Persistence in Stochastic Lotka–Volterra Food Chains with Intraspecific Competition

Abstract

This paper is devoted to the analysis of a simple Lotka–Volterra food chain evolving in a stochastic environment. It can be seen as the companion paper of Hening and Nguyen (J Math Biol 76:697–754, 2018b) where we have characterized the persistence and extinction of such a food chain under the assumption that there is no intraspecific competition among predators. In the current paper, we focus on the case when all the species experience intracompetition. The food chain we analyze consists of one prey and $n-1$ predators. The jth predator eats the $j-1$st species and is eaten by the $j+1$st predator; this way each species only interacts with at most two other species—the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on the invasion rates of the predators (which we can determine from the interaction coefficients of the system via an algorithm), which species go extinct and which converge to their unique invariant probability measure. We obtain stronger results than in the case with no intraspecific competition because in this setting we can make use of the general results of Hening and Nguyen (Ann Appl Probab 28:1893–1942, 2018a). Unlike most of the results available in the literature, we provide an in-depth analysis for both non-degenerate and degenerate noise. We exhibit our general results by analyzing trophic cascades in a plant–herbivore–predator system and providing persistence/extinction criteria for food chains of length $n\le 4$.

Appendix A: Proofs

The following result tells us that there is no ergodic invariant probability measure $\mu$ that has a gap in the chain of predators.

Lemma A.1

Suppose $\mu \in {\mathcal {M}}$ such that $I_\mu = \{n_1,\ldots ,n_k\}$. Then $I_\mu$ must be of the form $\{1,2,\ldots ,l\}$ for some $l\ge 1$.

Proof

We argue by contradiction. First, suppose that $n_1>1$. By (11)

$$\begin{aligned} \begin{aligned} \lambda _{n_1}(\mu )= 0&=-{\tilde{a}}_{n_1,0} + a_{n_1,n_1-1}\int _{ {\mathbb {R}}_+^n} x_{n_1-1}\mathrm{d}\mu - a_{n_1,n_1}\int _{ {\mathbb {R}}_+^n} x_{n_1}\mathrm{d}\mu \\&= -{\tilde{a}}_{n_1,0} - a_{n_1,n_1}\int _{ {\mathbb {R}}_+^n} x_{n_1}\mathrm{d}\mu \\&<0 \end{aligned} \end{aligned}$$

which is a contradiction.

Alternatively, suppose that there exists $\mu \in {\mathcal {M}}$ such that $I_\mu = \{1,\ldots , u^*, v^*,\ldots ,$ $n_k\}$ with $1\le u^*<v^*-1\le n_k\le n$. As a result one can see that $v^*-1\notin I^\mu$. Then by (11)

$$\begin{aligned} \begin{aligned} \lambda _{v^*}(\mu )= 0&=-{\tilde{a}}_{n_1,0} + a_{v^*,v^*-1}\int _{ {\mathbb {R}}_+^n} x_{v^*-1}\mathrm{d}\mu - a_{v^*,v^*}\int _{ {\mathbb {R}}_+^n} x_{v^*}\mathrm{d}\mu - a_{v^*,v^*+1}\\&\qquad \times \int _{ {\mathbb {R}}_+^n} x_{v^*+1}\mathrm{d}\mu \\&= -{\tilde{a}}_{v^*,0} - a_{v^*,v^*}\int _{ {\mathbb {R}}_+^n} x_{v^*}\mathrm{d}\mu - a_{v^*,v^*+1}\int _{ {\mathbb {R}}_+^n} x_{v^*+1}\mathrm{d}\mu \ \\&<0 \end{aligned} \end{aligned}$$

which is a contradiction. $\square$

For $i=1,\ldots ,n$, denote by ${\mathcal {M}}_i$ the set of all invariant probability measures $\mu$ of ${\mathbf {X}}$ satisfying $\mu ({\mathbb {R}}^{(i),\circ }_+)=1$. For $i=0$, define ${\mathcal {M}}_0=\{\varvec{\delta }^*\}$. By Lemma A.1, we have ${{\mathrm{Conv}}}({\mathcal {M}})={{\mathrm{Conv}}}(\cup _{i=0}^{n-1}{\mathcal {M}}_i)$ and ${{\mathrm{Conv}}}(\cup _{i=0}^{n}{\mathcal {M}}_i)$ is the set of all invariant probability measures of ${\mathbf {X}}$ on ${\mathbb {R}}^n_+$.

