \end{aligned}$$

The strong law of large numbers for martingales implies that for $\mu$ almost every ${\mathbf {z}}$ we have

$$\begin{aligned} \lim _{T\rightarrow \infty } \frac{1}{T} \sum _0^T\left( \log F_i({\mathbf {Z}}(t+1))-P \log F_i({\mathbf {Z}}(t))\right) =0, \,\text {when}~ {\mathbf {Z}}(0)={\mathbf {z}}. \end{aligned}$$

(A.2)

Having (A.2), we can follow the arguments by Benaïm and Schreiber (2019, Lemma 3 and Proposition 1) to obtain that if $\mu ({\mathcal {S}}^{I}_+)=1$ then $r_i(\mu )=0$ for any $i\in I$.

Finally, because of the boundedness (A.1), it is standard to show that if a function $\psi$ satisfies

$$\begin{aligned} \lim _{{\mathbf {z}}\rightarrow \infty } \frac{\psi ({\mathbf {z}})}{{\mathbb {E}}V({\mathbf {z}},\xi (t))h({\mathbf {z}},\xi (t))}=0 \end{aligned}$$

then $\psi$ is $\mu$-integrable for any invariant probability measure $\mu$. See Lemma 3.3 in Hening and Nguyen (2018a) for a similar proof. $\square$

Lemma A.2

There exist $M, C_2, \gamma _4>0,\rho _2\in (0,1)$ such that

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}\left[ V({\mathbf {Z}}(1))\prod _{i=1}^n X_i^{p_i}(1)\right] \le \left( {\varvec{1}}_{\{|{\mathbf {z}}|<M\}}(C_2-\rho _2)+\rho _2\right) V({\mathbf {z}})\prod _{i=1}^nx_i^{p_i}, ~{\mathbf {z}}\in {\mathcal {S}}\end{aligned}$$

for any ${\mathbf {p}}=(p_1,\ldots ,p_n)\in {\mathbb {R}}^n$ satisfying

$$\begin{aligned} |{\mathbf {p}}|_1:=\sum |p_i|\le \gamma _4. \end{aligned}$$

(A.3)

Proof

We note that for $w_1,\ldots ,w_n>0$

$$\begin{aligned} \begin{aligned} \prod _{i=1}^n w_i^{p_i}\le&\, \prod _{i=1}^n \left( w_i^{-1}\vee {w_i}\right) ^{|p_i|}\\ \le&\, \left( \max _{i=1,\ldots ,n}\left\{ w_i^{-1}\vee {w_i}\right\} \right) ^{\sum _i|p_i|}\\ \le&\, 1+\frac{\sum _i|p_i|}{\gamma _3}\left( \max _{i=1,\ldots ,n}\left\{ w_i^{-1}\vee {w_i}\right\} \right) ^{\gamma _3} \end{aligned} \end{aligned}$$

if $\sum _i |p_i|<\gamma _3.$ The last inequality follows from the inequality $x^p\le 1+px$ for $x\ge 0, p\in (0,1)$. Thus,

$$\begin{aligned} \prod _{i=1}^n F_i^{p_i}({\mathbf {z}},\xi ) \le 1+ \frac{|{\mathbf {p}}|_1}{\gamma _3} h({\mathbf {z}},\xi ). \end{aligned}$$

Since $\lim _{{\mathbf {z}}\rightarrow \infty } V({\mathbf {z}})=\infty$, we can select $\rho _1\in (\rho ,1)$, $M>0$ and $C_1>0$ such that

$$\begin{aligned} \rho V({\mathbf {z}})+C\le (C_1{\varvec{1}}_{\{|z|<M\}}+\rho _1)V({\mathbf {z}}). \end{aligned}$$

Let $\gamma _4\in (0,\gamma _3)$ be such that $\left( 1+\frac{\gamma _4}{\gamma _3}\right) \rho _1=:\rho _2<1.$ There exists $C_2>0$ satisfying

$$\begin{aligned} (C_1{\varvec{1}}_{\{|z|<M\}}+\rho _1)\left( 1+\frac{\gamma _4}{\gamma _3}\right) \le (C_2-\rho _2){\varvec{1}}_{\{|z|<M\}}+\rho _2. \end{aligned}$$

The above estimates together with (2.1), Lemma A.1, and (A.3) yield

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}\left[ V\left( {\mathbf {Z}}(1)\right) \prod _{i=1}^n X_i^{p_i}(1)\right] =&\,\prod _{i=1}^nx_i^{p_i} {\mathbb {E}}\left[ V({\mathbf {x}}\circ F({\mathbf {z}},\xi ), G({\mathbf {z}},\xi ))\prod _{i=1}^n F_i^{p_i}({\mathbf {z}},\xi )\right] \\ \le&\,\prod _{i=1}^nx_i^{p_i} {\mathbb {E}}\left[ V({\mathbf {x}}\circ F({\mathbf {z}},\xi ), G({\mathbf {z}},\xi ))\left( 1+\frac{|{\mathbf {p}}|_1}{\gamma _3}h({\mathbf {z}},\xi )\right) \right] \\ \le&\, \prod _{i=1}^nx_i^{p_i}(\rho V({\mathbf {z}})+C)\left( 1+\frac{|{\mathbf {p}}|_1}{\gamma _3}\right) \\ \le&\, \prod _{i=1}^nx_i^{p_i}(C_1{\varvec{1}}_{\{|z|<M\}}+\rho _1) V({\mathbf {z}})\left( 1+\frac{|{\mathbf {p}}|_1}{\gamma _3}\right) \\ \le&\,((C_2-\rho ){\varvec{1}}_{\{|z|<M\}}+\rho _2) \prod _{i=1}^nx_i^{p_i}V({\mathbf {z}}). \end{aligned}$$

$\square$

We will denote by $B_M :=\{{\mathbf {u}}\in {\mathcal {S}}:\Vert {\mathbf {u}}\Vert \le M\}$ the closed ball of radius $M>0$ around the origin. Let ${\mathcal {M}}$ be the set of ergodic invariant probability measures of ${\mathbf {X}}$ supported on the boundary ${\mathcal {S}}_0:=\partial {\mathbb {R}}^n_+\times {\mathbb {R}}^{\kappa _0}$. Remember that for a subset $\widetilde{\mathcal {M}}\subset {\mathcal {M}}$ we denote by ${{\,\mathrm{Conv}\,}}(\widetilde{\mathcal {M}})$ the convex hull of $\widetilde{\mathcal {M}}$, that is the set of probability measures $\pi$ of the form $\pi (\cdot )=\sum _{\mu \in \widetilde{\mathcal {M}}}p_\mu \mu (\cdot )$ with $p_\mu \ge 0,\sum _{\mu \in \widetilde{\mathcal {M}}}p_\mu =1$.

Consider $\mu \in {\mathcal {M}}$. Assume $\mu (\{0\}\times {\mathbb {R}}^{\kappa _0})=0$. Since $\mu$ is ergodic there exist $0<n_1<\cdots < n_k\le n$ such that ${{\,\mathrm{supp}\,}}(\mu )\subset S^\mu :={\mathbb {R}}^\mu _+\times {\mathbb {R}}^{\kappa _0}$ where

$$\begin{aligned} {\mathbb {R}}_+^\mu :=\{(x_1,\ldots ,x_n)\in {\mathbb {R}}^n_+: x_i=0\text { if } i\in I_\mu ^c\} \end{aligned}$$

for $I_\mu :={\mathcal {S}}(\mu )=\{n_1,\ldots , n_k\}$ and $I_\mu ^c:=\{1,\ldots ,n\}{\setminus }\{n_1,\ldots , n_k\}$.

$$\begin{aligned} {\mathbb {R}}_+^{\mu ,\circ }:=\{(x_1,\ldots ,x_n)\in {\mathbb {R}}^n_+: x_i=0\text { if } i\in I_\mu ^c\text { and }x_i>0\text { if }x_i\in I_\mu \} \end{aligned}$$

and $\partial {\mathbb {R}}_+^{\mu }:={\mathbb {R}}_+^\mu {\setminus }{\mathbb {R}}_+^{\mu ,\circ }.$

The following condition ensures persistence.

