Persistence and extinction for stochastic ecological models with internal and external variables

Abstract

The dynamics of species’ densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. To understand this interplay, we prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities i.e. stochastic persistence, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, in the case of persistence, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka–Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk–dove game, but inhibits coexistence in the rock–paper–scissors game and a Lotka–Volterra predator–prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight several challenges for future research.

Proofs

1.1 Proof of Proposition 1

We begin with the following fundamental lemma. Recall $\xi _t$ take values in a polish space $\varXi$ and have a common law $m(d\xi )$.

Lemma 3

Let $g : \mathcal {S}\times \varXi \mapsto \mathbb {R}$ be a measurable map such that $\sup _{z\in \mathcal {S}} \int g(z,\xi )^2 m(d\xi )<\infty .$ Define $\overline{g}(z) = \int g(z,\xi ) m(d\xi ).$ Then

• (i) For all $z \in \mathcal {S}$ and $Z_0 = z$

  $$\begin{aligned} \lim _{t \rightarrow \infty } \frac{\sum _{s = 0}^{t-1} g(Z_s,\xi _{s+1}) - \sum _{s = 0}^{t-1} \overline{g}(Z_s)}{t} = 0 \text{ with } \text{ probability } \text{ one. } \end{aligned}$$

• (ii) Let $\mu$ be an invariant (respectively ergodic) probability measure for $(Z_t)$, then there exists a bounded measurable map $\widehat{g}$ such that with probability one and for $\mu$-almost every z

  $$\begin{aligned} \lim _{t \rightarrow \infty } \frac{\sum _{s = 0}^{t-1} g(Z_s,\xi _{s+1})}{t} = \lim _{t \rightarrow \infty } \frac{\sum _{s = 0}^{t-1} \overline{g}(Z_s)}{t} = \widehat{g}(z) \text{ when } Z_0=z. \end{aligned}$$

  Furthermore

  $$\begin{aligned}&\int \overline{g}(z) \mu (dz) = \int \widehat{g}(z) \mu (dz) \text{(respectively } \widehat{g}(z)\\&\quad = \int \overline{g}(z) \mu (dz) \quad \mu -\text{ almost } \text{ surely } ). \end{aligned}$$

Proof

To prove the first assertion, let $C=\sup _{z\in \mathcal {S}}\int g(z,\xi )^2m(d\xi )$ and define the martingale

$$\begin{aligned} M_t=\sum _{s=0}^{t-1}\left( g(Z_s,\xi _{s+1})-\overline{g}(Z_s)\right) . \end{aligned}$$

with respect to the sigma-algebra ${\mathcal {F}}_t$ generated by $(Z_0,\xi _0),\dots ,(Z_t,\xi _t)$. $M_t$ has square integrable martingale differences as $\mathbb {E}[(M_t-M_{t-1})^2|{\mathcal {F}}_{t-1}]\le 2C^2.$ Define the previsible increasing process $\langle M\rangle _t$ by $\langle M\rangle _0=0$ and $\langle M\rangle _t=\langle M\rangle _{t-1}+\mathbb {E}[(M_t-M_{t-1})^2|{{\mathcal {F}}}_{t-1}]$. $\langle M\rangle _t$ is known as the angle-brackets process (Williams 1991, Sect. 12.12). By construction, we have $\langle M\rangle _t\le 2C^2 t$. Define $\langle M\rangle _\infty =\lim _{t\rightarrow \infty } \langle M\rangle _t$, which exists (possibly infinite) as $\langle M\rangle _t$ is increasing. On the event $\langle M\rangle _\infty <+\infty$, Williams (1991, Theorem 12.13a) implies that $\lim _{t\rightarrow \infty }M_t$ exists and is finite. In particular, $\lim _{t\rightarrow \infty }M_t/t=0$ on the event $\langle M\rangle _\infty <+\infty .$ On the event $\langle M\rangle _\infty =+\infty$, Williams (1991, Theorem 12.14a) implies that $\lim _{t\rightarrow \infty }M_t/\langle M\rangle _t=0$. In particular, as $\langle M\rangle _t\le 2C^2t$, $\lim _{t\rightarrow \infty }M_t/t=0$ on the event $\langle M\rangle _\infty =+\infty$. Thus, $\lim _{t\rightarrow \infty }M_t/t=0$ with probability one which completes the proof of the first assertion.