Lemma A.2

We have the following claims.

• If ${\mathcal {I}}_k\le 0$ then ${\mathcal {I}}_{k+1}<0$.

• If ${\mathcal {I}}_n\le 0$, there ${\mathbf {X}}$ has no invariant probability measure on ${\mathbb {R}}^{n,\circ }_+$.

Proof

If ${\mathcal {I}}_{k+1}= -{\tilde{a}}_{k+1,0} + a_{k+1,j} x^{(k)}_k\ge 0$, then $x^{(k)}_k>0$. We will show in Sect. 4 that $x^{(k)}_k$ has the same sign as ${\mathcal {I}}_k$. Thus, if ${\mathcal {I}}_{k+1}\ge 0$ then ${\mathcal {I}}_k>0$, which proves the first claim.

If ${\mathbf {X}}$ has an invariant probability measure $\mu$ on ${\mathbb {R}}^{n,\circ }_+$, then we must have $\int _{{\mathbb {R}}^n_+}x_n\mu (d{\mathbf {x}})=x^{(n)}_n$. As a result $x^{(n)}_n>0$, which leads to ${\mathcal {I}}_n>0$ since they have the same sign. The second claim is therefore proved. $\square$

Lemma A.3

We have the following claims.

• (1) For any initial condition ${\mathbf {X}}(0)={\mathbf {x}}\in {\mathbb {R}}^n_+$, the family $\{\widetilde{\Pi }_t(\cdot ), t\ge 1\}$ is tight in ${\mathbb {R}}^n_+$, and its $\hbox {weak}^*$-limit set, denoted by ${\mathcal {U}}={\mathcal {U}}(\omega )$ is a family of invariant probability measures of ${\mathbf {X}}$ with probability 1.

• (2) Suppose that there is a sequence $(T_k)_{k\in {\mathbb {N}}}$ such that $\lim _{k\rightarrow \infty }T_k=\infty$ and $(\widetilde{\Pi }_{T_k}(\cdot ))_{k\in {\mathbb {N}}}$ converge weakly to an invariant probability measure $\pi$ of ${\mathbf {X}}$ when $k\rightarrow \infty$ . Then for this sample path, we have $\int _{{\mathbb {R}}^n_+}h({\mathbf {x}})\widetilde{\Pi }_{T_k}(\mathrm{d}{\mathbf {x}})\rightarrow \int _{{\mathbb {R}}^n_+}h({\mathbf {x}})\pi (\mathrm{d}{\mathbf {x}})$ for any continuous function $h:{\mathbb {R}}^n_+\rightarrow {\mathbb {R}}$ satisfying $|h({\mathbf {x}})|<K_h(1+\Vert {\mathbf {x}}\Vert )\,,\,{\mathbf {x}}\in {\mathbb {R}}^n_+$, with $K_h$ a positive constant and $\delta \in [0,\delta _1)$.

• (3) For any ${\mathbf {x}}\in {\mathbb {R}}^{n,\circ }_+$

  $$\begin{aligned} {\mathbb {P}}_{\mathbf {x}}\left\{ \lim _{t\rightarrow \infty }\left( \dfrac{\ln X_i(t)}{t}-\lambda _i\left( {\tilde{\Pi }}_t\right) \right) =0,\, i=1,\ldots ,n\right\} =1 \end{aligned}$$

  (28)

  and

  $$\begin{aligned} {\mathbb {P}}_{\mathbf {x}}\left\{ \limsup _{t\rightarrow \infty } \dfrac{\ln X_i(t)}{t}\le 0, i=1,\ldots ,\right\} n=1. \end{aligned}$$