Assumption A.1

For any $\mu \in {{\,\mathrm{Conv}\,}}({\mathcal {M}})$ one has

$$\begin{aligned} \max _i r_i(\mu )>0, \end{aligned}$$

where

$$\begin{aligned} r_i(\mu ):=\int _{{\mathcal {S}}_0}\left[ {\mathbb {E}}\ln F_i({\mathbf {z}},\xi )\right] \mu (d{\mathbf {z}}). \end{aligned}$$

Remember that the occupation measures are defined as

$$\begin{aligned} \Pi _{t,{\mathbf {z}}}(\cdot )=\frac{1}{t}\sum _{s=0}^t{\mathbb {P}}_{\mathbf {z}}({\mathbf {Z}}(s)\in \cdot )\,ds,~{\mathbf {z}}\in {\mathcal {S}}, t\in {\mathbb {Z}}_+ \end{aligned}$$

Lemma A.3

Suppose the following

• The sequences $({\mathbf {z}}_k)_{k\in N}\subset {\mathbb {R}}_+^n\times {\mathbb {R}}^{\kappa _0}, (T_k)_{k\in {\mathbb {N}}}\subset {\mathbb {N}}$ are such that $\Vert {\mathbf {z}}_k\Vert \le M$, $T_k>1$ for all $k\in {\mathbb {N}}$ and $\lim _{k\rightarrow \infty }T_k=\infty$.

• The sequence $(\Pi _{T_k,{\mathbf {z}}_k})_{k\in {\mathbb {N}}}$ converges weakly to an invariant probability measure $\pi$.

• The function $h:{\mathbb {R}}^n_+\times {\mathbb {R}}^{\kappa _0}\rightarrow {\mathbb {R}}$ is any continuous function satisfying

  $$\begin{aligned} \lim _{{\mathbf {z}}\rightarrow \infty }\frac{|h({\mathbf {z}})|}{V({\mathbf {z}})\max _{i=1}^n\{(x_i^{\gamma _3}\wedge x^{-\gamma _3}_i)\}} =0 \end{aligned}$$

Then one has

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _{{\mathbb {R}}^n_+}h({\mathbf {x}})\Pi _{T_k,{\mathbf {z}}_k}(d{\mathbf {x}})= \int _{{\mathbb {R}}^n_+}h({\mathbf {x}})\pi (d{\mathbf {x}}). \end{aligned}$$

Proof

The proof is almost identical to that of Lemma 3.4 by Hening and Nguyen (2018a) and is therefore omitted. $\square$

It is shown in Schreiber (2011, Lemma 4) by the min–max principle that Assumption A.1 is equivalent to the existence of ${\mathbf {p}}>0$ such that

$$\begin{aligned} \min \limits _{\mu \in {\mathcal {M}}}\left\{ \sum _{i}p_i r_i(\mu )\right\} :=2r^*>0. \end{aligned}$$

(A.4)

By rescaling if necessary, we can assume that $|{\mathbf {p}}|_1=\gamma _4$.

Lemma A.4

Suppose that Assumption A.1 holds. Let ${\mathbf {p}}$ and $r^*$ be as in (A.3). There exists an integer $T^*>0$ such that, for any $T>T^*$, ${\mathbf {x}}\in \partial {\mathbb {R}}^n_+, {\mathbf {z}}= ({\mathbf {x}},{\mathbf {y}}) \in B_M$ one has

$$\begin{aligned} \sum _{t=0}^T{\mathbb {E}}_{\mathbf {z}}\left( \ln V({\mathbf {Z}}(t+1))-\ln V({\mathbf {Z}}(t))-\sum p_i\ln F_i({\mathbf {Z}}(t), \xi (t))\right) \le -r^*(T+1). \end{aligned}$$

(A.5)

Proof

In view of Lemma A.1,

$$\begin{aligned} \sup _{t\in {\mathbb {N}}, \Vert {\mathbf {z}}\Vert \le M} {\mathbb {E}}_{\mathbf {z}}V({\mathbf {Z}}(t))<\infty \end{aligned}$$

which implies, since

$$\begin{aligned} \frac{{\mathbb {E}}_{\mathbf {z}}\ln V({\mathbf {Z}}(T))}{T+1}\le \frac{{\mathbb {E}}_{\mathbf {z}}V({\mathbf {Z}}(T))}{T+1} \end{aligned}$$

that

$$\begin{aligned} \lim _{T\rightarrow \infty } \sup _{\Vert {\mathbf {z}}\Vert \le M} \frac{1}{T+1}\sum _{t=0}^T{\mathbb {E}}_{\mathbf {z}}\left( \ln V({\mathbf {Z}}(t+1))-\ln V({\mathbf {Z}}(t))\right) =0. \end{aligned}$$

With (A.4) and Lemma A.2 and Lemma A.3, we can argue by contradiction in the same manner as in Lemma 4.1 by Hening and Nguyen (2018a) to show that

$$\begin{aligned} \limsup _{T\rightarrow \infty } \frac{1}{T+1}\sup _{\Vert {\mathbf {z}}\Vert \le M} \sum _{t=0}^T{\mathbb {E}}_{\mathbf {z}}\left( -\sum p_i\ln F_i({\mathbf {Z}}(t), \xi (t))\right) \le -2r^*. \end{aligned}$$

Combining the two above limits finishes the proof. $\square$

Let $n^*\in {\mathbb {N}}$ be such that

$$\begin{aligned} \rho _2^{1-n^*}>C_2. \end{aligned}$$

(A.6)

Proposition A.1

Define $U:{\mathcal {S}}_+\rightarrow {\mathbb {R}}_+$ by

$$\begin{aligned} U({\mathbf {z}})=V({\mathbf {z}})\prod _{i=1}^nx_{i}^{-p_i} \end{aligned}$$

with ${\mathbf {p}}$ and $r^*$ satisfying (A.3) and $T^*>0$ satisfying the assumptions of Lemma A.4. There exist numbers $\theta \in \left( 0,\frac{\gamma _4}{2}\right)$, $K_\theta >0$, such that for any $T\in [T^*,n^*T^*]\cap {\mathbb {Z}}$ and ${\mathbf {z}}\in {\mathcal {S}}_+, \Vert {\mathbf {z}}\Vert \le M$,

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}U^\theta ({\mathbf {Z}}(T))\le U^\theta ({\mathbf {x}})\exp \left( -\frac{1}{2}\theta r^*T\right) +K_\theta . \end{aligned}$$

Proof

$$\begin{aligned} \begin{aligned} \ln U({\mathbf {Z}}(T))=&\ln U({\mathbf {Z}}(0)) + \sum _{t=0}^{T-1} \left( \ln U({\mathbf {Z}}(t+1))-\ln U({\mathbf {Z}}(t))\right) \\ =&\ln U({\mathbf {Z}}(0)) + G(T) \end{aligned} \end{aligned}$$

(A.7)

where

$$\begin{aligned} \begin{aligned} G(T)=\sum _{t=0}^{T-1} \left( \ln V({\mathbf {Z}}(t+1))-\ln V({\mathbf {Z}}(t))-\sum p_i\ln F_i({\mathbf {Z}}(t), \xi (t))\right) . \end{aligned} \end{aligned}$$

(A.8)

In view of (A.7) and Lemma A.2

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}\exp ( G(T))=\dfrac{{\mathbb {E}}_{\mathbf {z}}U({\mathbf {Z}}(T))}{U({\mathbf {z}})}\le (C_2)^T. \end{aligned}$$

(A.9)

Let ${\widehat{U}}(\cdot ):{\mathbb {R}}^{n,\circ }_+\times {\mathbb {R}}^{\kappa _0}\mapsto {\mathbb {R}}_+$ be defined by ${\widehat{U}}({\mathbf {z}})=V({\mathbf {z}})\prod _{i=1}^n x_i^{p_i}$. We also have

$$\begin{aligned} \dfrac{{\mathbb {E}}_{\mathbf {z}}{\widehat{U}}({\mathbf {Z}}(T))}{{\widehat{U}}({\mathbf {z}})}\le (C_2)^T. \end{aligned}$$

(A.10)

Note that

$$\begin{aligned} U^{-1}({\mathbf {z}})={\widehat{U}}({\mathbf {z}})\frac{1}{V^2({\mathbf {z}})}\le {\widehat{U}}({\mathbf {z}}). \end{aligned}$$

(A.11)

Using (A.11) and (A.10) yields

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}\exp (-G(T))=&\,\dfrac{{\mathbb {E}}_{\mathbf {z}}U^{-1}({\mathbf {Z}}(T))}{U^{-1}({\mathbf {z}})}\\ \le&\,\dfrac{{\mathbb {E}}_{\mathbf {z}}{\widehat{U}}({\mathbf {Z}}(T))}{V^2({\mathbf {z}})U^{-1}({\mathbf {x}})}\\ \le&\, \dfrac{{\mathbb {E}}_{\mathbf {z}}{\widehat{U}}({\mathbf {Z}}(T))}{{\widehat{U}}({\mathbf {z}})}\\ \le&\, (C_2)^T. \end{aligned} \end{aligned}$$

(A.12)

