Global asymptotic stability of stochastic competitive system with infinite delays

Abstract

In this paper, sufficient conditions for global asymptotic stability of a stochastic non-autonomous Lotka–Volterra competitive system with infinite delays are established. Some recent results are improved and generalized.

1 Introduction

In this paper we consider the global asymptotic stability (or GAS for short) of the positive solution of the following stochastic competitive system with infinite delays

$$\begin{aligned} \displaystyle dx_i(t)\displaystyle&= x_i(t)\bigg [r_i(t)-a_i(t) x_i(t)-\sum _{j=1}^nb_{ij}(t)x_j(t-\tau _{ij}(t))\nonumber \\&\displaystyle -\sum _{j=1}^nc_{ij}(t)\int _{-\infty }^0x_j(t+ \theta )d\nu _{ij}(\theta )\bigg ]dt\nonumber \\&\displaystyle +\, x_i(t)\sum _{k=1}^m\sigma _{ik} (t)dB_k(t),\quad 1\le i\le n, \end{aligned}$$

(1)

where $r_i(t),~a_i(t)\ge 0,~b_{ij}(t)\ge 0$, $\sigma _{ik}(t)$ and $c_{ij}(t)\ge 0$ are bounded and continuous functions on $[0,+\infty )$, $1\le i,j\le n,~1\le k\le m$; $(B_1(t),\ldots ,B_m(t))^T$ is an $m$-dimensional Wiener process defined on a complete probability space $(\Omega , \mathcal {F}, \{\mathcal {F}_t\}_{t\ge 0}, \mathcal {P})$; $\tau _{ij}(t)\ge 0$ is a bounded and continuously differentiable function on $[0,+\infty )$ with $\bar{\tau }:=\min _{1\le i,j\le n}\inf _{t\ge 0} (1-\dot{\tau }_{ij}(t))>0$, $\dot{\tau }_{ij}(t)=d\tau _{ij}(t)/dt$; $\nu _{ij}(\theta )$ is a probability measure on $(-\infty ,0]$, $1\le i,j\le n$. $c_{ij}(t)\int _{-\infty }^0 x_j(t+\theta ) d\nu _{ij}(\theta )$ means the effect of the infinite delay of species $x_j$ on the species $x_i$ (see e.g. [20]). Here we use an m-dimensional Wiener process not a standard Wiener process to describe the environmental perturbations because the noises on $r_i$ ($1\le i\le n$) may or may not correlate to each other.

GAS of the positive solution of population models has long been and will continue to be a dominant theme in biomathematics. Since in the natural world, many environmental fluctuations (for example, variation in intensity of sunlight, temperature, water level, etc.) can affect the growth of species [1], several scholars have considered the GAS of stochastic population systems. Jiang et al. did pioneering work in [2], where the authors investigated the GAS of the following stochastic logistic model

$$\begin{aligned} \displaystyle \ dx(t)=x(t)[r(t)-a(t)x(t)]{dt}+\sigma (t)x(t){dB(t)}, \end{aligned}$$

(2)

where $r(t), a(t)$ and $\sigma (t)$ are continuous $T$-periodic functions, and $a(t)>0$, $r(t)>0$, $B(t)$ is a standard Wiener process. The authors proved that if $\min _{t\in [0,T]}r(t)>\max _{t\in [0,T]}\sigma ^2(t),$ then system (2) is GAS. Then these results were extended and improved by Li and Mao in [3], where the authors considered the following stochastic Lotka–Volterra competitive system

$$\begin{aligned} \begin{array}{ll} \displaystyle dx_i(t)\displaystyle =x_i(t)\bigg [r_i(t)- \sum _{j=1}^nb_{ij}(t)x_j(t)\bigg ]dt+\sigma _i(t)x_i(t)dB_i(t),\quad 1\le i\le n, \end{array} \end{aligned}$$

(3)

where $b_{ij}(t)>0,~r_i(t),~\sigma _i(t)~(i,j=1,\ldots ,n)$ are continuous and bounded functions on $[0,+\infty )$, and $B_{i}(t)~(i =1,\ldots ,n)$ are independent standard Wiener processes. Li and Mao [3] proved that if for every $1\le i\le n$,

$$\begin{aligned} \inf _{t\ge 0}\bigg [b_{ii}(t)-\sum _{j=1,j\ne i}^nb_{ji}(t)\bigg ]>0, \end{aligned}$$

(4)

then model (3) is GAS. GAS of a stochastic competitive Lotka–Volterra system with Markovian switching was considered by Hu and Wang [4]. Bao et al. [5] established the sufficient conditions for GAS of a stochastic Lotka–Volterra competitive system with Lévy jumps. Liu and Wang [6, 7] investigated the GAS of stochastic predator-prey models and stochastic cooperative models respectively. Mandal [8] considered the GAS of a stochastic three-species competitor-competitor-cooperative model.

However, time delays should be incorporated into biological models to denote resource regeneration times, reaction times, feeding times, maturation periods, etc. [9]. At the same time, in the nature world it is a common phenomena that some species compete for limited resources. Therefore it is important to consider the GAS of stochastic competitive models with time delay. To the best of our knowledge, no result of this aspect has been reported. The aim of this paper is to investigate this problem. In Sect. 2, we establish our main results. Sufficient conditions for GAS of model (1) are established. Some recent results are extended and improved. In Sect. 3, a simulation figure is introduced to illustrate the main results. The conclusions are given in the last section.

