Dynamics of a nonautonomous Lotka-Volterra predator-prey dispersal system with impulsive effects

Abstract

By applying the comparison theorem, Lyapunov functional, and almost periodic functional hull theory of the impulsive differential equations, this paper gives some new sufficient conditions for the uniform persistence, global asymptotical stability, and almost periodic solution to a nonautonomous Lotka-Volterra predator-prey dispersal system with impulsive effects. The main results of this paper extend some corresponding results obtained in recent years. The method used in this paper provides a possible method to study the uniform persistence, global asymptotical stability, and almost periodic solution of the models with impulsive perturbations in biological populations.

MSC:34K14, 34K20, 34K45, 92D25.

1 Introduction

Because of the ecological effects of human activities and industry, more and more habitats are broken into patches and some of them are polluted. Negative feedback crowding or the effect of the past life history of the species on its present birth rate are common examples illustrating the biological meaning of time delays and justifying their use in these systems. Recently, diffusions have been introduced into Lotka-Volterra type systems. The effect of an environment change in the growth and diffusion of a species in a heterogeneous habitat is a subject of considerable interest in the ecological literature [1–7].

As was pointed out by Berryman [8], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. In recent years, the predator-prey system has been extensively studied by many scholars, many excellent results were obtained concerned with the persistent property and positive periodic solution of the system; see [9–15] and the references cited therein.

Considering the effect of almost periodically varying environment is an important selective forces on systems in a fluctuating environment, Meng and Chen [16] studied the case of combined effects: dispersion, time delays, almost periodicity of the environment. Namely, they investigated the following general nonautonomous Lotka-Volterra type predator-prey dispersal system:

$$\{\begin{array}{l}{\dot{x}}_1(t)=x_1(t)[r_1(t)-a_1(t)x_1(t)-b_1(t)x_1(t-{\tau }_1(t))\\ \phantom{{\dot{x}}_1(t)=}-{\int }_{-{\sigma }_1}^0{k}_1(t,s)x_1(t+s)\mathrm{d}s-\frac{c(t)y(t)}{1+\alpha (t)x_1(t)}]+{\sum }_{i=2}^n{D}_{i1}(t)[x_i(t)-x_1(t)],\\ {\dot{x}}_i(t)=x_i(t)[r_i(t)-a_i(t)x_i(t)-b_i(t)x_i(t-{\tau }_i(t))\\ \phantom{{\dot{x}}_i(t)=}-{\int }_{-{\sigma }_i}^0{k}_i(t,s)x_i(t+s)\mathrm{d}s]+{\sum }_{j=1}^n{D}_{ji}(t)[x_j(t)-x_i(t)],i=2,3,\dots ,n,\\ \dot{y}(t)=y(t)[-r_{n+1}(t)+\frac{f(t)x_1(t)}{1+\alpha (t)x_1(t)}-a_{n+1}(t)y(t)-b_{n+1}(t)y(t-{\tau }_{n+1}(t))\\ \phantom{\dot{y}(t)=}-{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t,s)y(t+s)\mathrm{d}s].\end{array}$$

(1.1)

By using the comparison theorem and functional hull theory of almost periodic system, the authors [16] obtained some sufficient conditions for the uniform persistence, global asymptotical stability, and almost periodic solution to system (1.1).

However, the ecological system is often deeply perturbed by human exploitation activities such as planting and harvesting and so on, which makes it unsuitable to be considered continually. To obtain a more accurate description of such systems, we need to consider impulsive differential equations. In recent years, the impulsive differential equations have been intensively investigated (see [17–29] for more details). To the best of the authors’ knowledge, in the literature, there are few papers concerning the permanence, global asymptotical stability, and almost periodic solution to the Lotka-Volterra type predator-prey dispersal system with impulsive effects. Therefore, we consider the following Lotka-Volterra type predator-prey dispersal system with impulsive effects:

$$\{\begin{array}{l}{\dot{x}}_1(t)=x_1(t)[r_1(t)-a_1(t)x_1(t)-b_1(t)x_1(t-{\tau }_1(t))\\ \phantom{{\dot{x}}_1(t)=}-{\int }_{-{\sigma }_1}^0{k}_1(t,s)x_1(t+s)\mathrm{d}s-\frac{c(t)y(t)}{1+\alpha (t)x_1(t)}]+{\sum }_{i=2}^n{D}_{i1}(t)[x_i(t)-x_1(t)],\\ {\dot{x}}_i(t)=x_i(t)[r_i(t)-a_i(t)x_i(t)-b_i(t)x_i(t-{\tau }_i(t))\\ \phantom{{\dot{x}}_i(t)=}-{\int }_{-{\sigma }_i}^0{k}_i(t,s)x_i(t+s)\mathrm{d}s]+{\sum }_{j=1}^n{D}_{ji}(t)[x_j(t)-x_i(t)],i=2,3,\dots ,n,\\ \dot{y}(t)=y(t)[-r_{n+1}(t)+\frac{f(t)x_1(t)}{1+\alpha (t)x_1(t)}-a_{n+1}(t)y(t)-b_{n+1}(t)y(t-{\tau }_{n+1}(t))\\ \phantom{\dot{y}(t)=}-{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t,s)y(t+s)\mathrm{d}s],t\ne t_k,\\ \mathrm{\Delta }{x}_j(t_k)=h_{jk}{x}_j(t_k),j=1,2,\dots ,n,\\ \mathrm{\Delta }y(t_k)=h_{n+1,k}y(t_k),k\in \mathbb{Z},\end{array}$$

(1.2)

where $x_1$ and y are population density of prey species x and predator species y in patch 1, and $x_i$ is density of prey species x in patch i; predator species y is confined to patch 1, while the prey species x can disperse among n patches; $D_{ij}(t)$ is the dispersion rate of the species from patch j to patch i, the terms $b_i(t)x_i(t-{\tau }_i(t))$ ($i=1,2,\dots ,n$), $b_{n+1}(t)y(t-{\tau }_{n+1}(t))$, ${\int }_{-{\sigma }_i}^0{k}_i(t,s)x_i(t+s)\mathrm{d}s$ ($i=1,2,\dots ,n$) and ${\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t,s)y(t+s)\mathrm{d}s$ represent the negative feedback crowding and the effect of all the past life history of the species on its present birth rate, respectively; $\mathrm{\Delta }{x}_i(t_k)=x_i(t_k^{+})-x_i(t_k^{-})$, $x_i(t_k^{+})$ and $x_i(t_k^{-})$ represent the right and the left limit of $x_i(t_k)$, $x_i(t_k^{-})=x_i(t_k)$, $k\in \mathbb{Z}$, $i=1,2,\dots ,n$. Related to a continuous function f, we use the following notations: $f^l={inf}_{s\in \mathbb{R}}f(s)$, $f^u={sup}_{s\in \mathbb{R}}f(s)$.

In system (1.2), we always assume that for all $i=1,2,\dots ,n+1$, $j=1,2,\dots ,n$:

(H₁) $r_i(t)$, $a_i(t)$, $b_i(t)$, $c(t)$, $f(t)$, $\alpha (t)$ and $D_{ij}(t)$ ($D_{ii}(t)=0$) are nonnegative and continuous almost periodic functions for all $t\in \mathbb{R}$, and $a_i^l+b_i^l>0$.

(H₂) $k_i(t,s)$ are defined on $\mathbb{R}\times (-\mathrm{\infty },0]$ and nonnegative and continuous almost periodic functions with respect to $t\in \mathbb{R}$ and integrable with respect to s on $(-\mathrm{\infty },0]$ such that ${\int }_{-{\sigma }_i}^0{k}_i(t,s)\mathrm{d}s$ is continuous and bounded with respect to $t\in \mathbb{R}$, $0<{\int }_{-{\sigma }_i}^0(-s)k_i^u(s)\mathrm{d}s<+\mathrm{\infty }$.

(H₃) ${\tau }_i(t)$ is continuous and differentiable bounded almost periodic functions on ℝ, and ${inf}_{t\in \mathbb{R}}\{1-{\dot{\tau }}_i(t)\}>0$.

