Asymptotic Properties of Multi-species Lotka–Volterra Models with Regime Switching Involving Weak and Strong Interactions

Abstract

This work focuses on multi-species Lotka–Volterra models with regime switching modulated by a continuous-time Markov chain involving a small parameter. The small parameter is used to reflect different rates of the switching among a large number of states representing the discrete events. Using perturbed Lyapunov function methods and the structure of the limit system as a bridge, stochastic permanence and extinction are obtained. Sufficient conditions under which the measures of the original system converge to the invariant measure of that of the limit system are provided. A couple of examples and numerical simulations are given to illustrate our results.

A Appendix: Proof of Theorem 3.1

To obtain the weak convergence, we first prove the tightness of $\{(x^{\varepsilon ,N}(\cdot ),{\bar{\gamma }}^\varepsilon (\cdot ))\}.$

Lemma A.1

$\{(x^{\varepsilon ,N}(\cdot ),{\bar{\gamma }}^\varepsilon (\cdot ))\}$ is tight in $D(\bar{\mathbb {R}}_+;{\mathbb {R}}^{r}_+\times {\bar{{\mathbb {S}}}})$.

Proof of Lemma A.1

Since $\{ {\bar{\gamma }}^\varepsilon (\cdot )\}$ is tight, it suffices to consider the tightness of $\{x^{\varepsilon ,N}(\cdot )\}$. Fix any $T>0.$ By (3.1) and the boundedness of $x^{\varepsilon ,N}(t)$, for any $\delta >0,$$0\le t\le T$, and $0<s\le \delta ,$ there is an ${{{\mathcal {F}}}}^\varepsilon _t$ measurable and bounded random variable ${\tilde{K}}(\omega )>0$ such that ${\mathbb {E}}_t^{\varepsilon }\Big |x^{\varepsilon ,N}(t+s)-x^{\varepsilon ,N}(t)\Big |^2 \le \tilde{K}(\omega ) \delta (\delta +1).$ Note that $\displaystyle \lim _{\delta \rightarrow 0}\overline{\lim _{\varepsilon \rightarrow 0}} {\mathbb {E}}{\tilde{K}}(\omega )\delta (\delta +1)=0.$ The desired assertion follows by virtue of the tightness criterion (see, e.g., Kushner 1984, Theorem 3, p. 47).

Proof of Theorem 3.1

In view of Lemma A.1, $\{(x^{\varepsilon ,N}(\cdot ),{\bar{\gamma }}^\varepsilon (\cdot ))\}$ is tight. By Prohorov’s theorem (see, e.g., Kushner 1984, Theorem 2, p. 28), we can extract a weakly convergent subsequence and denote the limit by $(x^{N}(\cdot ),{\bar{\gamma }}(\cdot ))$. For notational simplicity, still denote the subsequence by $\{(x^{\varepsilon ,N}(\cdot ),{\bar{\gamma }}^\varepsilon (\cdot ))\}$. Thus by Skorohod imbedding (see, e.g., Kushner 1984, p. 29, Theorem 3), we may assume that $(x^{\varepsilon ,N}(\cdot ),{\bar{\gamma }}^\varepsilon (\cdot ))$ converges to $(x^N(\cdot ),{\bar{\gamma }}(\cdot ))$ a.s. with a slight abuse of notation.

To prove $(x^N(\cdot ),{\bar{\gamma }}(\cdot ))$ is the solution to the martingale problem with operator ${\mathcal {L}}^{N}$ in (3.3), it suffices to show that for each $k\in \bar{\mathbb {S}}$, any $g(\cdot , k)\in C^2_0( {\mathbb {R}}^r,\bar{\mathbb {R}}),$ and any positive integer $\iota ,$$\phi _j(\cdot ,k)\in C_b({\mathbb {R}}^{r}_+;{\mathbb {R}})$ for $j\le \iota ,$ for any $t,s\ge 0,$ and any $0\le t_j\le t$ for $j\le \iota ,$

$$\begin{aligned}&{\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{N}(t_j),{\bar{\gamma }}(t_j))\Big (g( x^{N}(t+s),{\bar{\gamma }}(t+s))-g(x^{N}(t),{\bar{\gamma }}(t))\right. \Big .\nonumber \\&\left. \quad -\int _{t}^{t+s}{\mathcal {L}}_x^Ng(x^{N}(u),{\bar{\gamma }}(u))\hbox {d}u \Big )\right] =0. \end{aligned}$$