2 Main results

For the sake of convenience, we let $R^n_+=\{x\in R^n:x_i>0~\text{ for } \text{ all }~1\le i\le n\}$ and denote $BC((-\infty ,0],R_+^n)$ the family of bounded and continuous functions from $(-\infty ,0]$ to $R_+^n$ with the norm $||\xi ||=\sup _{\theta \le 0}|\xi (\theta )|$. Define $\hat{f}=\sup _{t\ge 0}f(t)$, $\check{f}=\inf _{t\ge 0}f(t)$. For the following stochastic differential equation

$$\begin{aligned} dx(t)=f(x(t),t)dt+g(x(t),t)dw(t), \end{aligned}$$

define

$$\begin{aligned} LV(x,t)=\frac{\partial V(x,t)}{\partial t}+V_x(x,t) f(x,t)+0.5trace\left[ g^T(x,t)V_{xx}(x,t)g(x,t)\right] , \end{aligned}$$

where $V\in C^{2,1}(R^n\times R_+;R_+)$.

Definition 1

Model (1) is said to be globally asymptotically stable (or globally attractive) if for any two solutions $x(t)=(x_1(t),\ldots ,x_n(t))^T$ and $y(t)=(y_1(t),\ldots ,y_n(t))^T$ of (1) with initial data $x(\theta )\in BC((-\infty ,0],R_+^n)$ and $y(\theta )\in BC\left( (-\infty ,0],R_+^n\right)$ respectively, $\lim \limits _{t\rightarrow +\infty }\sum _{i=1}^n|x_i(t)-y_i(t)|=0$ a.s.

Lemma 2

For any initial value $x(\theta )\in BC\left( (-\infty ,0],R_+^n\right)$, there is a unique global positive solution $x(t)$ on $t\in R$ to Eq. (1) almost surely (a.s.).

Proof

The proof is motivated by [10–12]. Note that the coefficients of Eq. (1) are locally Lipschitz continuous, then for any initial data $x(\theta )\in BC\left( (-\infty ,0],R_+^n\right)$, Eq. (1) has a unique local solution $x(t)$ on $t\in (-\infty ,\tau _e)$, where $\tau _e$ represents the explosion time [13]. To complete the proof, we only need to prove that $\tau _e=+\infty$. Let $k_0>0$ be sufficiently large such that each component of $x(0)$ lying within $[1/k_0,k_0]$. For every integer $k>k_0$, define

$$\begin{aligned} \tau _k=\inf \{t\in [0,\tau _e]:x_i(t)\notin (1/k,k)~\text{ for } \text{ some } ~1\le i\le n\}. \end{aligned}$$

Set $\tau _\infty =\lim \limits _{k\rightarrow +\infty }\tau _k$. Then $\tau _\infty \le \tau _e~a.s.$ To complete the proof all we need to show is that $\tau _\infty =\infty$. If this statement is not true, we can find positive constants $T$ and $\varepsilon \in (0,1)$ such that $\mathcal {P}\{\tau _\infty <T\}>\varepsilon .$ Therefore we can find an integer $k_1\ge k_0$ such that

$$\begin{aligned} \mathcal {P}\{\tau _k<T\}>\varepsilon , \quad k>k_1. \end{aligned}$$

(5)

Define

$$\begin{aligned} V_1(t)&= \hat{b}\sum _{i=1}^n\sum _{j=1}^n\int _{t-\tau _{ij}(t)}^t \frac{|x_j(s)|}{1-\dot{\tau }_{ij}\left( \phi _{ji}^{-1}(s)\right) }ds\\&+\,\, \hat{c}\sum _{i=1}^n\sum _{j=1}^n\int _{-\infty }^0\int _{t+ \theta }^t|x_j(s)|dsd\nu _{ij}(\theta )+V(x), \end{aligned}$$

where

$$\begin{aligned} V(x)=\sum _{i=1}^n[x_i-1-\ln x_i],~\hat{b}=\displaystyle \max _{1\le i,j\le n}\widehat{b_{ij}},~\hat{c}=\displaystyle \max _{1\le i,j\le n}\widehat{c_{ij}}, \end{aligned}$$

$\phi _{ji}^{-1}(t)$ is the inverse function of $\phi _{ji}(t)=t-\tau _{ji}(t).$ Then we have

$$\begin{aligned} EV_1(\tau _k\wedge T)=V_1(0)+E\int _{0}^{\tau _k\wedge T}LV_1(s)ds, \end{aligned}$$