  (29)

Proof

Let ${\tilde{c}}_1=1, {\tilde{c}}_i:=\prod _{j=2}^i\dfrac{a_{k-1,k}}{2a_{k,k-1}}=c_{i-1}\dfrac{a_{i-1,i-1}}{2a_{i,i-1}}, i\ge 2.$ Put

$$\begin{aligned} {\tilde{\gamma }}=\min _{i=1,\ldots ,n}\left\{ {c_i}\frac{a_{ii}}{2}\right\} \end{aligned}$$

we can easily verify that

$$\begin{aligned} \sum _{i=1}^n{\tilde{c}}_i f_i({\mathbf {x}})\le {\tilde{C}} -{\tilde{\gamma }}\sum _{i=1}^n x_i\,\text { for some positive constant }\, {\tilde{C}}. \end{aligned}$$

Thus, when $\Vert x\Vert$ is sufficiently large, $|\sum _{i=1}^n{\tilde{c}}_i f_i({\mathbf {x}})|\ge {\tilde{\gamma }}\sum _{i=1}^n x_i$, which implies

$$\begin{aligned} \liminf _{\Vert x\Vert \rightarrow \infty } \dfrac{\sum _{i=1}^n{\tilde{c}}_i |f_i({\mathbf {x}})|}{\sum _{i=1}^n x_i} \ge \liminf _{\Vert x\Vert \rightarrow \infty } \dfrac{\left| \sum _{i=1}^n{\tilde{c}}_i f_i({\mathbf {x}})\right| }{\sum _{i=1}^n x_i}\ge {\tilde{\gamma }}. \end{aligned}$$

As a result,

$$\begin{aligned} \liminf _{\Vert x\Vert \rightarrow \infty } \dfrac{\Vert {\mathbf {x}}\Vert ^\delta }{\sum _{i=1}^n |f_i({\mathbf {x}})|}=0\quad \text { for any }\,\delta \in (0,1). \end{aligned}$$

In other words, Assumption 1.4 of Hening and Nguyen (2018a) is satisfied by our model. Thus, the first and second claims of this lemma follow from (Hening and Nguyen 2018a, Lemmas 4.6, 4.7). By Itô’s formula and the definition of ${\tilde{\Pi }}_t$, we have

$$\begin{aligned} \left( \dfrac{\ln X_i(t)}{t}-\lambda _i\left( {\tilde{\Pi }}_t\right) \right) = \dfrac{\ln X_i(0)}{t}+\frac{E_i(t)}{ t}. \end{aligned}$$

By the strong law of large numbers for martingales,

$$\begin{aligned} \lim _{t\rightarrow \infty }\dfrac{\ln X_i(0)}{t}+\frac{E_i(t)}{ t}=0 \,\text { a.s.} \end{aligned}$$

which leads to (28).

(29) can be derived by using Eq. (4.22) of Hening and Nguyen (2018a) or by mimicking the proof of (Du and Sam 2006, Theorem 2.4). $\square$

Proof of Theorem 1.1 (i)

Since ${\mathcal {I}}_n>0$, it follows from Lemma A.2 that ${\mathcal {I}}_k>0$ for any $k=1,\ldots ,n$. By Lemma A.1, for any $\mu \in {{\mathrm{Conv}}}({\mathcal {M}})={{\mathrm{Conv}}}(\cup _{i=0}^{n-1}{\mathcal {M}}_i)$, we can decompose $\mu =\rho _1\mu _{i_1}+\cdots +\rho _k\mu _{i_k}$ where $0\le i_1<\cdots <i_k\le n-1$ and $\mu _{i_j}\in {\mathcal {M}}_{i_j}$, $\rho _j> 0$ for $j=1,\ldots ,k$ and $\sum \rho _j=1$. Since $i_1< i_j$ for $j=2,\ldots ,k$, we deduce from (11) that $\lambda _{i_1+1}(\mu _{i_j})=0$ for $j=2,\ldots ,k$. On the other hand, (4) and (12) imply

$$\begin{aligned} \lambda _{i_1+1}(\mu _{i_1})= -{\tilde{a}}_{i_1+1,0} + a_{i_1+1,i_1} x^{(i_1)}_{i_1}={\mathcal {I}}_{i_1+1}>0. \end{aligned}$$