The second assertion follows from Birkhoff’s ergodic theorem applied to stationary Markov Chains (see Meyn and Tweedie (2009), Theorem 17.1.2) $\square$

Equation (3.2) and the second assertions of Proposition 1 follow directly from the preceding lemma applied to $g(z,\xi ) = \log f^i(z,\xi )$. Assumption A4 implies that $\log f^i(z,\xi _t)$ are uniformly integrable (UI). Therefore, equation (3.3) follows from (3.2) and the UI convergence theorem. The first half of the third assertion follows from ergodicity. To prove the second half of the third assertion, let $i\in S(\mu )$. By assertion (i) of Proposition 1

$$\begin{aligned} \lim _{t \rightarrow \infty } \frac{\log X^i_t}{t} = {\widehat{r}}^{i}(z)\quad \text{ where } Z_0=z \end{aligned}$$

for $\mu$-almost $z \in \{(x,y)\in \mathcal {S}: x^i>0\}.$ Let $\mathcal {S}^{i,\eta } = \{(x,y) \in \mathcal {S}\, : x^i \ge \eta \}$ and $\eta ^*>0$ be such that $\mu (\mathcal {S}^{i,\eta })>0$ for all $\eta \le \eta ^*$. By Poincaré Recurrence Theorem, for $\mu$ almost all $z \in \mathcal {S}^{i,\eta }$

$$\begin{aligned} \mathbb {P}_x[ Z_t \in \mathcal {S}^{i,\eta } \text{ infinitely } \text{ often }] = 1 \end{aligned}$$

for $\eta \le \eta ^*$. Thus $\widehat{r}_{i}(z) = 0$ for $\mu$-almost all $z \in \mathcal {S}^{i,\eta }$ with $\eta \le \eta ^*$. Hence $\widehat{r}^{i}(z) = 0$ for $\mu$-almost all $z \in \bigcup _{n \in \mathbf {N}} \mathcal {S}^{i,1/n} = \{(x,y) \in \mathcal {S}\, : x^i > 0\}.$ This proves assertion (iii).

1.2 Proof of Proposition 2

We begin by showing there is $T\ge 1$ and $\alpha \in (0,1)$ such that inequality (3.5) holds i.e. $P_TV(z)-V(z)\le -\alpha$ for all $z\in \mathcal {S}_0$. Suppose to the contrary. Then there exists an increasing sequence of times $t_k\uparrow \infty$, a decreasing sequence of positive reals $\alpha _k\downarrow 0$ and a sequence of points $z_k$ in $\mathcal {S}_0$ such that $P_{t_k}V(z_k)-V(z_k)\ge -\alpha _k$ for all k. Define a sequence of Borel probability measures $\mu _k$ on $\mathcal {S}_0$ by

$$\begin{aligned} \int h(z) \mu _k(dz)= \frac{1}{t_k}\mathbb {E}_{z_k}\left[ \sum _{\tau =0}^{t_k-1} h(Z_\tau )\right] \text{ for } \text{ any } \text{ continuous } h:\mathcal {S}_0\rightarrow \mathbb {R}\text{. } \end{aligned}$$

Let $\mu$ be a weak* limit point of the $\mu _i$, which exists as $\mathcal {S}_0$ is compact. By a standard argument due to Khasminskii (see, e.g., Kifer (1988, Theorem 1.1)), $\mu$ is an invariant measure for the Markov chain and by weak* compactness, $\sum _i p^i r^i(\mu )\le 0$. By the ergodic decomposition theorem, there exists an ergodic measure $\nu$ supported on $\mathcal {S}_0$ such that inequality (3.4) is violated; a contradiction.

Let $\phi (x)=e^x-1-x$. For any real C,

$$\begin{aligned} |\phi (-\theta C)|\le \sum _{k\ge 2} \frac{|\theta C|^k}{k!} \le \theta ^2 e^C\quad \text{ whenever } |\theta |\le 1. \end{aligned}$$