where

$$\begin{aligned} LV_1&= \displaystyle \hat{b}\sum _{i=1}^n\sum _{j=1}^n \bigg (\frac{x_j(t)}{1-\dot{\tau }_{ij}\left( \phi _{ji}^{-1}(t)\right) }- x_j(t-\tau _{ij})(t)\bigg )\nonumber \\&\displaystyle +\,\, \hat{c}\sum _{i=1}^n\sum _{j=1}^n\bigg [x_j(t)- \int _{-\infty }^0x_j(t+\theta )d\nu _{ij}(\theta )\bigg ]\nonumber \\&\displaystyle +\sum _{i=1}^n(x_i-1)\bigg [r_i(t)-a_i(t)x_i(t)- \sum _{j=1}^nb_{ij}(t)x_j(t-\tau _{ij}(t))\nonumber \\&\displaystyle -\sum _{j=1}^nc_{ij}(t)\int _{-\infty }^0x_j(t+ \theta )d\nu _{ij}(\theta )\bigg ]+0.5\sum _{i=1}^n\sum _{k=1}^m \sigma ^2_{ik}(t)\nonumber \\&\le \displaystyle \hat{b}\sum _{i=1}^n\sum _{j=1}^n\bigg (\frac{x_j(t)}{1-\dot{\tau }_{ij}\left( \phi _{ji}^{-1}(t)\right) }-x_j(t-\tau _{ij})(t)\bigg )\nonumber \\&\quad \displaystyle +\,\, \hat{c}\sum _{i=1}^n\sum _{j=1}^n\bigg [x_j(t)-\int _{- \infty }^0x_j(t+\theta )d\nu _{ij}(\theta )\bigg ]\nonumber \\&\quad \displaystyle +\sum _{i=1}^n\bigg [r_i(t)x_i(t)+a_i(t)x_i(t)+ \sum _{j=1}^nb_{ij}(t)x_j(t-\tau _{ij}(t))\nonumber \\&\quad \displaystyle +\sum _{j=1}^nc_{ij}(t)\int _{-\infty }^0x_j(t+\theta ) d\nu _{ij}(\theta )-r_i(t)\bigg ]+0.5\sum _{i=1}^n\sum _{k=1}^m \sigma ^2_{ik}(t)\nonumber \\&=\displaystyle \hat{b}\sum _{i=1}^n\sum _{j=1}^n\frac{x_j(t)}{1 -\dot{\tau }_{ij}\left( \phi _{ji}^{-1}(t)\right) }+\hat{c}\sum _{i=1}^n\sum _{j=1}^nx_j(t) \nonumber \\&\quad -\sum _{i=1}^n\sum _{j=1}^n\bigg (\hat{b}-b_{ij}(t)\bigg )x_j(t-\tau _{ij}) (t)\nonumber \\&\quad \displaystyle -\sum _{i=1}^n\sum _{j=1}^n\int _{-\infty }^0\bigg (\hat{c} -c_{ij}(t)\bigg )x_j(t+\theta )d\nu _{ij}(\theta )\nonumber \\&\quad \displaystyle +\sum _{i=1}^n\bigg [r_i(t)x_i(t)+a_i(t)x_i(t)-r_i(t) \bigg ]+0.5\sum _{i=1}^n\sum _{k=1}^m\sigma ^2_{ik}(t)\nonumber \\&\le \displaystyle n(\hat{b}/\bar{\tau }+\hat{c})\sum _{i=1}^nx_i(t) +c_1\sum _{i=1}^nx_i(t)+c_2, \end{aligned}$$

(6)

where $c_1=\displaystyle \max _{1\le i\le n}\widehat{|r_i|}+ \max _{1\le i\le n}\widehat{a_i}$, $c_2= 0.5\displaystyle \sum _{i=1}^n\sum _{k=1}^m\widehat{\sigma ^2_{ik}} +\max _{1\le i\le n}\widehat{|r_i|}$. In the proof of the last inequality, we have used that

$$\begin{aligned} \bar{\tau }=\min _{1\le i,j\le n}\inf _{t\ge 0}(1-\dot{\tau }_{ij} (t))>0,~\hat{b}-b_{ij}(t)\ge 0,~\hat{c}-c_{ij}(t)\ge 0. \end{aligned}$$

It then follows that

$$\begin{aligned} \begin{array}{ll}&EV_1(\tau _k\wedge T) \displaystyle \le V_1(0)+E\int _{0}^{\tau _k \wedge T}(n\hat{b}/\bar{\tau }+n\hat{c}+c_1)\sum _{i=1}^nx_i(s)ds +c_2T. \end{array} \end{aligned}$$

An application of the basic inequality $u\le 2[u-1-\ln u]+2$ on $u>0$, one can see that

$$\begin{aligned} \begin{array}{ll} EV(x(\tau _k\wedge T)) &{}\displaystyle \le V_1(0)+E\int _{0}^{\tau _k \wedge T}(n\hat{b}/\bar{\tau }+n\hat{c}+c_1)[2V(x(s))+2]ds+c_2T\\ &{}\displaystyle \le 2(n\hat{b}/\bar{\tau }+n\hat{c}+c_1)\int _{0}^{T} EV(x(\tau _k\wedge s))ds+c_3, \end{array} \end{aligned}$$

where $c_3=V_1(0)+2(n\hat{b}/\bar{\tau }+n\hat{c}+c_1)T+c_2T$. It then follows from Gronwall’s inequality that

$$\begin{aligned} EV(x(\tau _k\wedge T))\le c_3e^{2(n\hat{b}/\bar{\tau }+n\hat{c}+c_1)T}<+\infty . \end{aligned}$$

The following proof is standard and hence is omitted. $\square$

Remark 3

There are three terms in $V_1(t)$. The first two terms are introduced for eliminating the delay terms.

