The ergodic property and positive recurrence of a multi-group Lotka–Volterra mutualistic system with regime switching

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Abstract

In this paper, we consider a stochastic multi-group Lotka–Volterra mutualistic system under regime switching. It is well known that the population is forced to expire when the perturbation is sufficiently large. The main aim here is to investigate its ergodic property and positive recurrence by stochastic Lyapunov functions under small perturbation, which can be used to explain some recurring phenomena in practice and thus provide a good description of permanence. The mean of the stationary distribution is estimated. Simulations are also carried out to confirm our analytical results.

Introduction

Taking the white and color noise into account, population systems described by stochastic differential equations with regime switching have recently been studied by many authors; see  [1], [2], [3], [4], [5], [6], [7], [8], [9], for example. It is well known that, when the perturbation is large, the population will be forced to expire whilst it remains stochastic permanent when the perturbation is small, which can provide some explanation of dynamical behavior from a biological perspective.

In practice, we may often observe the recurrence of higher and lower population levels of a permanent population system as time goes by. See  [10], [11], [12], [13], [14], [15], [16], [17], [18] and their references for pollination mutualism as examples. If we make a great number of records to investigate the dynamic behavior of a population system, we may find that the average of the records approaches a fixed positive point, but a single record may fluctuate around this fixed point even if the number of records is large. Then, how can we explain such biological phenomena? In such a case, stochastic permanence or limits of integral average do not seem adequate, so we need to investigate other dynamical properties to illustrate such biological phenomena. Therefore, in this paper, we concentrate on the ergodic property and positive recurrence of a multi-group Lotka–Volterra mutualistic population system to try to give a good explanation of the above biological phenomena (see Remark 3.1).

Consider a stochastic multi-group Lotka–Volterra system characterized by the following stochastic differential equation with color and white noise:dx(t)=diag(x1(t),,xn(t))[(b(r(t))+A(r(t))x(t)dt)+σ(r(t))dB(t)], where x=(x1,,xn)τ, B=(B1,,Bd)τ is a standard d-dimensional Brownian motion, {r(t),t0} is a right-continuous Markov chain independent of the Brownian motion B, taking values in a finite state space S={1,,N} with generator Γ=(γij)N×N given by P{r(t+δ)=j|r(t)=i}={γijδ+o(δ)if  ij,1+γi,iδ+o(δ)if  i=j, and, for any kS, b(k)=(b1(k),,bn(k))τ, A(k)=(aij(k))n×n, σ(k)=(σij(k))n×d. We also assume that, for kS, Rank(σ(k))=n, and aii(k)<0, aij(k)0, 1i,jn, ij. This means that Eq. (1.1) is a mutualistic system in which every species enhances the growth of each other  [19], [20], [21].

In this paper, we investigate the ergodic property and positive recurrence of Eq. (1.1) by stochastic Lyapunov functions with regime switching  [22], which implies the existence and uniqueness of a stationary distribution. We show that the ergodic property and positive recurrence can provide a biological perspective of cycling phenomena of a population system, and hence describe the permanence of a population system in practice.

The paper is organized as follows. In Section  2, we introduce some notation and assumptions, which are necessary for later discussion. In Section  3, we use a class of stochastic Lyapunov functions with regime switching to obtain the ergodic property and positive recurrence, which account for some recurring events of a population system. The mean of the stationary distribution of a population system is also investigated. In Section  4, we report results of computer simulations to confirm our theoretical analysis. In Section  5, we provide a concluding discussion to end this paper.

Section snippets

Preliminaries

Throughout this paper, unless otherwise specified, let (Ω,F,{Ft}t0,P) be a complete probability space with a filtration {Ft}t0 satisfying the usual conditions (i.e., it is right continuous and increasing while F0 contains all P-null sets). R+n denotes the positive zone in Rn, i.e.,  R+n={xRn;xi>0,1in}. If B is a symmetric n×n matrix, we recall the following notation:λmax+(B)=supxR+n,|x|=1xτBx, which is introduced in  [1], [23]. Thus, for any xR+n, we have xτBxλmax+(B)|x|2. We also

Ergodic property of positive recurrence

Before further discussion, we need to show that Eq. (1.1) has a unique positive solution, which is essential in modeling a population system. Since the coefficients of Eq. (1.1) do not satisfy the linear growth condition, the classical theory of stochastic differential equations is not applicable directly. In recent papers (see, e.g.,[24], [23], [25], [26]), there are some standard techniques to prove the existence and uniqueness of a positive solution to Eq. (1.1), so we give the following

Simulations and examples

In this section, we report the results of computer simulations to illustrate our analytical results. Consider a 2-group Lotka–Volterra mutualistic system, with the following coefficients: b(1)=(45),b(2)=(34),b(3)=(79),M(1)=(4265),M(2)=(2043),M(3)=(3604), and σ(1)=(1211),σ(2)=(1223),σ(3)=(2112). Assume that the Markov chain has the generator Γ=(211341112). Then its stationary distribution π=(715,15,13). Let λmax(M) be the largest eigenvalue of symmetric matrix M. Then we have λmax+(M)λ

Discussion and concluding remarks

To illustrate some recurrent biological phenomena of a population system and give a better description of permanence, we use stochastic Lyapunov functions with regime switching to investigate the ergodic property and positive recurrence of a mutualistic population system under small perturbation. We find that, if the perturbation is sufficiently small, the population system is ergodic and positive recurrent. By positive recurrence and the strong Markov property, higher and lower population

Acknowledgments

We are grateful to the anonymous referees for several helpful suggestions. This research is supported by the National Natural Science Foundation of China (No 11201060) and the Fundamental Research Funds for the Central Universities (N0. 11QNJJ002).

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