Our assumption that $\sup _{z,\xi }|\log f^i(z,\xi )|<\infty$ implies that there exists $C>0$ such that

$$\begin{aligned} |\sum _{\tau =0}^{T-1}\sum _i p^i \log f^i (z_\tau ,\xi _{\tau })|\le C \end{aligned}$$

for any $z_0,\dots ,z_{T-1}\in \mathcal {S}_0$ and $\xi _1,\dots ,\xi _T\in \varXi .$ Thus, for $z\in \mathcal {S}{\setminus } \mathcal {S}_0$

$$\begin{aligned} P_T V_\theta (z)= & {} \mathbb {E}_z\left[ \exp (-\theta \sum _i p^i \log X_T^i) \right] \\= & {} \mathbb {E}_z\left[ \exp \left( -\theta \sum _i p^i \left( \sum _{\tau =0}^{T-1}\log f^i(Z_\tau ,\xi _{\tau +1})+\log x^i\right) \right) \right] \\= & {} \mathbb {E}_z\left[ \exp \left( -\theta \sum _i p^i \sum _{\tau =0}^{T-1}\log f^i(Z_\tau ,\xi _{\tau +1}\right) \right] V_\theta (z)\\= & {} \mathbb {E}_z\left[ 1-\theta \sum _i p^i \sum _{\tau =0}^{T-1}\log f^i(Z_\tau ,\xi _{\tau +1})\right. \\&\left. +\phi \left( -\theta \sum _i p^i \sum _{\tau =0}^{T-1}\log f^i(Z_\tau ,\xi _{\tau +1})\right) \right] V_\theta (z)\\\le & {} (1-\theta \alpha +\theta ^2e^C )V_\theta (z). \end{aligned}$$

Choosing $\theta =\alpha \,e^{-C}/2$ and $\rho =1-\theta \alpha /2$ completes the proof of the proposition.

1.3 Proof of Theorem 1

In this section, we provide the details of the proof of Theorem 1 that are not presented in the main text. Through out this section, we assume that (3.4) holds. Let $\theta >0$, $T\ge 1$, $\rho \in (0,1)$, and $\beta >0$ be as given in Proposition 2.

We begin proving persistence in probability for the general case of $T\ge 1$. Given any $z_0\in \mathcal {S}{\setminus } \mathcal {S}_0$, any integer $t=\ell T+s\ge 1$ with $\ell \ge 0$ and $0\le s\le T-1$, and $\eta >0$, Proposition 2 implies

$$\begin{aligned} \mathbb {P}_{z_0}\left[ Z_{t}\in \mathcal {S}_\eta \right] \min _{z\in \mathcal {S}_\eta {\setminus }\mathcal {S}_0}V_\theta (z)\le & {} \int _\mathcal {S}V_\theta (z)(\delta _{z_0}P_{\ell T+s})(dz)\\\le & {} \rho ^\ell \int _\mathcal {S}V_\theta (z) (\delta _{z_0}P_s)(dz)+\frac{\beta }{1-\rho }\\\le & {} \rho ^\ell \max _{0\le \tau \le T-1}\int _\mathcal {S}V_\theta (z) (\delta _{z_0}P_\tau )(dz)+\frac{\beta }{1-\rho } \end{aligned}$$

As $\rho ^\ell \rightarrow 0$ as $t\rightarrow \infty$, inequality (3.7) implies

$$\begin{aligned} \limsup _{t\rightarrow \infty }\mathbb {P}_{z_0}[Z_t\in \eta ]\le a \eta ^b \text{ where } a=\frac{\beta }{a_0(1-\rho )} \end{aligned}$$

which completes the proof of persistence in probability.

To complete the proof of almost-sure persistence presented in the main text, we need the following lemma which follows the strategy of proof of Schreiber et al. (2011, Lemma 6).

Lemma 4

For all $z\in \mathcal {S}{\setminus } \mathcal {S}_0$, the weak* limit points $\mu$ of $\varPi _t$ almost surely satisfy $\mu (\mathcal {S}_0)=0$.

Proof

The process $\{Z_t\}_{t=0}^\infty$ being a (weak) Feller Markov chain over a compact set $\mathcal {S}$ implies that the set of weak* limit points of $\{\varPi _t\}_{t=0}^\infty$ is almost surely a non-empty compact subset of probability measures supported by $\mathcal {S}.$ Almost sure invariance of these weak* limit points follows from Lemma 3 (i).

Assertion (i) of Lemma 3 applied to $g(z,\xi ) = \sum _i p^i\log f^i(z,\xi )$ gives we have

$$\begin{aligned} \lim _{t \rightarrow \infty } \frac{\sum _i p^i\log X^i_t-\sum _ip^i\log x^i - \sum _{s=0}^{t-1}\sum _ip^i \overline{\log f^i}(Z_s )}{t} = 0 \end{aligned}$$

where $\overline{\log f^i}(z)=\int \log f^i (z,\xi )m(d\xi ).$ Since $\limsup _{t\rightarrow \infty }\frac{1}{t} \sum _ip^i\left( \log X^i_t-\log x^i\right) \le 0$ almost surely, we get that

$$\begin{aligned} \sum _ip^ir^i(\mu ) \le 0 \end{aligned}$$

(7.1)

almost surely for any weak* limit point $\mu$ of $\{\varPi _t\}_{t=0}^\infty$.