Lemma 4

For arbitrary $p>1$, there exists a constant $F=F(p)>0$ such that

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }\sum _{i=1}^n E\left( x_i^{2p}(t)\right) \le F. \end{aligned}$$

(7)

Proof

Define $V_2(x)=\sum _{i=1}^nx_i^{2p}$. Then

$$\begin{aligned} LV_2(x)\displaystyle \le 2p\sum _{i=1}^n\bigg [\bigg [r_i(t)+0.5 (2p-1)\sum _{k=1}^m\sigma _{ik}^2(t)\bigg ]x_i^{2p}-a_{i}(t) x_i^{2p+1}\bigg ]. \end{aligned}$$

It then follows that

$$\begin{aligned} \sum _{i=1}^nE\left[ x_i^{2p}(t)\right] \displaystyle \le V_2(0)+2p\sum _{i=1}^nE\int _0^tx_i^{2p}(s)\bigg [c_6-c_7 x_i(s)\bigg ]ds, \end{aligned}$$

where

$$\begin{aligned} c_6=\max _{1\le i\le n}\bigg [\widehat{|r_i|}+0.5(2p-1) \sum _{k=1}^m\widehat{\sigma _{ik}^2}\bigg ],~c_7=\min _{1\le i\le n}\check{a_{i}}. \end{aligned}$$

It then follows from Hölder’s inequality (see e.g. [14], page 5) that

$$\begin{aligned} \displaystyle \sum _{i=1}^nE\left[ x_i^{2p}(t)\right]&\le \displaystyle V_2(0)+ 2p c_6\int _0^t\sum _{i=1}^nE\left[ x_i^{2p}(s)\right] ds\\&-\, 2p c_7\sum _{i=1}^n \int _0^tE\left[ x^{2p+1}_i(s)\right] ds\\ \displaystyle&\le V_2(0)+2p c_6\int _0^t\sum _{i=1}^nE\left[ x_i^{2p} (s)\right] ds\\&-\, 2p c_7\sum _{i=1}^n\int _0^t\left\{ E\left[ x_i^{2p}(s)\right] \right\} ^{\frac{2p+ 1}{2p}}ds\\ \displaystyle&\le V_2(0)+2p c_6\int _0^t\sum _{i=1}^nE\left[ x_i^{2p} (s)\right] ds\\&-\, 2p c_7\int _0^tn^{-0.5/p}\bigg \{\sum _{i=1}^nE\left[ x_i^{2p} (s)\right] \bigg \}^{\frac{2p+1}{2p}}ds \end{aligned}$$

Let $z(t)=\sum _{i=1}^nE\left[ x_i^{2p}(t)\right]$, then one can see that

$$\begin{aligned} \frac{dz(t)}{dt}\le 2pz(t)\left[ c_6-c_7 n^{-0.5/p}z^{0.5/p}(t)\right] . \end{aligned}$$

According to the standard comparison theorem, we obtain

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }z(t)\le \bigg [\frac{c_6}{c_7 n^{-0.5/p}}\bigg ]^{2p}=:F. \end{aligned}$$

This completes the proof. $\square$

Lemma 5

Suppose that $x(t)=(x_1(t),\ldots ,x_n(t))^T$ is the solution of model (1) with initial data $x(\theta )\in BC\left( (-\infty ,0],R_+^n\right)$, then for each $1\le i\le n,$ almost every sample path of $x_i(t)$ is uniformly continuous.

Proof

According to (7), we can find a $T>0$ such that $\sum _{i=1}^nE\left( x_i^{2p}(t)\right) \le 1.5F$ for all $t\ge T.$ On the other hand, since $\sum _{i=1}^nE\left( x_i^{2p}(t)\right)$ is continuous and $x(\theta )$ is bounded, then we can find a $\tilde{F}>0$ such that $\sum _{i=1}^nE\left( x_i^{2p}(t)\right) \le \tilde{F}$ for $t\le T.$ Let $L=\max \{1.5F,\tilde{F}\}$, then one can see that

$$\begin{aligned} \sum _{i=1}^nE\left( x_i^{2p}(t)\right) \le L,~\text{ for } \text{ all }~t\in R. \end{aligned}$$

Note that Eq. (1) is equivalent to the following equation

$$\begin{aligned} x_i(t)&=x_i(0)+\displaystyle \int _0^tx_i(s)\bigg [r_i(s)-a_i(s) x_i(s)-\sum _{j=1}^nb_{ij}(s)x_j(s-\tau _{ij}(s))\\&\quad \displaystyle -\sum _{j=1}^nc_{ij}(s)\int _{-\infty }^0x_j(s+ \theta )d\nu _{ij}(\theta )\bigg ]ds+\sum _{k=1}^m\int _0^t \sigma _{ik}(s)x_i(s)dB_{k}(s). \end{aligned}$$