As a result,

$$\begin{aligned} \lambda _{i_1+1}(\mu )=\rho _1\lambda _{i_1+1}(\mu _{i_1})>0. \end{aligned}$$

Thus,

$$\begin{aligned} \max _{i=1,\ldots ,n}\lambda _i(\mu )>0,\quad \text { for any }\,\mu \in {{\mathrm{Conv}}}({\mathcal {M}}). \end{aligned}$$

(30)

In other words, Assumption 2.1 is satisfied. By Theorem 3.1 of Hening and Nguyen (2018a), there exist positive $p_1,\ldots , p_n, T$ and constants $\theta , \kappa \in (0,1)$ such that

$$\begin{aligned} {\mathbb {E}}_{{\mathbf {x}}} V^\theta (X( T))\le \kappa V^\theta (x)+ K \end{aligned}$$

(31)

where

$$\begin{aligned} V({\mathbf {x}}):=\dfrac{1+{\mathbf {c}}^\top {\mathbf {x}}}{\Pi _{i=1}^nx_i^{p_i}}\,\text { for }\, {\mathbf {x}}\in {\mathbb {R}}^{n,\circ }_+,\,\text { with } {\mathbf {c}}\,\text { defined in }\, (2.3),\,\text { and }\, \sum _{i=1}^np_i<1. \end{aligned}$$

Equation (31) and the Markov property of ${\mathbf {X}}$ lead to

$$\begin{aligned} {\mathbb {E}}_{{\mathbf {x}}} V^\theta (X(m T))\le \kappa ^m V^\theta (x)+K\sum _{j=1}^{m-1}\kappa ^j. \end{aligned}$$

Thus,

$$\begin{aligned} \limsup _{m\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}} V^\theta ({\mathbf {X}}(m T))\le \dfrac{ K}{1-\kappa },\quad {\mathbf {x}}\in {\mathbb {R}}^{n,\circ }_+. \end{aligned}$$

(32)

By (Hening and Nguyen 2018a, Lemma 2.1), there exists ${\widehat{K}}>0$ such that

$$\begin{aligned} {\mathbb {E}}_{{\mathbf {x}}} V^\theta ({\mathbf {X}}(t))\le \exp ({\widehat{K}}t)V^\theta ({\mathbf {x}}),\quad {\mathbf {x}}\in {\mathbb {R}}^{n,\circ }_+, \end{aligned}$$

which together with the Markov property implies

$$\begin{aligned} {\mathbb {E}}_{{\mathbf {x}}} V^\theta ({\mathbf {X}}(t))\le \exp ({\widehat{K}} T){\mathbb {E}}_{\mathbf {x}}V^\theta ({\mathbf {X}}(m T)) \text { for } t\in [m T,(m+1) T]. \end{aligned}$$

(33)

In view of (32) and (33), we have

$$\begin{aligned} \limsup _{t\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}} V^\theta ({\mathbf {X}}(t))\le \exp ({\widehat{K}} T)\dfrac{ K}{1-\kappa }. \end{aligned}$$

For any fixed $\varepsilon >0$, define $K:=\left\{ {\mathbf {x}}\in {\mathbb {R}}^{n,\circ }_+: V^\theta ({\mathbf {x}})\le \dfrac{1}{\varepsilon }\exp ({\widehat{K}} T)\dfrac{ K}{1-\kappa }\right\}$ then K is a compact subset of ${\mathbb {R}}^{n,\circ }_+$. The definition of K together with the last inequality yields

$$\begin{aligned} \limsup _{t\rightarrow \infty }{\mathbb {P}}_{{\mathbf {x}}} \{{\mathbf {X}}(t)\notin K\}\le \left( \varepsilon \exp (-\,{\widehat{K}} T)\dfrac{1-\kappa }{ K}\right) \limsup _{t\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}} V^\theta ({\mathbf {X}}(t))\le \varepsilon . \end{aligned}$$

(34)

The stochastic persistence in probability is therefore proved.