(H₄) The sequences $\{h_{ik}\}$ are almost periodic and $h_{ik}>-1$.

(H₅) The set of sequences $\{t_k^j\}$, $t_k^j=t_{k+j}-t_k$, $k\in \mathbb{Z}$, $j\in \mathbb{Z}$ is uniformly almost periodic and $\theta :={inf}_{k\in \mathbb{Z}}{t}_k^1>0$.

The main purpose of this paper is to establish some new sufficient conditions which guarantee the uniform persistence, global asymptotical stability, and almost periodic solution of system (1.2) by using the comparison theorem, the Lyapunov functional, and almost periodic functional hull theory of the impulsive differential equations [17, 18] (see Theorem 3.1, Theorem 4.1, and Theorem 5.1 in Sections 3-5).

The organization of this paper is as follows. In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections. In Section 3, by using the comparison theorem of the impulsive differential equations, we give the permanence of system (1.2). In Section 4, we study the global asymptotical stability of system (1.2) by constructing a suitable Lyapunov functional. In Section 5, some new sufficient conditions are obtained for the existence, uniqueness, and global asymptotical stability of the positive almost periodic solution of system (1.2).

2 Preliminaries

Now, let us state the following definitions and lemmas, which will be useful in proving our main result.

Let ${\mathbb{R}}^n$ be the n-dimensional Euclidean space with norm $\parallel x\parallel ={\sum }_{i=1}^n|x_i|$. By $\mathbb{I}$, $\mathbb{I}=\{\{t_k\}\in \mathbb{R}:t_k<t_{k+1},k\in \mathbb{Z},{lim}_{k\to \pm \mathrm{\infty }}{t}_k=\pm \mathrm{\infty }\}$, we denote the set of all sequences that are unbounded and strictly increasing with distance $\rho (\{t_k^{(1)}\},\{t_k^{(2)}\})$. Let $\mathrm{\Omega }\subset \mathbb{R}$, $\mathrm{\Omega }\ne \mathrm{\varnothing }$, $\tau :={sup}_{t\in \mathbb{R}}\{{\tau }_i(t):i=1,2,\dots ,n\}$, ${\xi }_0\in \mathbb{R}$, introduce the following notations:

$PC({\xi }_0)$ is the space of all functions $\varphi :[{\xi }_0-\tau ,{\xi }_0]\to \mathrm{\Omega }$ having points of discontinuity at ${\mu }_1,{\mu }_2,\dots \in [{\xi }_0-\tau ,{\xi }_0]$ of the first kind and being left continuous at these points.

For $J\subset \mathbb{R}$, $PC(J,\mathbb{R})$ is the space of all piecewise continuous functions from J to ℝ with points of discontinuity of the first kind $t_k$, at which it is left continuous.

Let ${\varphi }_i,\phi \in PC(0)$. Denote by $x_i(t)=x_i(t;0,{\varphi }_i)$, $y(t)=y(t;0,\phi )$, $x_i,y\in \mathrm{\Omega }$, $i=1,2,\dots ,n$ the solution of system (1.2) satisfying the initial conditions

$$\begin{array}{c}0\le x_i(s;0,{\varphi }_i)={\varphi }_i(s)<+\mathrm{\infty },s\in [-\tau ,0],x_i(0+0;0,{\varphi }_i)={\varphi }_i(0)>0;\hfill \\ 0\le y(s;0,\phi )=\phi (s)<+\mathrm{\infty },s\in [-\tau ,0],y(0+0;0,\phi )=\phi (0)>0.\hfill \end{array}$$

Remark 2.1 The problems of existence, uniqueness, and continuity of the solutions of impulsive differential equations have been investigated by many authors. Efficient sufficient conditions which guarantee the existence of the solutions of such systems are given in [17, 18].

Since the solution of system (1.2) is a piecewise continuous function with points of discontinuity of the first kind $t_k$, $k\in \mathbb{Z}$ we adopt the following definitions for almost periodicity.