For $p>2$, compute that

$$\begin{aligned}&E\displaystyle \bigg |x_i\bigg [r_i(t)-a_i(t)x_i-\sum _{j=1}^n b_{ij}(t)x_j(t-\tau _{ij}(t))\\&\qquad -\sum _{j=1}^nc_{ij}(t)\int _{- \infty }^0x_j(t+\theta )d\nu _{ij}(\theta )\bigg ]\bigg |^p\\&\quad \displaystyle =E\bigg [|x_i|^p\bigg |r_i(t)-a_i(t)x_i- \sum _{j=1}^nb_{ij}(t)x_j(t-\tau _{ij}(t))\\&\qquad -\sum _{j=1}^nc_{ij}(t)\int _{-\infty }^0x_j(t+\theta ) d\nu _{ij}(\theta )\bigg |^p\bigg ]\\&\quad \displaystyle \le 0.5E|x_i^{2p}|+0.5E\bigg |r_i(t)-a_i(t) x_i-\sum _{j=1}^nb_{ij}(t)x_j(t-\tau _{ij}(t))\\&\qquad -\sum _{j=1}^n c_{ij}(t)\int _{-\infty }^0x_j(t+\theta )d \nu _{ij}(\theta )\bigg |^{2p}\\&\quad \displaystyle =0.5L+0.5(2n+2)^{2p-1}\times \bigg [(\widehat{|r_i|}) ^{2p}+\widehat{a_{i}}^{2p}E|x_i^{2p}|\\&\qquad \displaystyle +\sum _{j=1}^n\widehat{b_{ij}}^{2p}E|x_j^{2p}(t- \tau _{ij}(t))|+\sum _{j=1}^n\widehat{c_{ij}}^{2p}E\bigg |\int _{- \infty }^0x_j(t+\theta )d\nu _{ij}(\theta )\bigg |^{2p}\bigg ]\\&\quad \displaystyle \le 0.5L+0.5(2n+2)^{2p-1}\times \bigg [(\widehat{ |r_i|})^{2p}+\widehat{a_{i}}^{2p}E|x_i^{2p}|\\&\qquad \displaystyle +\sum _{j=1}^n\widehat{b_{ij}}^{2p}E|x_j^{2p} (t-\tau _{ij}(t))|+\sum _{j=1}^n\widehat{c_{ij}}^{2p}\int _{- \infty }^0E\bigg |x_j^{2p}(t+\theta )\bigg |d\nu _{ij}(\theta ) \bigg ]\\&\quad \displaystyle \le 0.5L+0.5(2n+2)^{2p-1}\bigg [\widehat{|r_i|}^{2 p}+\bigg (\widehat{a_{i}}^{2p}+\sum _{j=1}^n\widehat{b_{ij}}^{2 p}+\sum _{j=1}^n\widehat{c_{ij}}^{2p}\bigg )L\bigg ]=:L_1. \end{aligned}$$

On the other hand, an application of of the moment inequality for stochastic integrals (see e.g. [14], Page 69) and Hölder’s inequality, one can observe that for $0\le t_1\le t_2$,

$$\begin{aligned}&\displaystyle E\bigg |\sum _{k=1}^m\int _{t_1}^{t_2}\sigma _{ik} (s)x_i(s)dB_k(s)\bigg |^p\\&\quad \displaystyle \le m^{p-1}\sum _{k=1}^mE\bigg |\int _{t_1}^{t_2} \sigma _{ik}(s)x_i(s)dB_k(s)\bigg |^p\\&\quad \displaystyle \le m^{p-1}\sum _{k=1}^m [\widehat{\sigma _{ik}^2}]^p \bigg [\displaystyle \frac{p(p-1)}{2}\bigg ]^{p/2}(t_2-t_1)^{(p-2)/2} \int _{t_1}^{t_2}E|x_i^p(s)|ds\\&\quad \displaystyle \le m^{p-1}\sum _{k=1}^m [\widehat{\sigma _{ik}^2}]^p \bigg [\displaystyle \frac{p(p-1)}{2}\bigg ]^{p/2}(t_2-t_1)^{(p-2)/2} \int _{t_1}^{t_2}(E|x_i^{2p}(s)|)^{0.5}ds\\&\quad \displaystyle \le m^{p-1}\sum _{k=1}^m [\widehat{\sigma _{ik}^2}]^p \bigg [\frac{p(p-1)}{2}\bigg ]^{p/2}(t_2-t_1)^{p/2}L^{0.5} \end{aligned}$$

Consequently, for $0<t_1<t_2<\infty ,~t_2-t_1\le 1,~1/p+1/q=1,~p>2$, we get

$$\begin{aligned} E(|x_i(t_2)-x_i(t_1)|^p)\displaystyle&=E\bigg |\int _{t_1}^{t_2} x_i(s)\bigg [r_i(s)-a_i(s)x_i(s)-\sum _{j=1}^nb_{ij}(s)x_j(s- \tau _{ij}(s))\\&\quad \displaystyle -\sum _{j=1}^nc_{ij}(s)\int _{-\infty }^0x_j(s +\theta )d\nu _{ij}(\theta )\bigg ]ds\\&\quad +\,\sum _{k=1}^m\int _{t_1}^{t_2} \sigma _{ik}(s)x_i(s)dB_k(s)\bigg |^p\\&\displaystyle \le 2^{p-1}E\bigg |\int _{t_1}^{t_2}x_i(s)\bigg [r_i (s)-a_i(s)x_i(s)\\&\quad -\,\sum _{j=1}^nb_{ij}(s)x_j(s-\tau _{ij}(s))\\&\quad \displaystyle -\sum _{j=1}^nc_{ij}(s)\int _{-\infty }^0x_j(s+ \theta )d\nu _{ij}(\theta )\bigg ]ds\bigg |^p\\&\quad +\,2^{p-1}E\bigg | \sum _{k=1}^m\int _{t_1}^{t_2}\sigma _{ik}(s)x_i(s)dB_k(s) \bigg |^p\\&\displaystyle \le 2^{p-1}(t_2-t_1)^{p/q}\int _{t_1}^{t_2}E \bigg |x_i(s)\bigg [r_i(s)-a_i(s)x_i(s)\\&\quad -\,\sum _{j=1}^nb_{ij} (s)x_j(s-\tau _{ij}(s))\\&\quad \displaystyle -\sum _{j=1}^nc_{ij}(s)\int _{-\infty }^0x_j (s+\theta )d\nu _{ij}(\theta )\bigg ]\bigg |^pds\\&\quad +\,2^{p-1}m^{p-1}\sum _{k=1}^m[\widehat{\sigma _{ik}^2}]^p \bigg [\frac{p(p-1)}{2}\bigg ]^{p/2}(t_2-t_1)^{p/2}L^{0.5}\\&\displaystyle \le 2^{p-1}(t_2-t_1)^{p/q+1}L_1\\&\quad +\, 2^{p-1}m^{p- 1}\sum _{k=1}^m[\widehat{\sigma _{ik}^2}]^p\bigg [\frac{p(p-1)}{2} \bigg ]^{p/2}(t_2-t_1)^{p/2}L^{0.5}\\&\displaystyle \le 2^{p-1}(t_2-t_1)^{p/2}\bigg [(t_2-t_1)^{p/2}+ \left( \displaystyle \frac{p(p-1)}{2}\right) ^{p/2}\bigg ]L_2\\&\displaystyle \le 2^{p-1}(t_2-t_1)^{p/2}\bigg [1+ \left( \frac{p(p-1)}{2}\right) ^{p/2}\bigg ]L_2, \end{aligned}$$