To prove (5), we need to show that for any initial value ${\mathbf {x}}\in {\mathbb {R}}^{n,\circ }_+$, the weak-limit points of ${\tilde{\Pi }}_{t}$ are a subset of ${\mathcal {M}}_n$ with probability 1.

Suppose the claim is false. Then, by part (i) of Lemma A.3, we can find ${\mathbf {x}}\in {\mathbb {R}}^{n,\circ }_+$ and ${\tilde{\Omega }}_{\mathbf {x}}\subset \Omega$ with ${\mathbb {P}}_{\mathbf {x}}({\tilde{\Omega }}_{\mathbf {x}})>0$ and such that for $\omega \in {\tilde{\Omega }}_{\mathbf {x}}$, there exists $t_k=t_k(\omega )$ satisfying that $\lim _{k\rightarrow \infty }t_k=\infty$ and ${\tilde{\Pi }}_{t_k}(\omega )$ converge weakly to $\mu (\omega )=\rho _1\mu _1+\rho _2\mu _2$ where $\mu _1\in {{\mathrm{Conv}}}({\mathcal {M}})$ and $\mu _2\in {\mathcal {M}}_n$ and $\rho _1>0$. By Lemma A.2, $\lambda _n(\mu _1)>0$. In view of (11), $\lambda _n(\mu _2)=0$. Thus, for almost all $\omega \in {\tilde{\Omega }}_{\mathbf {x}}$, we have from part (ii) of Lemma A.3 that

$$\begin{aligned} \lim _{k\rightarrow \infty }\dfrac{\ln X_n(t_k)}{t_k} =\lim _{k\rightarrow \infty }\lambda _n\left( {\tilde{\Pi }}_{t_k}\right) =\lambda _n(\mu )>0, \end{aligned}$$

which contradicts (29). Thus, with probability 1, the weak-limit points of ${\tilde{\Pi }}_{t}$ as $t\rightarrow \infty$ must be contained in ${\mathcal {M}}_n$. Then, (5) follows from (12).

When $\Sigma$ is positive definite, it follows from (Hening and Nguyen 2018a, Theorem 3.1) that the food chain ${\mathbf {X}}$ is strongly stochastically persistent and its transition probability converges to its unique invariant probability measure $\pi ^{(n)}$ on ${\mathbb {R}}_+^{n,\circ }$ exponentially fast in total variation. $\square$

Proof of Theorem 1.1 (ii)

We suppose there exists $j^*<n$ such that ${\mathcal {I}}_{j^*}>0$ and ${\mathcal {I}}_{j^*+1}<0$. By Lemma A.2 part (ii), there are no invariant probability measures on ${\mathbb {R}}^{(j),\circ }_+$ for $j=j^*+1,\ldots ,n$. Using Lemma A.1, we see that the set of invariant probability measures on ${\mathbb {R}}^n_+$ of ${\mathbf {X}}$ is ${{\mathrm{Conv}}}(\cup _{i=0}^{j^*}{\mathcal {M}}_i)$.

Note that $\lambda _{j^*+1}(\mu )=-{\tilde{a}}_{j^*+1}<0$ if $\mu \in {\mathcal {M}}_i$ for $i<j^*$ and $\lambda _{j^*+1}(\mu )={\mathcal {I}}_{j^*+1}<0$ if $\mu \in {\mathcal {M}}_{j^*}$. As a result, $\lambda _{j^*+1}(\mu )<0$ for any $\mu \in {{\mathrm{Conv}}}(\cup _{i=0}^{j^*}{\mathcal {M}}_i)$. Similarly, $\lambda _{j}(\mu )<0$ for any $j>j^*+1$ and $\mu \in {{\mathrm{Conv}}}(\cup _{i=0}^{j^*}{\mathcal {M}}_i)$. By (28) we have that

$$\begin{aligned} \lim _{t\rightarrow \infty }X_j(t)=0, j=j^*+1,\ldots ,n \,{\mathbb {P}}_{\mathbf {x}}-\text {a.s.} \end{aligned}$$