Let $T,P\in \mathbb{I}$, $s(T\cup P):\mathbb{I}\to \mathbb{I}$ be a map such that the set $s(T\cup P)$ forms a strictly increasing sequence and if $D\subset \mathbb{R}$ and $ϵ>0$, ${\theta }_{ϵ}(D)=\{t+ϵ:t\in D\}$, $F_{ϵ}(D)=\bigcap \{{\theta }_{ϵ}(D):ϵ>0\}$.

By $\varphi =(\phi (t),T)$ we denote the element from the space $PC\times \mathbb{I}$, and for every sequence of real numbers $\{{\alpha }_n\}$ we let ${\theta }_{{\alpha }_n}\varphi$ denote the sets $\{\phi (t-{\alpha }_n),T-{\alpha }_n\}\subset PC\times \mathbb{I}$, where $T-{\alpha }_n=\{t_k-{\alpha }_n:k\in \mathbb{Z},n=1,2,\dots \}$.

Definition 2.1 ([18])

The set of sequences $\{t_k^j\}$, $t_k^j=t_{k+j}-t_k$, $k\in \mathbb{Z}$, $j\in \mathbb{Z}$, $\{t_k\}\in \mathbb{I}$ is said to be uniformly almost periodic if for arbitrary $ϵ>0$ there exists a relatively dense set of ϵ-almost periods common for any sequences.

Definition 2.2 ([18])

The function $\phi \in PC(\mathbb{R},\mathbb{R})$ is said to be almost periodic, if the following hold:

• (1) The set of sequences $\{t_k^j\}$, $t_k^j=t_{k+j}-t_k$, $k\in \mathbb{Z}$, $j\in \mathbb{Z}$, $\{t_k\}\in \mathbb{I}$ is uniformly almost periodic.

• (2) For any $ϵ>0$ there exists a real number $\delta >0$ such that if the points $t^{\prime }$ and $t^{″}$ belong to one and the same interval of continuity of $\phi (t)$ and satisfy the inequality $|t^{\prime }-t^{″}|<\delta$, then $|\phi (t^{\prime })-\phi (t^{″})|<ϵ$.

• (3) For any $ϵ>0$ there exists a relatively dense set T such that if $\eta \in T$, then $|\phi (t+\eta )-\phi (t)|<ϵ$ for all $t\in \mathbb{R}$ satisfying the condition $|t-t_k|>ϵ$, $k\in \mathbb{Z}$. The elements of T are called ϵ-almost periods.

Lemma 2.1 ([18])

The set of sequences $\{t_k^j\}$, $t_k^j=t_{k+j}-t_k$, $k\in \mathbb{Z}$, $j\in \mathbb{Z}$, $\{t_k\}\in \mathbb{I}$ is uniformly almost periodic if and only if from each infinite sequence of shifts $\{t_k-{\alpha }_n\}$, $k\in \mathbb{Z}$, $n=1,2,\dots$ , ${\alpha }_n\in \mathbb{R}$, we can choose a subsequence which is convergent in $\mathbb{I}$.

Definition 2.3 ([18])

The sequence ${\varphi }_n$, ${\varphi }_n=({\phi }_n(t),T_n)\in PC\times \mathbb{I}$ is uniformly convergent to ϕ, $\varphi =(\phi (t),T)\in PC\times \mathbb{I}$ if and only if for any $ϵ>0$ there exists $n_0>0$ such that

$$\rho (T,T_n)<ϵ,\parallel {\phi }_n(t)-\phi (t)\parallel <ϵ$$

hold uniformly for $n\ge n_0$ and $t\in \mathbb{R}\setminus F_{ϵ}(s(T_n\cup T))$.

Definition 2.4 ([18])

The function $\varphi \in PC$ is said to be an almost periodic piecewise continuous function with points of discontinuity of the first kind from the set T if for every sequence of real numbers $\{{\alpha }_m^{\prime }\}$ there exists a subsequence $\{{\alpha }_n\}$ such that ${\theta }_{{\alpha }_n}\varphi$ is compact in $PC\times \mathbb{I}$.