where $L_2=\max \{L_1,m^{p-1}\sum _{k=1}^m[\widehat{\sigma _{i k}^2}]^pL^{0.5}\}$. According to the Kolmogorov continuity criterion (see e.g. [15]), we obtain that almost every sample path of $x_i(t)$ is uniformly continuous.

Lemma 6

[16] If $f$ is a non-negative, uniformly continuous and integrable function defined on $t\ge 0$, then $\lim \limits _{t\rightarrow +\infty }f(t)=0$.

Theorem 7

Suppose that there are positive constants $\gamma _1,\ldots ,\gamma _n$ and $\lambda$ such that for all $1\le i\le n$ and $t\ge 0$,

$$\begin{aligned} \gamma _ia_{i}(t)-\sum _{j=1}^n\gamma _j\frac{b_{ji}\left( \phi _{ji}^{-1} (t)\right) }{1-\dot{\tau }_{ji}\left( \phi _{ji}^{-1}(t)\right) }-\sum _{j=1}^n\gamma _j \int _{-\infty }^0c_{ji}(t-\theta )d\nu _{ji}(\theta )\ge \lambda , \end{aligned}$$

(8)

then Eq. (1) is GAS.

Proof

Define

$$\begin{aligned} V_3(t)&= \sum _{i=1}^n\gamma _i|\ln x_i(t)-\ln y_i(t)|;\\ V_4(t)&= \sum _{i=1}^n\sum _{j=1}^n\gamma _i\int _{t-\tau _{ij} (t)}^t\frac{b_{ij}\left( \phi _{ij}^{-1}(s)\right) }{1-\dot{\tau }_{ij} \left( \phi _{ij}^{-1}(s)\right) }|x_j(s)-y_j(s)|ds;\\ V_5(t)&= \sum _{i=1}^n\sum _{j=1}^n\gamma _i\int _{-\infty }^0 \int _{t+\theta }^tc_{ij}(s-\theta )|x_j(s)-y_j(s)|dsd\nu _{ij} (\theta ). \end{aligned}$$

Then we have

$$\begin{aligned} \displaystyle \frac{d^+V_3(t)}{dt}&=\sum _{i=1}^n\gamma _i \text{ sgn }(x_i-y_i)\bigg \{\displaystyle \bigg [\frac{dx_i}{x_i} -\frac{(dx_i)^2}{2x_i^2}\bigg ]-\bigg [\frac{dy_i}{y_i}- \frac{(dy_i)^2}{2y_i^2}\bigg ]\bigg \}\nonumber \\&=\sum _{i=1}^n\gamma _i\text{ sgn }(x_i-y_i) \bigg \{-a_{i}(t)(x_i-y_i)\nonumber \\&\qquad -\sum _{j=1}^nb_{ij}(t)\bigg (x_j(t-\tau _{ij}(t))-y_j(t- \tau _{ij}(t))\bigg )\bigg \}\nonumber \\&\qquad \displaystyle +\sum _{i=1}^n\gamma _i\text{ sgn }(x_i-y_i)\nonumber \\&\quad \quad \qquad \bigg \{\!\!-\sum _{j=1}^nc_{ij}(t)\int _{-\infty }^0\bigg [x_j(t+ \theta )-y_j(t+\theta )\bigg ]d\nu _{ij}(\theta )\bigg \}\nonumber \\&\displaystyle \le -\sum _{i{=}1}^n\gamma _i a_{i}(t)|x_i{-} y_i|{+}\sum _{i{=}1}^n\sum _{j{=}1}^n\gamma _ib_{ij}(t)\bigg |x_j(t- \tau _{ij}(t))-y_j(t-\tau _{ij}(t))\bigg |\nonumber \\&\qquad \displaystyle +\sum _{i=1}^n\sum _{j=1}^n\gamma _ic_{ij} (t)\int _{-\infty }^0\bigg |x_j(t+\theta )-y_j(t+\theta )\bigg |d \nu _{ij}(\theta ). \end{aligned}$$

(9)