By (A.9) and (A.12) the assumptions of Hening and Nguyen (2018a, Lemma 3.5) hold for the random variable G(T). Therefore, there exists ${\tilde{K}}_2\ge 0$ such that

$$\begin{aligned} 0\le \dfrac{d^2{\tilde{\phi }}_{{\mathbf {z}},T}}{d\theta ^2}(\theta )\le {\tilde{K}}_2\,\text { for all }\,\theta \in \left[ 0,\frac{1}{2}\right) ,\, {\mathbf {z}}\in {\mathbb {R}}^{n,\circ }_+\times {\mathbb {R}}^{\kappa _0}, \Vert {\mathbf {z}}\Vert \le M, T\in [T^*,n^*T^*]\cap {\mathbb {Z}}\end{aligned}$$

where

$$\begin{aligned} {\tilde{\phi }}_{{\mathbf {z}},T}(\theta )=\ln {\mathbb {E}}_{\mathbf {z}}\exp (\theta G(T)). \end{aligned}$$

In view of Lemma A.4 and the Feller property of $({\mathbf {Z}}(t))$, there exists a ${\tilde{\delta }}>0$ such that if $\Vert {\mathbf {z}}\Vert \le M$, $\mathrm {dist}({\mathbf {z}},\partial {\mathbb {R}}^n_+)<{\tilde{\delta }}$ and $T\in [T^*,n^*T^*]\cap {\mathbb {Z}}$ then

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}G(T)\le -\dfrac{3}{4}r^*T. \end{aligned} \end{aligned}$$

(A.13)

Another application of Hening and Nguyen (2018a, Lemma 3.5) yields

$$\begin{aligned} \dfrac{d{\tilde{\phi }}_{{\mathbf {z}},T}}{d\theta }(0)= & {} {\mathbb {E}}_{\mathbf {z}}G(T)\le -\dfrac{3}{4} r^*T\,\text { for }\, {\mathbf {z}}\in {\mathbb {R}}^{n,\circ }_+\times {\mathbb {R}}^{\kappa _0}, \Vert {\mathbf {z}}\Vert \le M, \mathrm {dist}({\mathbf {z}},\partial {\mathbb {R}}^n_+)\\< & {} {\tilde{\delta }}, T\in [T^*,n^*T^*]\cap {\mathbb {Z}}. \end{aligned}$$

By a Taylor expansion around $\theta =0$, for $\Vert {\mathbf {z}}\Vert \le M, \mathrm {dist}({\mathbf {z}},\partial {\mathbb {R}}^n_+)<{\tilde{\delta }}, T\in [T^*,n^*T^*]\cap {\mathbb {Z}}$ and $\theta \in \left[ 0,\frac{1}{2}\right)$ we have

$$\begin{aligned} {\tilde{\phi }}_{{\mathbf {x}},T}(\theta )\le -\dfrac{3}{4} r^*T\theta +\theta ^2{\tilde{K}}_2. \end{aligned}$$

If we choose any $\theta \in \left( 0,\frac{1}{2}\right)$ satisfying $\theta <\frac{r^*T^*}{4{\tilde{K}}_2}$, we obtain that

$$\begin{aligned} {\tilde{\phi }}_{{\mathbf {z}},T}(\theta )\le -\dfrac{1}{2} r^*T\theta \,\,\text { for all }\,{\mathbf {z}}\in {\mathbb {R}}^{n,\circ }\times {\mathbb {R}}^{\kappa _0},\Vert {\mathbf {z}}\Vert \le M, \mathrm {dist}({\mathbf {z}},\partial {\mathbb {R}}^n_+)<{\tilde{\delta }}, T\in [T^*,n^*T^*]\cap {\mathbb {Z}}.\nonumber \\ \end{aligned}$$

(A.14)

In light of (A.14), we have for all $\theta <\frac{r^*T^*}{4{\tilde{K}}_2}$, $\Vert {\mathbf {z}}\Vert \le M, 0<\mathrm {dist}({\mathbf {z}},\partial {\mathbb {R}}^n_+)<{\tilde{\delta }}, T\in [T^*,n^*T^*]$ that

$$\begin{aligned} \dfrac{{\mathbb {E}}_{\mathbf {z}}U^\theta ({\mathbf {Z}}(T))}{U^\theta ({\mathbf {z}})}=\exp {\tilde{\phi }}_{{\mathbf {z}},T}(\theta )\le \exp \left( -\frac{1}{2}r^*T\theta \right) . \end{aligned}$$

(A.15)

In view of Lemma A.2, we have for ${\mathbf {z}}$ satisfying $\Vert {\mathbf {z}}\Vert \le M, \mathrm {dist}({\mathbf {z}},\partial {\mathbb {R}}^n_+)\ge {\tilde{\delta }}$ and $T\in [T^*,n^*T^*]$ that

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}U^\theta ({\mathbf {Z}}(T))\le (C_2)^{\theta n^*T^*}\sup \limits _{\Vert {\mathbf {z}}\Vert \le M, \mathrm {dist}({\mathbf {z}},\partial {\mathbb {R}}^n_+)\ge {\tilde{\delta }}}\{U^\theta ({\mathbf {z}})\}=:K_\theta <\infty . \end{aligned}$$

(A.16)

Combining (A.15) and (A.16) we are done. $\square$

Theorem A.1

Suppose that Assumption A.1 holds. Let $\theta$ be as in Proposition A.1, $T^*$ as in Lemma A.4 and $n^*$ as in (A.6). There exist numbers $\kappa =\kappa (\theta ,T^*)\in (0,1)$ and ${\tilde{K}}={\tilde{K}}(\theta ,T^*)>0$ such that

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}U^\theta ({\mathbf {Z}}(n^*T^*))\le \kappa U^\theta ({\mathbf {z}})+\tilde{K}\,\text { for all }\, {\mathbf {z}}\in {\mathbb {R}}^{n,\circ }_+\times {\mathbb {R}}^{\kappa _0}. \end{aligned}$$

(A.17)

We have

$$\begin{aligned} \limsup _{t\rightarrow \infty }{\mathbb {P}}_{\mathbf {z}}\{|X_i(t)|\vee |X^{-1}_i(t)|>m \text { for some } i=1,\ldots , n\}\le c_2 m^{-c_3} \end{aligned}$$

(A.18)

for some positive $c_2, c_3>0.$ Moreover, for any compact set $K\subset {\mathbb {R}}^{n,\circ }\times {\mathbb {R}}^{\kappa _0}$,

$$\begin{aligned} {\mathbb {P}}_{\mathbf {z}}(\tau _K>k)\le c_KU^\theta ({\mathbf {z}})\kappa ^k \end{aligned}$$

(A.19)

If the Markov chain ${\mathbf {Z}}(t)$ is irreducible and aperiodic on ${\mathbb {R}}^{n,\circ }_+\times {\mathbb {R}}^{\kappa _0}$, and a compact set is petite, then there is $c_4>1$ such that

$$\begin{aligned} c_4^t\Vert P_t({\mathbf {z}},\ldots )-\pi \Vert _{TV}\rightarrow 0\text { as } t\rightarrow \infty . \end{aligned}$$

Proof

Define

$$\begin{aligned} \tau =\inf \{t\ge 0: \Vert {\mathbf {Z}}(t)\Vert \le M\}. \end{aligned}$$

(A.20)

By Lemma A.2 for all ${\mathbf {z}}\in {\mathcal {S}}$

$$\begin{aligned} PU({\mathbf {z}})\le \rho _2 U({\mathbf {z}}), ~\Vert {\mathbf {z}}\Vert \ge M. \end{aligned}$$

This implies that the process $\rho ^{-t}_2 U({\mathbf {Z}}(t))$ is a supermartingale and therefore

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}&\left[ \rho _2^{-\theta (\tau \wedge n^*T^*)}U^\theta ({\mathbf {Z}}(\tau \wedge n^*T^*))\right] \le U^\theta ({\mathbf {z}}), {\mathbf {z}}\in {\mathcal {S}}. \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} U^\theta ({\mathbf {z}})\ge&{\mathbb {E}}_{\mathbf {z}}\left[ \rho _2^{-\theta (\tau \wedge n^*T^*)}U^\theta ({\mathbf {Z}}(\tau \wedge n^*T^*))\right] \\ =&{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \le (n^*-1)T^*\}}\rho _2^{-\theta (\tau \wedge n^*T^*)}U^\theta ({\mathbf {Z}}(\tau \wedge n^*T^*))\right] \\&+{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{ (n^*-1)T^*<\tau<n^*T^*\}}\rho _2^{-\theta (\tau \wedge n^*T^*)}U^\theta ({\mathbf {Z}}(\tau \wedge n^*T^*))\right] \\&+ {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \ge n^*T^*\}}\rho _2^{-\theta (\tau \wedge n^*T^*)}U^\theta ({\mathbf {Z}}(\tau \wedge n^*T^*))\right] \\ \ge&{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \le (n^*-1)T^*\}}U^\theta ({\mathbf {Z}}(\tau ))\right] \\&+\rho _2^{-\theta (n^*-1)T^*}{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{ (n^*-1)T^*<\tau <n^*T^*\}}U^\theta ({\mathbf {Z}}(\tau ))\right] ~~\\&+\rho _2^{-\theta n^*T^*}{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \ge n^*T^*\}}U^\theta ({\mathbf {Z}}(n^*T^*))\right] .\\ \end{aligned} \end{aligned}$$