(A.1)

Since $(x^{\varepsilon ,N}(\cdot ),{\bar{\gamma }}^\varepsilon (\cdot ))$ solves the martingale problem with operator (3.2), we have

$$\begin{aligned}&{\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{\varepsilon ,N}(t_j),{\bar{\gamma }}^\varepsilon (t_j) )\Big (g( x^{\varepsilon ,N}(t+s),{\bar{\gamma }}^\varepsilon (t+s))-g(x^{\varepsilon ,N}(t),{\bar{\gamma }}^\varepsilon (t))\right. \Big .\nonumber \\&\quad \left. \Big .-\int _{t}^{t+s}{\mathcal {L}}_x^{\varepsilon ,N}g(x^{\varepsilon ,N}(u),{\bar{\gamma }}^\varepsilon (u))\hbox {d}u \Big )\right] =0. \end{aligned}$$

(A.2)

It follows from weak convergence and Skorohod imbedding that

$$\begin{aligned}&{\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{\varepsilon ,N}(t_j),{\bar{\gamma }}^\varepsilon (t_j) )\Big (g( x^{\varepsilon ,N}(t+s),{\bar{\gamma }}^\varepsilon (t+s))-g(x^{\varepsilon ,N}(t),{\bar{\gamma }}^\varepsilon (t)) \Big )\right] \nonumber \\&\quad \rightarrow {\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{N}(t_j),{\bar{\gamma }}(t_j) )\Big (g( x^{N}(t+s),{\bar{\gamma }}(t+s))-g(x^{N}(t),{\bar{\gamma }}(t))\Big )\right] . \end{aligned}$$

(A.3)

Next, we deal with the integration term of the operator ${\mathcal {L}}_x^{\varepsilon ,N}$ in (A.2). From the definitions of $\bar{b}_i(\cdot ),~{\bar{a}}_{ij}(\cdot ),$ and $\bar{\sigma }_{i\iota }(\cdot )\bar{\sigma }_{j\iota }(\cdot )$ for $1\le i,j\le r,$$1\le \iota \le d,$

$$\begin{aligned}&{b_i}(\gamma ^\varepsilon (u))-\bar{b}_i({\bar{\gamma }}^\varepsilon (u))=\sum _{k=1}^{l}\sum _{v=1}^{m_k} {b_i}(s_{kv})\Big [I_{\{\gamma ^\varepsilon (u)=s_{kv}\}}-\pi ^{k}_{v}I_{\{\gamma ^\varepsilon (u)\in {\mathbb {S}}_k\}}\Big ], \end{aligned}$$

(A.4)

$$\begin{aligned}&a_{ij}(\gamma ^\varepsilon (u))-{\bar{a}}_{ij}({\bar{\gamma }}^\varepsilon (u))=\sum _{k=1}^{l}\sum _{v=1}^{m_k} a_{ij}(s_{kv})\Big [I_{\{\gamma ^\varepsilon (u)=s_{kv}\}}-\pi ^{k}_{v}I_{\{\gamma ^\varepsilon (u)\in {\mathbb {S}}_k\}}\Big ], \end{aligned}$$

(A.5)

and

$$\begin{aligned}&{\sigma }_{i\iota }(\gamma ^\varepsilon (u)){\sigma }_{j\iota }(\gamma ^\varepsilon (u))-\bar{\sigma }_{i\iota }({\bar{\gamma }}^\varepsilon (u))\bar{\sigma }_{j\iota }({\bar{\gamma }}^\varepsilon (u))\nonumber \\&\quad =\sum _{k=1}^{l}\sum _{v=1}^{m_k} {\sigma }_{i\iota }(s_{kv}){\sigma }_{j\iota }(s_{kv})\Big [I_{\{\gamma ^\varepsilon (u)=s_{kv}\}}-\pi ^{k}_{v}I_{\{\gamma ^\varepsilon (u)\in {\mathbb {S}}_k\}}\Big ]. \end{aligned}$$