At the same time

$$\begin{aligned} \displaystyle \frac{dV_4(t)}{dt}\displaystyle&=\sum _{i=1}^n \sum _{j=1}^n\gamma _i\frac{b_{ij}\left( \phi _{ij}^{-1}(t)\right) }{1- \dot{\tau }_{ij}\left( \phi _{ij}^{-1}(t)\right) }|x_j(t)-y_j(t)|\nonumber \\&\quad \displaystyle -\sum _{i=1}^n\sum _{j=1}^n\gamma _i b_{ij}(t)\bigg | x_j(t-\tau _{ij}(t))-y_j(t-\tau _{ij}(t))\bigg |\nonumber \\&\displaystyle =\sum _{i=1}^n|x_i(t)-y_i(t)|\sum _{j=1}^n\gamma _j \frac{b_{ji}\left( \phi _{ji}^{-1}(t)\right) }{1-\dot{\tau }_{ji}\left( \phi _{j i}^{-1}(t)\right) }\nonumber \\&\quad \displaystyle -\sum _{i=1}^n\sum _{j=1}^n\gamma _i b_{ij}(t) \bigg |x_j(t-\tau _{ij}(t))-y_j(t-\tau _{ij}(t))\bigg |.\end{aligned}$$

(10)

$$\begin{aligned} \displaystyle \frac{dV_5(t)}{dt}\displaystyle&= \sum _{i=1}^n\sum _{j=1}^n\gamma _i\int _{-\infty }^0c_{ij}(t -\theta )|x_j(t)-y_j(t)|d\nu _{ij}(\theta )\nonumber \\&\quad \displaystyle -\sum _{i=1}^n\sum _{j=1}^n\gamma _i \int _{-\infty }^0c_{ij}(t)|x_j(t+\theta )-y_j(t+\theta )|d \nu _{ij}(\theta )\nonumber \\&\displaystyle =\sum _{i=1}^n|x_i(t)-y_i(t)|\sum _{j=1}^n \gamma _j\int _{-\infty }^0c_{ji}(t-\theta )d\nu _{ji}(\theta )\nonumber \\&\quad \displaystyle -\sum _{i=1}^n\sum _{j=1}^n\gamma _i\int _{- \infty }^0c_{ij}(t)|x_j(t+\theta )-y_j(t+\theta )|d\nu _{ij}(\theta ). \end{aligned}$$

(11)

Define $V_6(t)=V_3(t)+V_4(t)+V_5(t).$ According to (9–11), one can get

$$\begin{aligned} \displaystyle \frac{d^+V_6(t)}{dt}&\displaystyle \le -\sum _{i= 1}^n\gamma _ia_{i}(t)|x_i(t)-y_i(t)|\\&\quad +\sum _{i=1}^n|x_i(t)-y_i (t)|\sum _{j=1}^n\gamma _j\frac{b_{ji}\left( \phi _{ji}^{-1}(t)\right) }{1- \dot{\tau }_{ji}\left( \phi _{ji}^{-1}(t)\right) }\nonumber \\&\quad \displaystyle +\sum _{i=1}^n|x_i(t)-y_i(t)|\sum _{j=1}^n \gamma _j\int _{-\infty }^0|c_{ji}(t-\theta )|d\nu _{ji} (\theta )\nonumber \\&\displaystyle =-\sum _{i=1}^n\bigg [\gamma _ia_{i}(t)- \sum _{j=1}^n\gamma _j\frac{b_{ji}\left( \phi _{ji}^{-1}(t)\right) }{1-\dot{\tau }_{ji}\left( \phi _{ji}^{-1}(t)\right) }\\&\quad -\sum _{j=1}^n\gamma _j\int _{-\infty }^0c_{ji}(t-\theta )d \nu _{ji}(\theta )\bigg ]|x_i(t)-y_i(t)|\nonumber \\&\displaystyle \le -\lambda \sum _{i=1}^n|x_i(t)-y_i(t)|. \end{aligned}$$

That is to say

$$\begin{aligned} V_6(t)\le V_6(0)-\lambda \sum _{i=1}^n\int _0^t|x_i(s)-y_i(s)|ds. \end{aligned}$$

In other words,

$$\begin{aligned} V_6(t)+\lambda \sum _{i=1}^n\int _0^t|x_i(s)-y_i(s)|ds\le V_6(0)<\infty . \end{aligned}$$

Nota that $V_6(t)\ge 0$, then $|x_i(t)-y_i(t)|$ is integrable on $[0,+\infty )$. By Lemmas 5 and 6, we obtain the required assertion. $\square$

For Eq. (3), by Theorem 7, we obtain the following result.

Corollary 8

Suppose that there are positive constants $\gamma _1,\ldots ,\gamma _n$ and $\lambda >0$ such that for all $1\le i\le n$ and $t\ge 0$,

$$\begin{aligned} \gamma _ib_{ii}(t)-\sum _{j=1,j\ne i}^n\gamma _jb_{ji}(t)\ge \lambda . \end{aligned}$$

(12)

then model (3) is GAS.