Since

$$\begin{aligned} \int _{{\mathbb {R}}^n_+}x_i'\mu (\mathrm{d}{\mathbf {x}}')= {\left\{ \begin{array}{ll} x^{(j^*)}_i &{}\quad \text { if}\; i=1,\ldots ,j^*,\\ 0 &{}\quad \text {if}\, i=j^*+1,\ldots , n. \end{array}\right. } \quad \text {for }\; \mu \in {\mathcal {M}}_{j^*}, \end{aligned}$$

(35)

we have

$$\begin{aligned} \lambda _{i}(\mu )= {\left\{ \begin{array}{ll} {\mathcal {I}}_{j^*+1}&{}\quad \text { if}\; i=j^*+1\\ -\,{\tilde{a}}_{i0}&{}\quad \text {if}\; i>j^*+1. \end{array}\right. } \quad \text {for}\;\mu \in {\mathcal {M}}_{j^*}. \end{aligned}$$

(36)

Using (29) and a contradiction argument similar to that in the proof of part (i), we can show that with probability 1, the weak-limit points of ${\tilde{\Pi }}_{t}$ as $t\rightarrow \infty$ must be contained in ${\mathcal {M}}_{j^*}$. Thus, for ${\mathbf {x}}\in {\mathbb {R}}^{n,\circ }_+$, we have from (35), (36), and Lemma A.2 that

$$\begin{aligned} \lim _{t\rightarrow \infty }\dfrac{1}{t}\int _0^t X_i(s)\mathrm{d}s= {\left\{ \begin{array}{ll} x^{(j^*)}_i&{}\quad \text {if}\; i=1,\ldots ,j^*,\\ 0&{}\quad \text {if}\; i=j^*+1,\ldots , n \end{array}\right. }\quad {\mathbb {P}}_{\mathbf {x}}-\text {a.s}. \end{aligned}$$

and

$$\begin{aligned} \lim _{t\rightarrow \infty }\dfrac{\ln X_i(t)}{t}= {\left\{ \begin{array}{ll} {\mathcal {I}}_{j^*+1}&{}\quad \text {if}\; i=j^*+1\\ -\,{\tilde{a}}_{i0}&{}\quad \text {if}\; i>j^*+1. \end{array}\right. } \quad {\mathbb {P}}_{\mathbf {x}}-\text {a.s}. \end{aligned}$$

To prove the persistence in probability of $(X_1,\ldots ,X_{j^*})$, we define

$$\begin{aligned} {\mathbb {R}}^{(j^*),\diamond }= & {} \left\{ {\mathbf {x}}=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n_+: x_j>0 \,\text { for }\, j=1,\ldots ,j^*\right\} ,\,\text { and }\, \partial {\mathbb {R}}^{(j^*),\diamond }\\= & {} {\mathbb {R}}^n_+{\setminus }{\mathbb {R}}^{(j^*),\diamond }. \end{aligned}$$

We have proved that ${{\mathrm{Conv}}}(\bigcup _{j=0}^{j^*}{\mathcal {M}}_j)$ is the set of invariant probability measures of ${\mathbf {X}}$ on ${\mathbb {R}}^n_+$. Note that ${{\mathrm{Conv}}}(\bigcup _{j=0}^{j^*-1}{\mathcal {M}}_j)$ is the set of invariant probability measures of ${\mathbf {X}}$ on $\partial {\mathbb {R}}^{(j^*),\diamond }$. Since ${\mathcal {I}}_{j^*}>0$, applying (30) with n replaced by $j^*$ we obtain

$$\begin{aligned} \max _{i=1,\ldots ,j^*}\lambda _i(\mu )>0,\quad \text {for any}\;\mu \in {{\mathrm{Conv}}}\left( \cup _{j=0}^{j^*-1}{\mathcal {M}}_j\right) . \end{aligned}$$