(A.21)

By the strong Markov property of ${\mathbf {Z}}(t)$ and Proposition A.1, we obtain

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}&\left[ {\varvec{1}}_{\{\tau \le (n^*-1)T^*\}}U^\theta ({\mathbf {Z}}(n^*T^*))\right] \\&\le {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \le (n^*-1)T^*\}}\big [K_\theta +e^{-\frac{1}{2}\theta r^*(n^*T^*-\tau )}U^\theta ({\mathbf {Z}}(\tau ))\big ]\right] \\&\le K_\theta + \exp \left( -\frac{1}{2}\theta r^*T^*\right) {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \le (n^*-1)T^*\}}U^\theta ({\mathbf {Z}}(\tau ))\right] \end{aligned} \end{aligned}$$

(A.22)

Similarly, the strong Markov property of ${\mathbf {Z}}(t)$, Jensen’s inequality and Lemma A.2 imply

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}&\left[ {\varvec{1}}_{\{(n^*-1)T^*<\tau<n^*T^*\}}U^\theta ({\mathbf {Z}}(n^*T^*))\right] \\&\le {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{(n^*-1)T^*<\tau<n^*T^*\}}C_2^{\theta (n^*T^*-\tau )}U^\theta ({\mathbf {Z}}(\tau ))\right] \\&\le C_2^{\theta T^*}{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{(n^*-1)T^*<\tau <n^*T^*\}}U^\theta ({\mathbf {Z}}(\tau ))\right] . \end{aligned} \end{aligned}$$

(A.23)

Applying (A.22) and (A.23) to (A.21) yields

$$\begin{aligned} \begin{aligned} U^\theta (x) \ge&{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \le (n^*-1)T^*\}}U^\theta ({\mathbf {Z}}(\tau ))\right] \\&+\rho _2^{-\theta (n^*-1)T^*}{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{ (n^*-1)T^*<\tau<n^*T^*\}}U^\theta ({\mathbf {Z}}(\tau ))\right] \\&+\rho _2^{-\theta n^*T^*} {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \ge n^*T^*\}}U^\theta ({\mathbf {Z}}(n^*T^*))\right] \\ \ge&\exp \left( \frac{1}{2}\theta r^*T^*\right) {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \le (n^*-1)T^*\}}U^\theta ({\mathbf {Z}}(n^*T^*))\right] -\exp \left( \frac{1}{2}\theta r^*T^*\right) K_\theta \\&+C_2^{-\theta T^*}\rho _2^{-\theta (n^*-1)T^*}{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{ (n^*-1)T^*<\tau <n^*T^*\}}U^\theta ({\mathbf {Z}}(n^*T^*))\right] \\&+\rho _2^{-\theta n^*T^*} {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \ge n^*T^*\}}U^\theta ({\mathbf {Z}}(n^*T^*))\right] \\ \ge&\kappa ^{-1}{\mathbb {E}}_{\mathbf {z}}U^\theta ({\mathbf {Z}}(n^*T^*))-K_\theta \exp \left( \frac{1}{2}\theta \rho ^*T^*\right) \end{aligned} \end{aligned}$$

(A.24)

where $\kappa =\max \left\{ \exp \left( -\frac{1}{2}\theta r^*T^*\right) , C^{\theta T^*}\rho _2^{\theta (n^*-1)T^*}, \rho _2^{\theta n^*T^*}\right\} <1$ by (A.6). The proof of (A.17) is complete by taking

$$\begin{aligned} {\tilde{K}}=K_\theta \exp \left( \frac{1}{2}\theta \rho ^*T^*\right) \kappa . \end{aligned}$$

Having (A.17), the claims (A.19) and (A.18) follow by Benaïm and Schreiber (2019, Proposition 3.3 and Theorem 3.1). $\square$

Appendix B. Extinction Proofs

For $I\subset \{1,\ldots ,n\}$, denote by ${\mathcal {M}}^I, {\mathcal {M}}^{I,+}, {\mathcal {M}}^{I,\partial }$ the set of ergodic probability measures on ${\mathcal {S}}^I, {\mathcal {S}}^{I}_+, {\mathcal {S}}_0^I$ respectively.

Lemma B.1

Assume that there exists a function $\phi :{\mathcal {S}}\rightarrow {\mathbb {R}}_+$ and constants $C, \delta _\phi >0$ such that for all ${\mathbf {z}}\in {\mathcal {S}}$

$$\begin{aligned} P V({\mathbf {z}})\le V({\mathbf {z}})-\phi ({\mathbf {z}})+C \end{aligned}$$

(B.1)

and

$$\begin{aligned} {\mathbb {E}}_{{\mathbf {z}}}\left( V({\mathbf {Z}}(1))-PV({\mathbf {z}})\right) ^2+{\mathbb {E}}\left| \log F({\mathbf {z}},\xi (1))-{\mathbb {E}}\log F({\mathbf {z}},\xi (1))\right| ^2\le \delta _\phi \phi ({\mathbf {z}}).\nonumber \\ \end{aligned}$$

(B.2)

Then, the family of random occupation measures $({\widetilde{\Pi }}_t)_{t\in {\mathbb {N}}}$ is tight. and with probability one

$$\begin{aligned} \lim _{T\rightarrow \infty }\frac{1}{T}\sum _{t=0}^T\left( \log F({\mathbf {Z}}(t),\xi (t))-{\mathbb {E}}\left[ \log F({\mathbf {Z}}(t),\xi (t))\big |{\mathcal {F}}_t\right] \right) =0. \end{aligned}$$

(B.3)

where $({\mathcal {F}}_t)_{t\in {\mathbb {N}}}$ is the filtration generated by the process ${\mathbf {Z}}$.

Proof

Suppose ${\mathbf {Z}}(0)={\mathbf {z}}\in {\mathcal {S}}$. We have

$$\begin{aligned} \begin{aligned} V({\mathbf {Z}}(t+1))\le&V({\mathbf {Z}}(t))-\phi ({\mathbf {Z}}(t))+C +(V({\mathbf {Z}}(t+1))-PV({\mathbf {Z}}(t))\\ \le&V({\mathbf {Z}}(t))-\frac{1}{2}\phi ({\mathbf {Z}}(t))+2C -\left( \frac{1}{2}\phi ({\mathbf {Z}}(t))+C -(V({\mathbf {Z}}(t+1))-PV({\mathbf {Z}}(t)))\right) \end{aligned}\nonumber \\ \end{aligned}$$

(B.4)

We see from (B.2) that the quadratic variation of the martingale $V({\mathbf {Z}}(t+1))-PV({\mathbf {Z}}(t))$ is bounded by $\delta \phi ({\mathbf {Z}}(t))$. As a result we can use the strong law of large numbers for martingales and the bound (B.2), to get

$$\begin{aligned} \lim _{T\rightarrow \infty } \frac{\frac{1}{T}\sum _{t=0}^T(V({\mathbf {Z}}(t+1))-PV({\mathbf {Z}}(t)))}{\frac{1}{T}\sum _{t=0}^T \left( \phi ({\mathbf {Z}}(t))+C\right) }=\lim _{T\rightarrow \infty } \frac{\sum _{t=0}^T(V({\mathbf {Z}}(t+1))-PV({\mathbf {Z}}(t)))}{\sum _{t=0}^T \left( \phi ({\mathbf {Z}}(t))+C\right) }=0\,\text { a.s.} \end{aligned}$$

which implies

$$\begin{aligned} \limsup _{T\rightarrow \infty } \frac{1}{T}\sum _{t=0}^T \left( \frac{1}{2}\phi ({\mathbf {Z}}(t))+C -(V({\mathbf {Z}}(t+1))-PV({\mathbf {Z}}(t)))\right) \le 0 \text { a.s.} \end{aligned}$$

(B.5)