Lemma 2.2 ([18])

Let $\{t_k\}\in \mathbb{I}$. Then there exists a positive integer A such that on each interval of length 1, we have no more than A elements of the sequence $\{t_k\}$, i.e.,

$$i(s,t)\le A(t-s)+A,$$

where $i(s,t)$ is the number of the points $t_k$ in the interval $(s,t)$.

Lemma 2.3 Let $\{t_k\}\in \mathbb{I}$. Then

$$i(s,t)\ge \frac{t-s}{\theta }-1,$$

where $i(s,t)$ is the number of the points $t_k$ in the interval $(s,t)$.

Proof The proof of this lemma is easy and we omit it. This completes the proof. □

3 Uniform persistence

In this section, we establish a uniform persistence result for system (1.2).

Lemma 3.1 ([17])

Assume that $x\in PC(\mathbb{R})$ with points of discontinuity at $t=t_k$ and is left continuous at $t=t_k$ for $k\in {\mathbb{Z}}^{+}$, and

$$\{\begin{array}{l}\dot{x}(t)\le f(t,x(t)),t\ne t_k,\\ x(t_k^{+})\le I_k(x(t_k)),k\in {\mathbb{Z}}^{+},\end{array}$$

(3.1)

where $f\in C(\mathbb{R}\times \mathbb{R},\mathbb{R})$, $I_k\in C(\mathbb{R},\mathbb{R})$ and $I_k(x)$ is nondecreasing in x for $k\in {\mathbb{Z}}^{+}$. Let $u^{\ast }(t)$ be the maximal solution of the scalar impulsive differential equation

$$\{\begin{array}{l}\dot{u}(t)=f(t,u(t)),t\ne t_k,\\ u(t_k^{+})=I_k(u(t_k))\ge 0,k\in {\mathbb{Z}}^{+},\\ u(t_0^{+})=u_0\end{array}$$

(3.2)

existing on $[t_0,\mathrm{\infty })$. Then $x(t_0^{+})\le u_0$ implies $x(t)\le u^{\ast }(t)$ for $t\ge t_0$.

Remark 3.1 If the inequalities (3.1) in Lemma 3.1 is reversed and $u_{\ast }(t)$ is the minimal solution of system (3.2) existing on $[t_0,\mathrm{\infty })$, then $x(t_0^{+})\ge u_0$ implies $x(t)\ge u_{\ast }(t)$ for $t\ge t_0$.

Lemma 3.2 Assume that $a\theta >{\xi }^l$, $b>0$, $h_k>-1$, and $x(t)>0$ is a solution of the following impulsive logistic equation:

$$\{\begin{array}{l}\dot{x}(t)=x(t)[a-bx(t)],t\ne t_k,\\ \mathrm{\Delta }x(t_k)=h_{k}x(t_k),k\in \mathbb{Z},\end{array}$$

(3.3)

then

$$lim\underset{t\to +\mathrm{\infty }}{sup}x(t)\le \frac{e^{{\xi }^l}(a\theta -{\xi }^l)}{b\theta },$$

where ${\xi }^l:=ln{inf}_{k\in \mathbb{Z}}\frac{1}{1+h_k}$.

Proof Let $u=\frac{1}{x}$, then system (3.3) changes to

$$\{\begin{array}{l}\frac{\mathrm{d}u(t)}{\mathrm{d}t}=-au(t)+b,t\ne t_k,\\ u(t_k^{+})=\frac{u(t_k)}{1+h_k},k\in \mathbb{Z}.\end{array}$$