Since $\mathcal {S}_0$ and $\mathcal {S}{\setminus }\mathcal {S}_0$ are invariant, there exists $\alpha \in (0,1]$ such that $\mu =(1-\alpha )\nu _0+\alpha \nu _1$ where $\nu _0$ is an invariant probability measure with $\nu _0( \mathcal {S}_0)=1$ and $\nu _1$ is an invariant probability measure with $\nu _1(\mathcal {S}_0)=0$. By Proposition 1, $\sum _ip^ir^i(\nu _1) = 0$. Thus, $(1-\alpha ) \sum _ip^ir^i(\nu _0) \le 0$. Since by assumption $\sum _ip^ir^i(\nu _0) > 0$, $\alpha$ must be 1.

$\square$

It follows that with probability one, the weak* limit points $\mu$ of $\varPi _t$ (given $Z_0=z\in \mathcal {S}{\setminus }\mathcal {S}_0$) are invariant probability measures satisfying $\mu (\mathcal {S}_0)=0$. To complete, the proof of almost-sure persistence, we need to provide uniform upperbounds to the amount of weight that invariant probability measures $\mu$ with $\mu (\mathcal {S}_0)=0$ place near $\mathcal {S}_0$. To this end, let $\mu$ be an invariant probability measure with $\mu (\mathcal {S}_0)=0$. As, in general, we can not assume that $\int V_\theta (z)\mu (dz)$ is well-defined and finite, we present a slightly longer argument than shown in the main text, to deal with this issue. This argument follows Hairer (2006, Proposition 4.24). Let $M>0$ be any positive real. For two real numbers a, b, let $a\wedge b$ denote $\min \{a,b\}.$ Then invariance, Jensen’s inequality and Proposition 2 imply

$$\begin{aligned} \begin{aligned} \int (V_\theta \wedge M)(z) \mu (dz)=&\int (P_{T}(V_\theta \wedge M))(z) \mu (dz) \\ =&\int \mathbb {E}_z[V_\theta (Z_T)\wedge M]\mu (dz) \le \int \mathbb {E}_z[V_\theta (Z_T)]\wedge M \mu (dz)\\ \le&\int (\rho V_\theta (z)+\beta )\wedge M \mu (dz). \end{aligned} \end{aligned}$$

Iterating this inequality $k\ge 1$ times yields

$$\begin{aligned} \int (V_\theta \wedge M)(z) \mu (dz)\le \int ((\rho )^k V_\theta (z)+\beta /(1-\rho ))\wedge M \mu (dz). \end{aligned}$$

(7.2)

By the dominated convergence theorem, taking the limit $k\rightarrow \infty$ yields

$$\begin{aligned} \int (V_\theta \wedge M)(z) \mu (dz)\le \frac{\beta }{1-\rho }. \end{aligned}$$

(7.3)

By the dominated convergence theorem, taking the limit $M\rightarrow \infty$ yields

$$\begin{aligned} \int V_\theta (z) \mu (dz)\le \frac{\beta }{1-\rho }. \end{aligned}$$

(7.4)

Thus, as in the main text, for any $\eta >0$,

$$\begin{aligned} \mu (\mathcal {S}_\eta )\min _{z\in \mathcal {S}_\eta }V_\theta (z)\le \int V_\theta (z) \mu (dz) \le \frac{\beta }{1-\rho }. \end{aligned}$$

Inequality (3.7) implies that

$$\begin{aligned} \mu (\mathcal {S}_\eta )\le a (\eta )^b \text{ for } \text{ all } \eta \le 1 \text{ where } a=\frac{\beta }{a_0(1-\rho )} \end{aligned}$$

which completes the proof of almost-sure persistence.