(A.6)

From (Yin and Zhang 1998, p. 187), it is known that there is a constant $k_0>0$ such that for any $u\ge t$, $1\le k\le l,$$1\le v\le m_k,$

$$\begin{aligned} {\mathbb {E}}_t^{\varepsilon }\Big [I_{\{\gamma ^\varepsilon (u)=s_{kv}\}}-\pi ^{k}_{v}I_{\{\gamma ^\varepsilon (u)\in {\mathbb {S}}_k\}}\Big ]=O(\varepsilon +\hbox {e}^{-k_0(u-t)/\varepsilon }). \end{aligned}$$

(A.7)

(A.4)–(A.7) together with $(x^{\varepsilon ,N}(\cdot ),{\bar{\gamma }}^\varepsilon (\cdot )) \Rightarrow (x^N(\cdot ),{\bar{\gamma }}(\cdot ))$ implies

$$\begin{aligned}&{\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{\varepsilon ,N}(t_j),{\bar{\gamma }}^\varepsilon (t_j) ) \right. \nonumber \\&\left. \quad \int _{t}^{t+s} g_x(x^{\varepsilon ,N}(u),{\bar{\gamma }}^\varepsilon (u))\hbox { diag}(x_1^{\varepsilon ,N}(u),\dots ,x_r^{\varepsilon ,N}(u))\Big . q_N(x^{\varepsilon ,N}(u)){b}(\gamma ^\varepsilon (u))\hbox {d}u\right] \nonumber \\&\quad \rightarrow {\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{N}(t_j),{\bar{\gamma }}(t_j) ) \right. \nonumber \\&\left. \quad \int _{t}^{t+s} g_x(x^{N}(u),{\bar{\gamma }}(u))\hbox { diag}(x_1^{N}(u),\dots ,x_r^{N}(u))\Big . q_N(x^{N}(u))\bar{b}({\bar{\gamma }}(u))\hbox {d}u\right] . \end{aligned}$$

(A.8)

Similarly, we deduce that

$$\begin{aligned}&{\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{\varepsilon ,N}(t_j),{\bar{\gamma }}^\varepsilon (t_j) )\int _{t}^{t+s} g_x(x^{\varepsilon ,N}(u),{\bar{\gamma }}^\varepsilon (u))\hbox { diag}(x_1^{\varepsilon ,N}(u),\dots ,x_r^{\varepsilon ,N}(u))\right. \nonumber \\&\qquad \left. \times q_N(x^{\varepsilon ,N}(u)){A}(\gamma ^\varepsilon (u))x^{\varepsilon ,N}(u)\hbox {d}u\right] \nonumber \\&\quad \rightarrow {\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{N}(t_j),{\bar{\gamma }}(t_j) )\int _{t}^{t+s} g_x(x^{N}(u),{\bar{\gamma }}(u)) \hbox { diag}(x_1^{N}(u),\dots ,x_r^{N}(u))\right. \nonumber \\&\qquad \left. \times q_N(x^{N}(u))\bar{A}(\bar{g}(u)) x^{N}(u)\hbox {d}u\right] , \end{aligned}$$

(A.9)