Taking sums in (B.4), noting that

$$\begin{aligned} \liminf _{T\rightarrow \infty }\frac{1}{T} \sum _0^T (V({\mathbf {Z}}(t+1)) - V({\mathbf {Z}}(t)))= \liminf _{T\rightarrow 0}\frac{1}{T} (V({\mathbf {Z}}(T+1)) - V({\mathbf {z}})) \ge 0\text { a.s.} \end{aligned}$$

(because V is nonnegative) and using (B.5) yields that with probability one

$$\begin{aligned} \liminf _{T\rightarrow \infty } \frac{1}{T}\sum _0^T \left( -\frac{1}{2}\phi ({\mathbf {Z}}(t))+2C\right) \ge 0. \end{aligned}$$

As a result

$$\begin{aligned} \limsup _{T\rightarrow \infty } \frac{1}{T}\sum _0^T \phi ({\mathbf {Z}}(t))\le 4C, \end{aligned}$$

(B.6)

almost surely. Since $\lim _{|{\mathbf {z}}|\rightarrow \infty }\phi ({\mathbf {z}})=\infty$, the boundedness of $\widetilde{\Pi }_T\phi$ in (B.6) implies that the family of randomized occupation measures $\{\widetilde{\Pi }_t, t\in {\mathbb {N}}\}$ is tight. Moreover, the strong law of large numbers together with (B.6) and (B.2) implies (B.3). $\square$

Theorem B.1

If $E_1$ is nonempty, then for any $I\in E_1$, there exists $\alpha _I>0$ such that, for any a compact set ${\mathcal {K}}^I\subset {\mathcal {S}}^{I}_+$, we have

$$\begin{aligned} \lim _{\mathrm {dist}({\mathbf {z}},{\mathcal {K}}^I)\rightarrow 0, {\mathbf {z}}\in S^\circ }{\mathbb {P}}_{\mathbf {z}}\left\{ \lim _{t\rightarrow \infty }\dfrac{\ln X_i(t)}{t}\le -\alpha _I, i\in I^c\right\} =1. \end{aligned}$$

Theorem B.2

If $E_2$ is empty or $\max _{i}\{ r_i(\nu )\}>0$ for any $\nu$ with $\mu ({\mathcal {S}}^{J}_+)=1$ for some $J\in E_2$ and $\cup _{I\in E_1}{\mathcal {S}}^{I}_+$ is accessible then

$$\begin{aligned} \sum _{I\in E_1} p_{{\mathbf {z}},I}=1 \end{aligned}$$

where

$$\begin{aligned} p_{{\mathbf {z}},I}= & {} {\mathbb {P}}_{\mathbf {z}}\left\{ \emptyset \ne {\mathcal {U}}(\omega )\subset {{\,\mathrm{Conv}\,}}\{{\mathcal {M}}^{I,+}\} ~\text {and}~\lim _{t\rightarrow \infty }\right. \\&\left. \frac{\ln X_j(t)}{t}\in \{r_j(\mu ):\mu \in {{\,\mathrm{Conv}\,}}({\mathcal {M}}^{I,+})\},\right. \\&\left. j\in I^c\right\} . \end{aligned}$$

Proof

Once Theorem B.1 is proved, Theorem B.2 can be obtained using the fact that any weak limit of a family of random occupation measures is an invariant probability measure supported on $S_0$ (Benaim 2018; Hening and Nguyen 2018a) and by using the arguments from Lemma 5.8, Lemma 5.9 and Theorem 5.2 by Hening and Nguyen (2018a). $\square$

Fix $I\in E_1$. Since by Lemma B.1 the family $({\widetilde{\Pi }}_t)_{t\in {\mathbb {N}}}$ of random occupation measures is tight, condition (2.9) is equivalent to the existence of $0<{\widehat{p}}_i<\gamma _3/n, i\in I$

$$\begin{aligned} \inf _{\nu \in {{\,\mathrm{Conv}\,}}({\mathcal {M}}^{I,\partial })}\sum _{i\in I}{\widehat{p}}_i r_i(\nu )>0. \end{aligned}$$

As a result, there exists a small ${\check{p}}\in (0,\gamma _3/n)$ such that

$$\begin{aligned} \begin{aligned} \sum _{i\in I}{\widehat{p}}_i r_i(\nu )-{\check{p}}\max _{i\notin I}\{ r_i(\nu )\}>0 \text { for any }\nu \in {{\,\mathrm{Conv}\,}}({\mathcal {M}}^{I,\partial }). \end{aligned} \end{aligned}$$

(B.7)

Define ${\tilde{p}}_i$ by ${\tilde{p}}_i={\widehat{p}}_i$ if $i\in I_\mu$ and ${\tilde{p}}_i=-{\check{p}}$ if $i\in I_\mu ^c$. In view of (B.7), (2.8) and Lemma B.1, there is $r_e>0$ such that for any $\nu \in {{\,\mathrm{Conv}\,}}({\mathcal {M}}^I)$,

$$\begin{aligned} \sum _{i\in I}{\widehat{p}}_i r_i(\nu )-{\check{p}}\max _{i\in I^c}\left\{ r_i(\nu )\right\} >3r_e. \end{aligned}$$

(B.8)

Lemma B.2

Let $I\in E_1$ and suppose that Assumption 2.1 holds. Suppose ${\widehat{p}}_i, {\check{p}}, r_e$ are the quantities from (B.8) and $n^*$ is defined by (A.6). There exist constants $T_e\ge 0$, $\delta _e>0$ such that, for any $T\in [T_e,n^*T_e]\cap {\mathbb {Z}}$, $\Vert {\mathbf {z}}\Vert \le M, x_i<\delta _e, i\in I^c$, we have

$$\begin{aligned} \begin{aligned}&\sum _{t=0}^T{\mathbb {E}}_{\mathbf {z}}\left( \ln V({\mathbf {Z}}(t+1))-\ln V({\mathbf {Z}}(t))-\sum _{i\in I} \ln F_i({\mathbf {Z}}(t), \xi (t))+{\check{p}} \max _{i\in I^c} \ln F_i({\mathbf {Z}}(t), \xi (t))\right) dt\\&\quad \le -r_e(T+1). \end{aligned} \end{aligned}$$

(B.9)

Proof

This is very similar to the proof of Lemma A.4 and is therefore omitted. $\square$

Proposition B.1

Let $I\in E_1$ and suppose that Assumption 2.1 holds. There exists $\theta \in (0,1)$ such that for any $T\in [T_e,n^*T_e]\cap {\mathbb {Z}}$ and ${\mathbf {z}}\in {\mathcal {S}}_+$ satisfying $\Vert {\mathbf {z}}\Vert \le M,$ $x_i<\delta _e,$ $i\in I^c$ one has

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}W_\theta ({\mathbf {Z}}(T))\le \exp \left( -\frac{1}{2}\theta r_eT\right) W_\theta ({\mathbf {z}}) \end{aligned}$$

where $M, T_e, {\widehat{p}}_i,{\check{p}}, \delta _e, n^*$ are as in Lemma B.2 and

$$\begin{aligned} W_\theta ({\mathbf {z}}):=\sum _{i\in I^c}\left[ V({\mathbf {z}})\dfrac{x_i^{\check{p}}}{\prod _{j\in I} x_j^{{\widehat{p}}_j}}\right] ^\theta , {\mathbf {z}}\in {\mathcal {S}}_+. \end{aligned}$$

Proof

For $i\in I^c$, let $W({\mathbf {z}},i):=V({\mathbf {z}})\dfrac{x_i^{{\check{p}}}}{\prod _{j\in I} x_j^{{\widehat{p}}_j}}.$ Similarly to Proposition A.1, by making use of Lemma B.2, one can find a $\theta >0$ such that for $T\in [T_e,n^*T_e]\cap {\mathbb {Z}}$, ${\mathbf {z}}\in {\mathbb {R}}^{n,\circ }_+\times {\mathbb {R}}^{\kappa _0}$ with $\Vert {\mathbf {z}}\Vert \le M,$ and $x_i<\delta _e$ we have

$$\begin{aligned} {\mathbb {E}}W^\theta ({\mathbf {Z}}(T),i)\le \exp \left( -\frac{1}{2}\theta r_eT\right) W^\theta ({\mathbf {z}}, i). \end{aligned}$$

The proof is complete by noting that

$$\begin{aligned} W_\theta ({\mathbf {z}})=\sum _{i\in I^c}W^\theta ({\mathbf {z}},i). \end{aligned}$$

$\square$

Proof of Theorem B.1

Let $I\in E_1$. By Lemma A.2 for any $i\in I^c$

$$\begin{aligned} P W({\mathbf {z}},i)\le \rho _2 W({\mathbf {z}},i), |{\mathbf {z}}|\ge M. \end{aligned}$$