1.4 Proof of Theorem 2 and Corollary 1

The strategy of this proof is based on the proof of Theorem 3.3 by Benaïm and Lobry (2016). Define $V(x,y)=\sum _i p^i \log x^i$. As in the proof of Theorem 1, assumption (4.1) and Proposition 1 imply there exists $T\ge 1$, $\eta >0$ and $\alpha >0$ such that

$$\begin{aligned} P_T V(z)-V(z)\le -\alpha \quad \text{ for } \text{ all } z\in \mathcal {S}_\eta . \end{aligned}$$

(7.5)

Moreover, using the same argument as found in Proposition 2, there exists $\rho \in (0,1)$ and $\theta >0$ such that (choosing $\eta >0$ to be smaller if necessary)

$$\begin{aligned} P_T V_\theta (z)\le \rho V_\theta (z)\quad \text{ for } \text{ all } z\in \mathcal {S}_\eta \text{ where } V_\theta (z)=\exp (\theta V(z)). \end{aligned}$$

(7.6)

Define the stopping time $\tau =\inf \{k: Z_{kT}\notin \mathcal {S}_\eta \}$ and the event $\mathcal {A}=\{\limsup _{t\rightarrow \infty } V(Z_t)/t\le -\alpha \}$. We will show that there exists a function $q(\varepsilon )\in (0,\eta )$ such that $\mathbb {P}_z(\mathcal {A})\ge q(\varepsilon )$ for $z\in \mathcal {S}_\varepsilon$ and $q(\varepsilon )\uparrow 1$ as $\varepsilon \downarrow 0.$ To this end, define $W_k=V_\theta (Z_{kT})$. Equation(7.6) implies that $W_{k\wedge \tau }$ is a super martingale. Hence, if we define $C(\varepsilon )=\sup _{z\in \mathcal {S}_\varepsilon } \exp (V(z))$ for any $\varepsilon >0$ (note that $C(\varepsilon )\downarrow 0$ as $\varepsilon \downarrow 0$), then

$$\begin{aligned} \mathbb {E}\left[ W_{k\wedge \tau }\mathbf {1}_{\tau <\infty }\right] \le W_0 \le C(\varepsilon )^\theta \quad \text{ whenever } Z_0=z\in \mathcal {S}_\varepsilon . \end{aligned}$$

Taking the limit as $k\rightarrow \infty$ and defining $D=\min _{z\in \mathcal {S}{\setminus } \mathcal {S}_\eta } \exp (V(z))>0$, the dominated convergence theorem implies that

$$\begin{aligned} \mathbb {P}_z[\tau<\infty ]D^\theta \le \mathbb {E}_z[W_\tau \mathbf {1}_{\tau <\infty }]\le C(\varepsilon )^\theta \quad \text{ whenever } z\in \mathcal {S}_\varepsilon . \end{aligned}$$

Thus,

$$\begin{aligned} \mathbb {P}_z[\tau =\infty ]\ge 1-\left( \frac{C(\varepsilon )}{D} \right) ^\theta =:q(\varepsilon )\quad \text{ whenever } z\in \mathcal {S}_\varepsilon \end{aligned}$$

and where $q(\varepsilon )\uparrow 1$ as $\varepsilon \downarrow 0.$

Next, consider the martingale, $M_n=\sum _{\ell =1}^n V(\ell T)-P_TV((\ell -1)T)$. By the strong law for martingales and inequality (7.5), $\limsup _{n\rightarrow \infty } V(nT)/n\le -\alpha$ on the event $\tau =\infty .$ As we have assumed that $|\log f^i|$ is uniformly bounded on $\mathcal {S}\times \varXi$, $\limsup _{t\rightarrow \infty }V(t)/t \le -\alpha$ on the event $\tau =\infty$. Thus, as claimed, we have shown that

$$\begin{aligned} \mathbb {P}_z[\mathcal {A}]\ge q(\varepsilon )\quad \text{ for } z\in \mathcal {S}_\varepsilon . \end{aligned}$$

As $V(Z_t)\downarrow -\infty$ implies that $\mathrm {dist}(Z_t,\mathcal {S}_0)\downarrow 0$, the proof of Theorem 2 is complete.