Remark 1

Let us compare our Corollary 8 with the results in Li and Mao [3]. Clearly, (12) is weaker than (4). For example, consider the following model

$$\begin{aligned} \left\{ \begin{array}{lll} \!\!dx_1(t)\displaystyle =x_1(t)\bigg [0.4-(0.3+0.04\sin t)x_1(t)- (0.25+0.05\sin t)x_2(t)\bigg ]dt\\ \,\,\quad \quad \quad \quad +\,0.5x_1(t)dB_1(t),\\ \!\!dx_2(t)\displaystyle =x_2(t)\bigg [0.3-(0.27+0.1\sin t)x_1(t)-(0.35+0.05\sin t)x_2(t)\bigg ]dt\\ \,\,\quad \qquad \quad +\,0.6x_2(t)dB_2(t). \end{array}\right. \end{aligned}$$

(13)

Compute that $\min _{t\ge 0}[b_{11}(t)-b_{21}(t)]=-0.03<0, ~b_{22}(t)-b_{12}(t)\equiv 0.1.$ It is easy to see that (4) does not hold. Then the results in [3] can not be used. However, choose $\gamma _1=1.1,~\gamma _2=1,$ then

$$\begin{aligned} \min _{t\ge 0}\{\gamma _1b_{11}(t)-\gamma _2b_{21}(t)\}=0.004>0,~~ \min _{t\ge 0}\{\gamma _2b_{22}(t)-\gamma _1b_{12}(t)\}=0.07>0. \end{aligned}$$

It then follows from our Corollary 8 that model (13) is GAS.

3 Numerical simulations

In this section, we introduce a numerical figure to validate the main results by using the Milstein method (see e.g., [17]). For the sake of simplicity, we choose $n=m=2$, $b_{12}(t)=b_{21}(t)=c_{11}(t)=c_{22}(t)=0$, $\sigma _{21}(t)=0,$ $\nu _{ij}(\theta )=e^{\theta }$, $\theta \in (-\infty ,0],~1\le i,j\le 2.$ Hence, Eq. (1) becomes

$$\begin{aligned} \left\{ \begin{array}{ll} \!\!\displaystyle dx_1(t)\displaystyle =x_1(t)\bigg [r_1(t)-a_{1}(t)x_1(t)-b_{11}(t)x_1(t-\tau _{11}(t))\\ ~~~~~~~~~~~~~~ -c_{12}(t)e^{-t}\int _{-\infty }^0e^{\theta }x_2(\theta )d\theta \displaystyle -c_{12}(t)e^{-t}\int _{0}^te^{\theta } x_2(\theta )d\theta \bigg ]dt \\ ~~~~~~~~~~~~~~ +\sigma _{11}(t)x_1(t)dB_1(t)+\sigma _{12}(t)x_1(t)dB_2(t),\\ \!\!dx_2(t)\displaystyle =x_2(t)\bigg [r_2(t)-a_{2}(t)x_2(t)-b_{22}(t)x_2(t-\tau _{22}(t))\\ ~~~~~~~~~~~~~~ -c_{21}(t)e^{-t}\int _{-\infty }^0e^{\theta }x_1(\theta )d\theta \\ ~~~~~~~~~~~~~~\displaystyle -c_{21}(t)e^{-t}\int _{0}^te^{\theta }x_1(\theta )d\theta \bigg ]dt +\sigma _{22}(t)x_2(t)dB_2(t), \end{array}\right. \end{aligned}$$

(14)

In Fig. 1, we choose $r_1(t)=0.4+0.1\sin t$, $r_2(t)=0.5+0.02\sin t$, $a_1(t)\equiv 0.7$, $a_2(t)\equiv 0.6$, $b_{11}(t)\equiv 0.3$, $b_{22}(t)\equiv 0.15$, $c_{12}(t)\equiv 0.2$, $c_{21}(t)\equiv 0.41$, $\tau _{11}(t)=\tau _{22}(t)=0.2t$, $\sigma _{11}(t)=\sigma _{12}(t) \equiv 0.2,~\sigma _{22}(t)\equiv 0.3$. Let $\gamma _1=1$, $\gamma _2=0.7$, then

$$\begin{aligned} \gamma _1 \bigg (a_1-\frac{b_{11}}{0.75}\bigg )-\gamma _2c_{21}=0.013 >0,~\gamma _2\bigg (a_2-\frac{b_{22}}{0.75}\bigg )-\gamma _1c_{12}=0.08>0. \end{aligned}$$

Then by Theorem 7, model (14) is GAS. Figure 1 confirms this.

Fig. 1

Solution of model (14) for $r_1(t)=0.4+0.1\sin t$, $r_2(t)=0.5+0.02\sin t$, $a_1(t)\equiv 0.7$, $a_2(t)\equiv 0.6$, $b_{11}(t)\equiv 0.3$, $b_{22}(t)\equiv 0.15$, $c_{12}(t)\equiv 0.2$, $c_{21}(t)\equiv 0.41$, $\tau _{11}(t)=\tau _{22}(t)=0.2t$, $\sigma _{11}(t)=\sigma _{12}(t)\equiv 0.2,~\sigma _{22}(t)\equiv 0.3$, step size $\Delta t=0.001$, initial data $x_1(\theta )=0.8e^\theta ,~x_2(\theta )=0.3e^\theta ,$ $y_1(\theta )=0.6e^\theta ,~y_2(\theta )=0.1e^\theta , \theta \in (-\infty ,0]$

4 Conclusions

This paper is concerned with a stochastic non-autonomous Lotka–Volterra competitive system with infinite delays. Sufficient conditions for GAS are obtained. Some recent results are extended and improved. In recent years, GAS of the Lotka–Volterra competitive model with time delays has been investigated extensively (see e.g. [18–33] and the references cited therein), but mainly in deterministic case. This paper is the first one, to the best of our knowledge, to consider the stochastic case.

Some interesting problems deserve further investigation. It is of interest to consider the permanence of model (1). Another problem of interest is to consider other environmental noises, for example, telephone noises (see e.g. [4]) and Lévy jump noises (see e.g. [5]).