(37)

Using this condition, we can imitate the proofs in (Hening and Nguyen 2018a,Sect. 3) to construct a Lyapunov function $U({\mathbf {x}}):{\mathbb {R}}^{(j^*),\diamond }_+\mapsto {\mathbb {R}}_+$ of the form

$$\begin{aligned} U({\mathbf {x}})=\dfrac{1+{\mathbf {c}}^\top {\mathbf {x}}}{\Pi _{i=1}^{j^*}x_i^{{\tilde{p}}_i}},\quad {\tilde{p}}_i>0, i=1,\ldots ,j^* \end{aligned}$$

satisfying

$$\begin{aligned} {\mathbb {E}}_{{\mathbf {x}}} U^{{\tilde{\theta }}}(X( T))\le {\tilde{\kappa }} U^{{\tilde{\theta }}}(x)+{\tilde{K}},\quad \text {for}\;{\mathbf {x}}\in {\mathbb {R}}^{(j^*),\diamond }_+ \end{aligned}$$

(38)

and

$$\begin{aligned} {\mathbb {E}}_{{\mathbf {x}}} U^{{\tilde{\theta }}}(X( t))\le \exp ({\overline{K}} t) U^{{\tilde{\theta }}}(x)\quad \text {for}\;{\mathbf {x}}\in {\mathbb {R}}^{(j^*),\diamond }_+ , \end{aligned}$$

(39)

where ${\tilde{p}}_i>0$ for $i=1,\ldots ,j^*$, $\sum _{i=1}^{j^*}{\tilde{p}}_i<1$, ${\tilde{\theta }}, {\tilde{\kappa }}$ are some constants in (0, 1), and ${\tilde{T}}, {\tilde{K}}, {\overline{K}}$ are positive constants. Using (38) and (39), we can obtain the persistence in probability of $(X_1,\ldots ,X_{j^*})$ in the same manner as (34). The proof is complete. $\square$

Proof of Theorem 1.1 (iii)

Let $f:{\mathbb {R}}^n_+\mapsto {\mathbb {R}}$ be a continuous function and $\sup _{{\mathbf {x}}\in {\mathbb {R}}^n_+}|f({\mathbf {x}})|\le 1$. Fix ${\mathbf {x}}_0\in {\mathbb {R}}^{n,\circ }_+$. We have to show that

$$\begin{aligned} \lim _{t\rightarrow \infty }\left| \int _{{\mathbb {R}}^n_+} f({\mathbf {x}}')\pi _{j^*}(\mathrm{d}{\mathbf {x}}')-\int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(t, {\mathbf {x}}_0, \mathrm{d}{\mathbf {x}}')\right| =0. \end{aligned}$$

(40)

In part (ii), we have proved that $(X_1,\ldots , X_{j^*})$ is persistent in probability. Thus, for any $\varepsilon >0$, there exist $T_1>0$ and $H>1$ such that

$$\begin{aligned} {\mathbb {P}}_{{\mathbf {x}}_0}\left\{ H^{-1}\le X_j(t)\le H, j=1,\ldots , j^*\right\} >1-\varepsilon \quad \text {for any}\;t\ge T_1.\end{aligned}$$

(41)

For $\delta \ge 0$ define

$$\begin{aligned} K_\delta= & {} \{{\mathbf {x}}=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n_+: H^{-1}\le x_j\le H,\text { for } j=1,\ldots , j^*, x_j\le \delta ,\text { for }\\ j= & {} j^*+1,\ldots ,n\}. \end{aligned}$$

Let ${\overline{f}}=\int _{{\mathbb {R}}^n_+} f({\mathbf {x}}')\pi _{j^*}(\mathrm{d}{\mathbf {x}}')$. In view of (6), there exists $T_2>0$ such that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(T_2, {\mathbf {x}}, \mathrm{d}{\mathbf {x}}')-{\overline{f}}\right| <\varepsilon \quad \text {for any}\; {\mathbf {x}}\in K_0 \end{aligned}$$