To prove Corollary 1, define the event

$$\begin{aligned} \mathcal {E}=\left\{ \lim _{t\rightarrow \infty } \mathrm {dist}(Z_t,\mathcal {S}_0)=0\right\} . \end{aligned}$$

Choose $\varepsilon >0$ such that $q(\varepsilon )>1/2.$ Define the stopping time

$$\begin{aligned} {{\widetilde{\tau }}} = \inf \{t\ge 0 \ : Z_t \in \mathcal {S}_\epsilon \}. \end{aligned}$$

Since $\mathcal {S}_0$ is accessible, there exists $\gamma >0$ such that $\mathbb {P}_{z}[{{\widetilde{\tau }}}<\infty ]>\gamma$ for all $z\in \mathcal {S}$. The strong Markov property implies that for all $z\in \mathcal {S}$

$$\begin{aligned} \mathbb {P}_z\left[ \mathcal {E}\right]= & {} \mathbb {E}_{z}\left[ \mathbb {P}_{Z_{{\widetilde{\tau }}}} \left[ \mathcal {E} \right] \mathbf {1}_{\{ {{\widetilde{\tau }}}<\infty \}} \right] +\mathbb {E}_{z}\left[ \mathbb {P}_{Z_{{\widetilde{\tau }}}} \left[ \mathcal {E} \right] \mathbf {1}_{\{ {{\widetilde{\tau }}}=\infty \}} \right] \\= & {} \mathbb {E}_{z}\left[ \mathbb {P}_{Z_{{\widetilde{\tau }}}} \left[ \mathcal {E} \right] \mathbf {1}_{\{ {{\widetilde{\tau }}}<\infty \}} \right] \ge \gamma /2. \end{aligned}$$

Let $\mathcal {F}_t$ be the $\sigma$-algebra generated by $\{Z_1,\dots ,Z_t\}$. The Lévy zero-one law implies that for all $z\in \mathcal {S}$, $\lim _{t\rightarrow \infty } \mathbb {E}_{z}\left[ \mathbf {1}_{\mathcal {E}} | \mathcal {F}_t\right] = \mathbf {1}_{\mathcal {E}}$ almost surely. On the other hand, the Markov property implies that $\mathbb {E}_{z}\left[ \mathbf {1}_{\mathcal {E}} | \mathcal {F}_t\right] =\mathbb {E}_z[\mathbb {P}_{Z_t}[{{\mathcal {E}}}]]\ge \gamma /2$ for all $z\in \mathcal {S}$. Hence $\mathbb {P}_{z}[\mathcal {E}]=1$ for all $z\in \mathcal {S}$.

1.5 Proof of Theorem 3

This proof follows the strategy of Theorem 2, but using a V function introduced by Hening and Nguyen (2018a).

Let $I\subset \{1,2,\dots ,n\}$ be the subset of species and $\{p^i\}_{i\in I}$ the set of positive reals such that $\sum _{i\in I} p^i r^i(\mu )>0$ for every ergodic $\mu$ supported on $\mathcal {S}^I_0=\{z=(x,y)\in \mathcal {S}^I: \prod _{i\in I}x^i=0\}$ where $\mathcal {S}^{I}=\{z=(x,y)\in \mathcal {S}: x^i=0$ for all $i\notin I\}.$ Assume that $r^i(\mu )<0$ for all $i\notin I$ and ergodic $\mu$ such that $\mu (\mathcal {S}^I_+)=1$ where $\mathcal {S}^I_+=\mathcal {S}^I{\setminus } \mathcal {S}^I_0.$ Choose $\delta >0$ and $\alpha >0$ such that $-\sum _{i\in I} p^i r^i(\mu )+\delta \max _{i\notin I} r^i(\mu )\le -2\alpha$ for all ergodic probability measures $\mu$ such that $\mu (\mathcal {S}^I)=1.$

Define $V(x,y)=-\sum _{i\in I} p^i \log x^i+\delta \max _{i\notin I}\log x^i$ and

$$\begin{aligned} V_\theta (x,y)=e^{\theta V(x,y)}=\left( \prod _{i\in I}(x^i)^{-\theta p^i}\right) \max _{i\notin I}(x^i)^{\theta \delta }. \end{aligned}$$

As in the proof of Theorem 1 and Proposition 1, there exists $T\ge 1$, $\eta \in (0,1]$ such that

$$\begin{aligned} P_T V(z)-V(z)\le -\alpha \text{ for } \text{ all } z\in K_\eta \end{aligned}$$

(7.7)

where $K_\eta =\{(x,y)\in \mathcal {S}: \max _{i\notin I} x^i \le \eta \}.$ Moreover, using the same argument as found in Proposition 2, there exists $\rho \in (0,1)$ and $\theta >0$ such that (choosing $\eta >0$ to be smaller if necessary)

$$\begin{aligned} P_T V_\theta (z)\le \rho V_\theta (z) \text{ for } \text{ all } z\in K_\eta {\setminus }\mathcal {S}^I_0. \end{aligned}$$