$$\begin{aligned}&{\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{\varepsilon ,N}(t_j),{\bar{\gamma }}^\varepsilon (t_j) )\int _{t}^{t+s} \frac{1}{2}\hbox {trace}\left[ q_N(x^{\varepsilon ,N}(u))\sigma ^\mathrm{T}(\gamma ^\varepsilon (u))\hbox { diag}(x_1^{\varepsilon ,N}(u),\dots ,x_r^{\varepsilon ,N}(u)) \right. \right. \nonumber \\&\qquad \left. \left. \times g_{xx}(x^{\varepsilon ,N}(u),{\bar{\gamma }}^\varepsilon (u))\hbox { diag}(x_1^{\varepsilon ,N}(u),\dots ,x_r^{\varepsilon ,N}(u))\sigma (\gamma ^\varepsilon (u))\right] \hbox {d}u\right] \nonumber \\&\quad \rightarrow {\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{N}(t_j),{\bar{\gamma }}(t_j) )\int _{t}^{t+s} \frac{1}{2} \hbox {trace}\left[ q_N(x^{N}(u))\bar{\sigma }^\mathrm{T}({\bar{\gamma }}(u))\hbox { diag}(x_1^{N}(u),\dots ,x_r^{N}(u)) \right. \right. \nonumber \\&\qquad \left. \left. \times g_{xx}(x^{N}(u),{\bar{\gamma }}(u))\hbox { diag}(x_1^{N}(u),\dots ,x_r^{N}(u))\bar{\sigma }({\bar{\gamma }}(u))\right] \hbox {d}u\right] , \end{aligned}$$

(A.10)

and

$$\begin{aligned}&{\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{\varepsilon ,N}(t_j),{\bar{\gamma }}^\varepsilon (t_j) )\int _{t}^{t+s} {\hat{Q}} \bar{g}(x^{\varepsilon ,N}(u),\cdot )(\gamma ^\varepsilon (u))\hbox {d}u\right] \nonumber \\&\quad \rightarrow {\mathbb {E}}\left[ \prod _{j=1}^{\iota }\phi _j(x^{N}(t_j),{\bar{\gamma }}(t_j) )\int _{t}^{t+s} {\bar{Q}}g(x^{N}(u),\cdot )({\bar{\gamma }}(u))\hbox {d}u\right] . \end{aligned}$$

(A.11)

(A.8)–(A.11) together with (A.3) implies (A.1). Therefore, $(x^N(\cdot ),{\bar{\gamma }}(\cdot ))$ is the solution to the martingale problem with operator (3.3). By Assumption  1,

$$\begin{aligned} \lambda _{\max }^{+}(\bar{C}\bar{A}(k)+\bar{A}^\mathrm{T}(k)\bar{C})\le \bar{\lambda }(k):=\sum _{v=1}^{m_k}\pi ^k_v\lambda _{kv}\le 0,~~\forall k\in \bar{\mathbb {S}}, \end{aligned}$$

(A.12)

holds, which implies the solution of (3.4) with any given initial value is unique (see Li et al. 2009). Let $\bar{x}(\cdot )$ be the solution of (3.4) given $\bar{x}(0)=\bar{x}_0,{\bar{\gamma }}(0)={\bar{\gamma }}_0.$ Let $P(\cdot )$ and $P^N(\cdot )$ be the probability measures induced by $(\bar{x}(\cdot ),\bar{\gamma }(\cdot ))$ and $(x^N(\cdot ),\bar{\gamma }(\cdot ))$, respectively, on the Borel subsets of $D(\bar{\mathbb {R}}_+;{\mathbb {R}}^{r}_+\times \bar{\mathbb {S}})$. Then, $P(\cdot )$ is unique, and for any $T>0,$$P(\cdot )$ must agree with $P^N(\cdot )$ on all Borel subsets of the set of paths in $D(\bar{\mathbb {R}}_+;{\mathbb {R}}^{r}_+\times \bar{\mathbb {S}})$ whose values are in $\bar{B}_N(0)\times \bar{\mathbb {S}}$ for each $t\le T.$ By $\lim _{N\rightarrow \infty }P(\sup _{t\le T} |\bar{x}(t)|\le N)=1$ and $(x^{\varepsilon ,N}(\cdot ),\bar{\gamma }^{\varepsilon }(\cdot )) \Rightarrow (x^N(\cdot ),\bar{\gamma }(\cdot )),$ we have $(x^{\varepsilon }(\cdot ),\bar{\gamma }^{\varepsilon }(\cdot )) \Rightarrow (\bar{x}(\cdot ),\bar{\gamma }(\cdot )).$ Since the limit is unique, this result is independent of the chosen subsequence. $\square$