(42)

Since ${\mathbf {X}}$ is a Markov–Feller process on ${\mathbb {R}}^n_+$, we can find a sufficiently small $\delta =\delta (\varepsilon )>0$ such that

$$\begin{aligned}&\left| \int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(T_2, {\mathbf {x}}_1, \mathrm{d}{\mathbf {x}}')-\int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(T_2, {\mathbf {x}}_2, \mathrm{d}{\mathbf {x}}')\right| <\varepsilon \nonumber \\&\qquad \text {given that}\; \Vert {\mathbf {x}}_1-{\mathbf {x}}_2\Vert \le \delta . \end{aligned}$$

(43)

Thus, (42) and (43) imply

$$\begin{aligned} \left| \int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(T_2, {\mathbf {x}}, \mathrm{d}{\mathbf {x}}')-{\overline{f}}\right| <2\varepsilon \quad \text {for any} \;{\mathbf {x}}\in K_\delta . \end{aligned}$$

(44)

Since $X_{j^*+1},\ldots , X_n$ converges to 0 almost surely, there exists $T_3>T_1$ such that

$$\begin{aligned} {\mathbb {P}}_{{\mathbf {x}}_0}\left\{ X_j(t)\le \delta , j=j^*+1,\ldots ,n\right\} >1-\varepsilon \quad \text {for any}\;t\ge T_3.\end{aligned}$$

(45)

We deduce from (41) and (45) that

$$\begin{aligned} P(t, {\mathbf {x}}_0,K_\delta )={\mathbb {P}}_{{\mathbf {x}}_0}\left\{ {\mathbf {X}}_j(t)\in K_\delta \right\} >1-2\varepsilon \quad \text {for any }\;t\ge T_3. \end{aligned}$$

(46)

For any $t\ge T_3+T_2$, we have from the Chapman–Kolmogorov equation, (44), (46) and $|f({\mathbf {x}})|\le 1$ that

$$\begin{aligned} \begin{aligned} \left| \int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(t, {\mathbf {x}}_0, \mathrm{d}{\mathbf {x}}')-{\overline{f}}\right|&= \left| \int _{{\mathbb {R}}^n_+}\left( \int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(T_2, {\mathbf {x}}, \mathrm{d}{\mathbf {x}}')-{\overline{f}}\right) P(t-T_2,{\mathbf {x}}_0,\mathrm{d}{\mathbf {x}})\right| \\&\le \left| \int _{K_\delta }\left( \int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(T_2, {\mathbf {x}}, \mathrm{d}{\mathbf {x}}')-{\overline{f}}\right) P(t-T_2,{\mathbf {x}}_0,\mathrm{d}{\mathbf {x}})\right| \\&\quad +\, \left| \int _{{\mathbb {R}}^n_+\setminus K_\delta }\left( \int _{{\mathbb {R}}^n_+}f({\mathbf {x}}')P(T_2, {\mathbf {x}}, \mathrm{d}{\mathbf {x}}')-{\overline{f}}\right) P(t-T_2,{\mathbf {x}}_0,\mathrm{d}{\mathbf {x}})\right| \\&\le 2\varepsilon (1-\varepsilon )+2(2\varepsilon )\le 6\varepsilon , \end{aligned} \end{aligned}$$

which leads to (A.13). The proof is complete. $\square$

Proof of Theorem 1.1 (iv)

If $\Sigma _{j^*}$ is positive definite, then by Theorem 1.1 part (i) for ${\mathbf {x}}\in {\mathbb {R}}^{(j^*),\circ }_+$ one has that as $t\rightarrow \infty$ the transition probability $P(t, x, \cdot )$ converges in total variation to a unique invariant probability measure $\pi _{j^*}$. Moreover, the convergence is uniform in each compact set of ${\mathbb {R}}^{(j^*),\circ }_+$ (due to the property of the Lyapunov function constructed in the proof). As a result (6) is satisfied and the conclusion follows by part (iii) of Theorem 1.1. $\square$