(7.8)

Choose ${\widetilde{\eta }}>0$ such that $\{z: V_\theta (z)\le {\widetilde{\eta }}\}\subset K_\eta .$ Define the stopping time $\tau =\{k: V_\theta (Z_{kT})\ge {\widetilde{\eta }}\}$ and the event $\mathcal {A}=\{\limsup _{t\rightarrow \infty } V(Z_t)/t\le -\alpha \}$. Define $W_k=V_\theta (Z_{kT})$ and ${\widetilde{W}}_k = {\widetilde{\eta }} \wedge W_k$. Equation (7.8) implies that $W_{k\wedge \tau }$ is a super martingale. Thus, for any k, concavity of $t\mapsto \delta \wedge t$ and Jensen’s inequality implies that

$$\begin{aligned} \begin{aligned} \mathbb {E}\left[ {\widetilde{W}}_{k\wedge \tau }\mathbf {1}_{\tau<\infty }\right] \le&{\widetilde{\eta }}\wedge \mathbb {E}\left[ W_{k\wedge \tau }\mathbf {1}_{\tau <\infty }\right] \\ \le&{\widetilde{\eta }} \wedge W_0 = {\widetilde{\eta }}\wedge \frac{\max _{i\notin I}(x^i)^{\theta \delta }}{\prod _{i\in I}(x^i)^{\theta p^i}} =:C(z)\\&\text{ whenever } Z_0=z=(x,y)\in K_\eta {\setminus }\mathcal {S}^I_0. \end{aligned} \end{aligned}$$

Taking the limit as $k\rightarrow \infty$, the dominated convergence theorem implies that

$$\begin{aligned} \mathbb {P}_z[\tau<\infty ]{\widetilde{\eta }} \le \mathbb {E}_z[{\widetilde{W}}_\tau \mathbf {1}_{\tau <\infty }]\le C(z)\quad \text{ for } \text{ all } z\in K_\eta {\setminus }\mathcal {S}^I_0. \end{aligned}$$

Thus,

$$\begin{aligned} \mathbb {P}_z[\tau =\infty ]\ge 1-\frac{C(z)}{{\widetilde{\eta }}}\quad \text{ for } \text{ all } z\in K_\eta {\setminus }\mathcal {S}^I_0. \end{aligned}$$

Next, consider the martingale, $M_n=\sum _{\ell =1}^n V(\ell T)-P_TV((\ell -1)T)$. By the strong law for martingales and (7.7), $\limsup _{n\rightarrow \infty } V(nT)/n\le -\alpha$ on the event $\tau =\infty .$ As we have assumed that $|\log f^i|$ is uniformly bounded on $\mathcal {S}\times \varXi$, $\limsup _{t\rightarrow \infty }V(t)/t \le -\alpha$ on the event $\tau =\infty$. Thus, we have shown that

$$\begin{aligned} \mathbb {P}_z[\mathcal {A}]\ge 1-\frac{C(z)}{{\widetilde{\eta }}}\quad \text{ for } \text{ all } z\in K_\eta {\setminus }\mathcal {S}^I_0. \end{aligned}$$

Let $M=\sup _{(x,y)\in \mathcal {S}} \max _i x^i>0.$ As $\limsup _{t\rightarrow \infty } -\frac{\log X^i_t}{t}\ge \limsup _{t\rightarrow \infty } -\frac{\log M}{t} =0$ with probability one whenever $X_0^i>0$,

$$\begin{aligned} \begin{aligned} -\alpha \ge \limsup _{t\rightarrow \infty }\frac{V(Z_t)}{t}=&\limsup _{t\rightarrow \infty }\frac{1}{t}\left( -\sum _{i\in I} p^i \log X_t^i + \delta \max _{i\notin I}\log X_t^i \right) \\ \ge&\limsup _{t\rightarrow \infty }\frac{\delta }{t} \max _{i\notin I}\log X_t^i \end{aligned} \end{aligned}$$

almost surely on the event $\mathcal {A}$ whenever $Z_0=z\in K_\eta {\setminus } \mathcal {S}_0^I$. Hence, on this event, $\lim$$\mathrm {dist}(Z_t,\mathcal {S}_0^I)\downarrow 0$ and the proof of Theorem 3 is complete.