Then using Jensen’s inequality, we have

$$\begin{aligned} P W_\theta ({\mathbf {z}})\le \rho _2^\theta W_\theta ({\mathbf {z}}) \text { if } |{\mathbf {z}}|\ge M. \end{aligned}$$

(B.10)

Define the constants

$$\begin{aligned} C_U:=\sup \left\{ \dfrac{\prod _{i\in I} x_i^{{\widehat{p}}_i}}{V({\mathbf {z}})}: {\mathbf {z}}\in {\mathbb {R}}^{n,\circ }_+\times {\mathbb {R}}^{\kappa _0}\right\} <\infty , \\ \varsigma :=\dfrac{\delta _e^{{\check{p}}\theta }}{C_U^\theta } \end{aligned}$$

and the stopping time

$$\begin{aligned} \eta :=\inf \left\{ t\ge 0: W_\theta ({\mathbf {Z}}(t))\ge \varsigma \right\} . \end{aligned}$$

Clearly, if $W_\theta ({\mathbf {z}})<\varsigma$, then $\eta >0$ and for any $i\in I^c$, we get

$$\begin{aligned} X_i(t)\le \delta _e\,, t\in [0,\eta ). \end{aligned}$$

(B.11)

Let

$$\begin{aligned} \widetilde{W}_\theta ({\mathbf {z}}):=\varsigma \wedge W_\theta ({\mathbf {z}}). \end{aligned}$$

We have from the concavity of $x\mapsto x\wedge \varsigma$ that

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}\widetilde{W}_\theta ({\mathbf {Z}}(T))\le \varsigma \wedge {\mathbb {E}}_{\mathbf {z}}W_\theta ({\mathbf {Z}}(T)). \end{aligned}$$

Let $\tau$ be defined as in (A.20). By (B.10) we have that

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}\left[ \rho _2^{-\theta (\tau \wedge \eta \wedge n^*T_e)}W_\theta ({\mathbf {Z}}(\theta \gamma _b(\tau \wedge \xi \wedge n^*T_e))\right] \le W_\theta ({\mathbf {z}}), {\mathbf {z}}\in {\mathcal {S}}_+. \end{aligned}$$

As a result for all ${\mathbf {z}}\in {\mathcal {S}}_+$

$$\begin{aligned} \begin{aligned} W_\theta ({\mathbf {z}})\ge&\, {\mathbb {E}}_{\mathbf {z}}\left[ \rho _2^{-\theta (\tau \wedge \eta \wedge n^*T_e)}W_\theta ({\mathbf {Z}}(\tau \wedge \eta \wedge n^*T_e))\right] \\ \ge&\, {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\tau \}}W_\theta ({\mathbf {Z}}(\tau ))\right] \\&+\, {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\eta \}}W_\theta ({\mathbf {Z}}(\eta ))\right] \\&+\,\rho _2^{-\theta (n^*-1)T_e} {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{(n^*-1)T_e<\tau \wedge \eta \le n^*T_e\}}W_\theta ({\mathbf {Z}}(\tau \wedge \eta ))\right] \\&+\,\rho _2^{-\theta n^*T_e} {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta > n^*T_e\}}W_\theta ({\mathbf {Z}}(n^*T_e))\right] .\\ \end{aligned} \end{aligned}$$

(B.12)

By the strong Markov property of $({\mathbf {Z}}(t))$ and Proposition B.1 (which we can use because of (B.11)) we see that for all ${\mathbf {z}}\in {\mathcal {S}}_+$

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}&\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\tau \}}W_\theta ({\mathbf {Z}}(n^*T_e))\right] \\&\le {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\tau \}}\exp \left( -\frac{1}{2}\theta r_e(n^*T_e-\tau )\right) W_\theta ({\mathbf {Z}}(\tau ))\right] \\&\le \exp \left( -\frac{1}{2}\theta r_eT_e)\right) {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\tau \}}W_\theta ({\mathbf {Z}}(\tau ))\right] . \ \end{aligned} \end{aligned}$$

(B.13)

Similarly, by the strong Markov property of $({\mathbf {Z}}(t))$ and Lemma A.2, we obtain for any ${\mathbf {z}}\in {\mathcal {S}}_+$ that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}_{\mathbf {z}}&\left[ {\varvec{1}}_{\{(n^*-1)T_e<\tau \wedge \eta \le n^*T_e\}}W_\theta ({\mathbf {Z}}(n^*T_e))\right] \\&\le {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{(n^*-1)T_e<\tau \wedge \eta \le n^*T_e\}}C_2^{\theta (n^*T_e-\tau \wedge \eta )}W_\theta ({\mathbf {Z}}(\tau \wedge \eta ))\right] \\&\le C_2^{\theta T_e}{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{(n^*-1)T_e<\tau \wedge \eta \le n^*T_e\}}W_\theta ({\mathbf {Z}}( \tau \wedge \eta ))\right] . \end{aligned} \end{aligned}$$

(B.14)

If $W_\theta ({\mathbf {z}})<\varsigma$ then applying (B.13), (B.14) and the inequality $\widetilde{W}_\theta ({\mathbf {Z}}(n^*T_e))\le W_\theta ({\mathbf {Z}}(n^*T_e\wedge \eta ))$ to (B.12) yields

$$\begin{aligned} \begin{aligned} \widetilde{W}_\theta ({\mathbf {z}})=W_\theta ({\mathbf {z}}) \ge&{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\tau \}}W_\theta ({\mathbf {Z}}(\tau ))\right] \\&+ {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\eta \}}W_\theta ({\mathbf {Z}}(\eta ))\right] \\&+\rho _2^{-\theta (n^*-1)T_e} {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{(n^*-1)T_e<\tau \wedge \eta \le n^*T\}}W_\theta ({\mathbf {Z}}(\tau \wedge \eta ))\right] \\&+\rho _2^{-\theta n^*T_e} {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta> n^*T_e\}}W_\theta ({\mathbf {Z}}(n^*T_e))\right] \\ \ge&\exp \left( \frac{1}{2}\theta r_eT_e)\right) {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\tau \}}W_\theta ({\mathbf {Z}}(n^*T_e))\right] \\&+ {\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta \wedge (n^*-1)T_e=\eta \}}\widetilde{W}_\theta ({\mathbf {Z}}(n^*T_e))\right] \\&+\rho _2^{-\theta (n^*-1)T_e}C_2^{-\theta T_e}{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{(n^*-1)T_e<\tau \wedge \eta \le n^*T_e\}}W_\theta ({\mathbf {Z}}(n^*T_e))\right] \\&+\rho _2^{-\theta n^*T_e}{\mathbb {E}}_{\mathbf {z}}\left[ {\varvec{1}}_{\{\tau \wedge \eta > n^*T_e\}}W_\theta ({\mathbf {Z}}(n^*T_e))\right] \\ \ge&{\mathbb {E}}_{\mathbf {z}}\widetilde{W}_\theta ({\mathbf {Z}}(n^*T_e)) \,\quad \text { (since } \widetilde{W}_\theta (\cdot )\le W_\theta (\cdot )). \end{aligned} \end{aligned}$$

(B.15)

Clearly, if $W_\theta ({\mathbf {z}})\ge \varsigma$ then

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}\widetilde{W}_\theta ({\mathbf {Z}}(n^*T_e)) \le \varsigma =\widetilde{W}_\theta ({\mathbf {z}}). \end{aligned}$$

(B.16)

As a result of (B.15), (B.16) and the Markov property of $({\mathbf {Z}}(t))$, the sequence $\{w_2(k): k\ge 0\}$ where $w_2(k):=\widetilde{W}_\theta ({\mathbf {Z}}(kn^*T_e))$ is a supermartingale. Define the discrete stopping time

$$\begin{aligned} \eta ^*:=\inf \{k\in {\mathbb {N}}: W_\theta ({\mathbf {Z}}(kn^*T_e))\ge \varsigma \}. \end{aligned}$$

Moreover, we also deduce from (B.15) that

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}{\varvec{1}}_{\{\eta ^*>1\}}W_\theta ({\mathbf {Z}}(n^*T_e))\le \kappa _e W_\theta ({\mathbf {z}}), {\mathbf {z}}\in {\mathcal {S}}_+ \end{aligned}$$

where $\kappa _e^{-1}=\min \left\{ \rho _2^{-\theta (n^*-1)T_e}C_2^{-\theta T_e},\exp \left( \frac{1}{2}\theta r_eT_e)\right) ,\rho _2^{-\theta n^*T_e}\right\} >1.$ As a result, $\{w_3(k): k\ge 0\}$ with

$$\begin{aligned} w_3(k):=\kappa _e^{-k}{\varvec{1}}_{\{\eta ^*>k\}}W_\theta ({\mathbf {Z}}(kn^*T_e)) \end{aligned}$$

is also a supermartingle. For any $\varepsilon \in (0,1)$, if $W_\theta ({\mathbf {z}})\le \varsigma \varepsilon$ we have

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}w_2(k\wedge \eta ^*)\le {\mathbb {E}}_{\mathbf {z}}w_2(0)=W_\theta ({\mathbf {z}})\le \varsigma \varepsilon \,, k\ge 0. \end{aligned}$$

(B.17)

Subsequently, (B.17) combined with the Markov inequality and the fact that $w_2(\eta )=W_\theta ({\mathbf {Z}}(\eta n^*T_e))\ge \varsigma$ yields

$$\begin{aligned} {\mathbb {P}}_{\mathbf {z}}\{\eta ^*<k\}\le \varsigma ^{-1}{\mathbb {E}}_{\mathbf {z}}w_2(k\wedge \eta ^*)\le \varepsilon , ~\text {if}~k\in {\mathbb {N}}, W_\theta ({\mathbf {z}})\le \varsigma \varepsilon . \end{aligned}$$

Next, let $k\rightarrow \infty$ to get

$$\begin{aligned} {\mathbb {P}}_{\mathbf {z}}\{\eta ^*<\infty \}\le \varepsilon \,\text { if }\,W_\theta ({\mathbf {z}})\le \varsigma \varepsilon . \end{aligned}$$

(B.18)

Let $\kappa _3\in (\kappa _e,1)$. Since $P_t \widetilde{W}_\theta ({\mathbf {z}})\le C_2^{\theta t} \widetilde{W}_\theta ({\mathbf {z}})$ for any $t>0, {\mathbf {z}}\in {\mathcal {S}}_+$, we have that

$$\begin{aligned} {\mathbb {P}}_{\mathbf {z}}\left\{ \widetilde{W}_\theta ({\mathbf {Z}}(t))\le c, t=1,\ldots , n^*T_e\right\} \le \dfrac{n^*T_e C_2^{\theta n^*T_e}\widetilde{W}_\theta ({\mathbf {z}})}{c}. \end{aligned}$$

The last inequality, the Markov property of ${\mathbf {Z}}(t)$, and the fact that $(w_3(k))_{k\ge 0}$ is a supermartingale imply that for any ${\mathbf {z}}\in {\mathcal {S}}_+$

$$\begin{aligned} \begin{aligned} {\mathbb {P}}_{\mathbf {z}}&\left\{ {\varvec{1}}_{\{\eta ^*>k\}}\widetilde{W}_\theta ({\mathbf {Z}}(t))\le \left( \frac{\kappa _e}{\kappa _3}\right) ^{-k}, t\in [kn^*T_e+1, (k+1)n^*T_e]\right\} \\&\le n^*T_e C_2^{\theta n^*T_e}{\mathbb {E}}_{\mathbf {z}}w_3(k)\dfrac{\kappa _e^{-k}}{\kappa _3^{-k}}\le n^*T_e C_2^{\theta n^*T_e}\dfrac{{\mathbb {E}}_{\mathbf {z}}w_3(0)}{\kappa _3^{-k}}\le n^*T_e C_2^{\theta n^*T_e}\dfrac{W_\theta ({\mathbf {z}})}{\kappa _3^{-k}} \end{aligned} \end{aligned}$$

Since $\sum _k\left( n^*T_e C_2^{\theta n^*T_e}\dfrac{W_\theta ({\mathbf {z}})}{\kappa _3^{-k}}\right) <\infty$, we deduce from the Borel–Cantelli lemma that

$$\begin{aligned} {\mathbb {P}}_{\mathbf {z}}\left\{ \limsup _{k\rightarrow \infty } \sup _{t\in [kn^*T_e+1, (k+1)n^*T_e]}{\varvec{1}}_{\{\eta ^*>k\}}\widetilde{W}_\theta ({\mathbf {Z}}(t))\left( \frac{\kappa _3}{\kappa _e}\right) ^{k}<1\right\} =1. \end{aligned}$$

This and (B.18) imply that if $W_\theta ({\mathbf {z}})\le \varsigma \varepsilon$ then

$$\begin{aligned} {\mathbb {P}}_{\mathbf {z}}\left\{ \limsup _{t\rightarrow \infty } \left( \left( \frac{\kappa _3}{\kappa _e}\right) ^{\frac{t}{n^*T_e+1}} W_\theta ({\mathbf {Z}}(t))\right) <1\right\} \ge 1-\varepsilon . \end{aligned}$$

Since $\sup _{{\mathbf {z}}\in {\mathbb {R}}^n\times {\mathbb {R}}^{\kappa _0}}\frac{x_i^{\theta \check{p}}}{W_\theta ({\mathbf {z}})}<\infty , i\in I^c$, we can easily obtain the extinction result from Theorem 2.5.

$\square$

Appendix C. Robustness proofs

Let $\circ$ denote the element-wise product and ${\varvec{1}}_n$ be the vector in ${\mathbb {R}}^n$ whose components are all 1. Assume that the function V from Assumption A3) satisfies the robust estimate

$$\begin{aligned} {\mathbb {E}}\left[ V({\mathbf {x}}^\top (F({\mathbf {z}},\xi )\circ ({\varvec{1}}_n+\widetilde{\varepsilon }_1)), G({\mathbf {z}},\xi )+\widetilde{\varepsilon }_2))h({\mathbf {z}},\xi )\right] \le \rho V({\mathbf {z}})+C \end{aligned}$$

for any vectors $\widetilde{\varepsilon }_1,\widetilde{\varepsilon }_2\in {\mathbb {R}}^n$ such that $|\widetilde{\varepsilon }_1|\vee |\widetilde{\varepsilon }_2|<\delta$. Note that

$$\begin{aligned} {\tilde{h}}({\mathbf {z}},\xi )=\max _{i=1}^n \left\{ \widetilde{F}_i({\mathbf {z}},\xi ),\frac{1}{\widetilde{F}_i({\mathbf {z}},\xi )}\right\} ^{\gamma _3}. \end{aligned}$$

Consequently ${\tilde{h}}({\mathbf {z}},\xi )\le e^\delta h({\mathbf {z}},\xi )$ and

$$\begin{aligned} {\mathbb {E}}\left[ V({\mathbf {x}}^\top (\widetilde{F}({\mathbf {z}},\xi )), \widetilde{G}({\mathbf {z}},\xi ))\widetilde{h}({\mathbf {z}},\xi )\right] \le \rho V({\mathbf {z}})+C \end{aligned}$$

if $\delta >0$ is sufficiently small. This shows that there exist $C_2>0,\rho _2\in (0,1),\gamma _4\in (0,\gamma _3)$ such that

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}V\left( \widetilde{\mathbf {Z}}(1)\prod _{i=1}^n\widetilde{X}_i^{p_i}(1)\right) \le \left( {\varvec{1}}_{\{|{\mathbf {z}}|<M\}}(C_2-\rho _2)+\rho _2\right) V^{\frac{1}{2}}({\mathbf {z}})\prod _{i=1}^nx_i^{p_i} \end{aligned}$$

for any ${\mathbf {p}}=(p_1,\ldots ,p_n)\in {\mathbb {R}}^n$ satisfying

$$\begin{aligned} |{\mathbf {p}}|_1:=\sum |p_i|\le \gamma _4. \end{aligned}$$

Analogously to Lemma A.1 one can show the following uniform bound

$$\begin{aligned} \sup _{|{\mathbf {z}}|\le M, t\in {\mathbb {N}}} \left( {\mathbb {E}}_{\mathbf {z}}V(\widetilde{\mathbf {Z}}(t))+{\mathbb {E}}_{{\mathbf {z}}}\tilde{h}(\widetilde{\mathbf {Z}}(t),\xi (t))\right) =K_{M}<\infty \end{aligned}$$

(C.1)

when $\delta$ is sufficiently small.

Slight changing in the factor on the right hand side of A.5, we have

Lemma C.1

Suppose that Assumption A.1 holds. Let ${\mathbf {p}}$ and $r^*$ be as in (A.4). There exists a $\widetilde{T}^*>0$ such that, for any $T>\widetilde{T}^*$, ${\mathbf {z}}\in {\mathcal {S}}_0, \Vert {\mathbf {z}}\Vert \le M$ one has

$$\begin{aligned} \sum _{t=0}^T{\mathbb {E}}_{\mathbf {z}}\left( \ln V({\mathbf {Z}}(t+1))-\ln V({\mathbf {Z}}(t))-\sum p_i\ln F_i({\mathbf {Z}}(t), \xi (t))\right) \le -1.5r^*(T+1). \end{aligned}$$

(C.2)

On the other hand, it obviously follows from (2.11) that for any $\varepsilon>0, M>0, T>0, n_0>0$, we have

$$\begin{aligned} {\mathbb {E}}_{\mathbf {z}}\left\{ {\varvec{1}}_{\{|\widetilde{\mathbf {Z}}(t))|\vee |{\mathbf {Z}}(t)|<n_0\}}|\widetilde{\mathbf {Z}}(t))-{\mathbf {Z}}(t)|\right\} \le \varepsilon ,\, {\mathbf {z}}\le M, t\in \{0,\ldots , T\} \end{aligned}$$

when $\delta$ is sufficiently small. This and the uniform integrability (C.1) imply that, for any $\varepsilon , T>0$, there exists $\delta >0$ such that for any $\delta$-pertubation of (3.1), we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}_{{\mathbf {z}}}|\ln \widetilde{F}_i(\widetilde{\mathbf {Z}}(t))-\ln F_i({\mathbf {Z}}(t))| \le {\mathbb {E}}_{{\mathbf {z}}}|\ln \widetilde{F}_i(\widetilde{\mathbf {Z}}(t))-\ln \widetilde{F}_i({\mathbf {Z}}(t))|\\&\quad +{\mathbb {E}}_{{\mathbf {z}}}|\ln \widetilde{F}_i({\mathbf {Z}}(t))-\ln F_i({\mathbf {Z}}(t))| \varepsilon ,\, {\mathbf {z}}\in {\mathcal {S}}_0,|{\mathbf {z}}|\le M, t\in \{0,\ldots , T\}. \end{aligned} \end{aligned}$$

(C.3)

By virtue of (C.2) and (C.3), for sufficiently small $\delta$,

$$\begin{aligned} \sum _{t=0}^T{\mathbb {E}}_{\mathbf {z}}\left( \ln V(\widetilde{\mathbf {Z}}(t+1))-\ln V(\widetilde{\mathbf {Z}}(t))-\sum p_i\ln F_i(\widetilde{\mathbf {Z}}(t)), \xi (t))\right) \le -r^*(T+1) \end{aligned}$$

(C.4)

for any ${\mathbf {z}}\in {\mathcal {S}}_0, |{\mathbf {z}}|\le M$ and $T\ge \widetilde{T}^*.$ With this estimate, it follows from the arguments in Appendix A that Theorem A.1 holds for $\widetilde{\mathbf {Z}}(t)$. Similarly, Theorem 2.5 and Theorem 2.6 hold.

Appendix D. Proof of Lemma 4.1

Lemma D.1

Suppose $\{(E_i(t), S(t))_{i=1,\ldots , n}\}, t\in {\mathbb {N}}$ is a sequence of $n+1$-dimensional random variables, i.i.d. over t such that ${\mathbb {E}}\left[ S(t)e^{E_j(t)}\right] ^2<\infty .$ Then the model given by (2.4) and (4.11) satisfies Assumption 2.1 by taking a small enough $\gamma _3>0$, and

$$\begin{aligned} V({\mathbf {z}}) = \sum _j z_j + 1. \end{aligned}$$

Assumption 2.2 holds with

$$\begin{aligned} \phi ({\mathbf {z}}) = \delta V({\mathbf {z}}) \end{aligned}$$

for some $\delta >0$. Moreover, if the support of $\ln S(t) + max_j (E_j(t)\ln \delta _j)$ contains values less than 0 then the boundary is accessible.

Proof

We denote by K below a positive generic constant. Using (4.11)

$$\begin{aligned} \sum _j N_j(t)e^{E_j(t)-D_j(t)}\le \sum _j N_j(t) e^{E_j(t)}\exp \{-\alpha _{j} N_j(t) e^{E_j(t)}\}< K<\infty .\nonumber \\ \end{aligned}$$

(D.1)

On the other hand

$$\begin{aligned} \left( 1-\delta _j+ S(t)e^{E_j(t)-D_j(t)}\right) \vee \left( 1-\delta _j+ S(t)e^{E_j(t)-D_j(t)}\right) ^{-1} \le K+\sum _j S(t)e^{E_j(t)-D_j(t))} \end{aligned}$$

Using this and the inequality

$$\begin{aligned} (a+b)^\gamma \le 2^\gamma (a^\gamma +b^\gamma ) \end{aligned}$$

we see that for any $\varepsilon >0$ there is $\gamma _3>0$ sufficiently small such that

$$\begin{aligned} \begin{aligned} h(t):=&\max _j\left( \left( 1-\delta _j)+ S(t)e^{E_j(t)-D_j(t)}\right) ^{\gamma _3}\vee \left( 1-\delta _j+S(t) e^{E_j(t)-D_j(t)}\right) ^{-\gamma _3}\right) \\ \le&\left( 1+\varepsilon +\left( S(t)\sum e^{E_j(t)-D_j(t)}\right) ^{\gamma _3}\right) \end{aligned} \end{aligned}$$

Using (D.1) and the fact that ${\mathbb {E}}S(t)<\infty$ we get

$$\begin{aligned} {\mathbb {E}}^t\sum _j \left( (1-\delta _j)N_j(t)(1+\varepsilon +\left( S(t)e^{E_j(t)-D_j(t)}\right) ^{\gamma _3})\right) \le (1-0.5{\check{d}}) \sum _j N_j(t) +K \end{aligned}$$

where ${\mathbb {E}}^t[\cdot ]:={\mathbb {E}}[\cdot ~|~N_1(t),\ldots , N_n(t)]$ and ${\check{d}}=\min \{\delta _i\}$. Since $N_j(t) e^{E_j(t)}\exp \{-a_{j}(1+\gamma _3) N_j(t) e^{E_j(t)}$ is bounded above by a nonrandom constant, then

$$\begin{aligned} N_j(t)e^{(1+\gamma _3)(E_j(t)-D_j(t))}\le e^{\gamma _3 E_j(t)} N_j(t) e^{E_j(t)}\exp \{-a_{j}(1+\gamma _3) N_j(t) e^{E_j(t)}\}< Ke^{\gamma _3 E_j(t)} \end{aligned}$$

Therefore, since ${\mathbb {E}}\left[ S^2(t)e^{\gamma _3 E_j(t)}\right] <\infty$ we have

$$\begin{aligned} {\mathbb {E}}^t S^{1+\gamma _3}(t)N_j(t)e^{(1+\gamma _3)(E_j(t)-D_j(t))} \le K{\mathbb {E}}^t \left( S^{1+\gamma _3}(t)e^{\gamma _3 E_j(t)}\right) \le K_2. \end{aligned}$$

As a result,

$$\begin{aligned}&{\mathbb {E}}^t \sum _j \left( 1-\delta _j+S(t)e^{E_j(t)-D_j(t)}\right) N_j(t)h(t) \le \sum _j (1-0.5\delta _j) N_j(t) \\&\quad + K_2 \le (1-0.25{\check{d}}) \sum _j N_j(t)+K_2. \end{aligned}$$

This implies that Assumption A3) is satisfied with $V({\mathbf {z}})=\sum _j N_j+1=\sum _j z_j+1$ and small $\gamma _3>0$. Note that

$$\begin{aligned} \sum _j N_j(t+1)-P_1 \sum _j N_j(t)=S(t)\sum _j e^{E_j(t)-D_j(t)} N_j(t)- {\mathbb {E}}^t S(t) \sum _j e^{E_j(t)-D_j(t)} N_j(t) \end{aligned}$$

This shows that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}^t \left( \sum _j N_j(t+1)-P_1 \sum _j N_j(t)\right) ^2\le&\, {\mathbb {E}}^t \left( S(t)\sum _j e^{E_j(t)-D_j(t)} N_j(t)\right) ^2\\ \le&\, K_3^2{\mathbb {E}}^t \left[ S(t)e^{E_j(t)}\right] ^2 = K_3^2 {\mathbb {E}}^t \left[ S(t)e^{E_j(t)}\right] ^2 \end{aligned} \end{aligned}$$

since $\sum _j e^{-D_j(t)} N_j(t)$ is bounded by a constant $K_3$. If ${\mathbb {E}}\left[ S(t)e^{E_j(t)}\right] ^2<\infty$ then Assumption 2.2 is satisfied with $\phi ({\mathbf {z}})=\delta (|{\mathbf {z}}|+1)$ for some $\delta >0$.

Moreover, if support of $\ln S(t) + \max _j \{E_j(t)\ln \delta _j\}$ contains values less than 0 then it is clear that the boundary is accessible since when $\ln S(t) + \max _j \{E_j(t) \ln \delta _j\}$ is less than a negative constant, $N_j(t+1)\le \rho N_j(t)$ for $\rho <1$. $\square$