Similar to the proof in [18], we can obtain from Lemma 2.3

$$\begin{array}{rcl}u(t)& =& W(t,0)u(0)+b{\int }_0^{t}W(t,s)\mathrm{d}s\\ =& \prod _{t_k\in [0,t]}\frac{1}{1+h_k}{e}^{-at}u(0)+b{\int }_0^t\prod _{t_k\in [s,t]}\frac{1}{1+h_k}{e}^{-a(t-s)}\mathrm{d}s\\ =& {\left[\frac{1}{1+h_k}\right]}^{\frac{t}{\theta }-1}{e}^{-at}u(0)+b{\int }_0^t{\left[\frac{1}{1+h_k}\right]}^{\frac{t-s}{\theta }-1}{e}^{-a(t-s)}\mathrm{d}s\\ \ge & e^{-{\xi }^l}{e}^{-(a-\frac{{\xi }^l}{\theta })t}u(0)+b{\int }_0^t{e}^{-{\xi }^l}{e}^{-(a-\frac{{\xi }^l}{\theta })(t-s)}\mathrm{d}s\\ =& e^{-{\xi }^l}{e}^{-(a-\frac{{\xi }^l}{\theta })t}u(0)+\frac{b{e}^{-{\xi }^l}[1-e^{-(a-\frac{{\xi }^l}{\theta })t}]}{a-\frac{{\xi }^l}{\theta }},\end{array}$$

(3.4)

where

$$W(t,s)=\{\begin{array}{ll}{e}^{-a(t-s)},& t_{k-1}<s<t<t_k;\\ {\prod }_{j=m}^{k+1}\frac{1}{1+h_j}{e}^{-a(t-s)},& t_{m-1}<s\le t_m<t_k<t\le t_{k+1}.\end{array}$$

Then

$$lim\underset{t\to +\mathrm{\infty }}{sup}x(t)=lim\underset{t\to +\mathrm{\infty }}{sup}{[u(t)]}^{-1}\le \frac{e^{{\xi }^l}(a\theta -{\xi }^l)}{b\theta }.$$

This completes the proof. □

Lemma 3.3 Assume that $a>{\xi }^{u}A$, $b>0$, $h_k>-1$ and $x(t)>0$ is a solution of the following impulsive logistic equation:

$$\{\begin{array}{l}\dot{x}(t)=x(t)[a-bx(t)],t\ne t_k,\\ \mathrm{\Delta }x(t_k)=h_{k}x(t_k),k\in \mathbb{Z},\end{array}$$

(3.5)

then

$$lim\underset{t\to +\mathrm{\infty }}{inf}x(t)\ge \frac{a-{\xi }^{u}A}{b{e}^{{\xi }^{u}A}},$$

where A is defined as that in Lemma  2.2, ${\xi }^u:=ln{sup}_{k\in \mathbb{Z}}\frac{1}{1+h_k}$.

Proof Let $u=\frac{1}{x}$, then system (3.5) changes to

$$\{\begin{array}{l}\frac{\mathrm{d}u(t)}{\mathrm{d}t}=-au(t)+b,t\ne t_k,\\ u(t_k^{+})=\frac{u(t_k)}{1+h_k},k\in \mathbb{Z}.\end{array}$$

Similar to the proof as that in (3.4), we can obtain from Lemma 2.2

$$\begin{array}{rcl}u(t)& =& W(t,0)u(0)+b{\int }_0^{t}W(t,s)\mathrm{d}s\\ \le & \prod _{t_k\in [0,t]}\frac{1}{1+h_k}{e}^{-at}u(0)+b{\int }_0^t\prod _{t_k\in [s,t]}\frac{1}{1+h_k}{e}^{-a(t-s)}\mathrm{d}s\\ \le & {\left[\frac{1}{1+h_k}\right]}^{At+A}{e}^{-at}u(0)+b{\int }_0^t{\left[\frac{1}{1+h_k}\right]}^{A(t-s)+A}{e}^{-a(t-s)}\mathrm{d}s\\ \le & e^{{\xi }^{u}A}{e}^{-(a-{\xi }^{u}A)t}u(0)+b{\int }_0^t{e}^{{\xi }^{u}A}{e}^{-(a-{\xi }^{u}A)(t-s)}\mathrm{d}s\\ =& e^{{\xi }^{u}A}{e}^{-(a-{\xi }^{u}A)t}u(0)+\frac{b{e}^{{\xi }^{u}A}[1-e^{-(a-{\xi }^{u}A)t}]}{a-{\xi }^{u}A},\end{array}$$

which implies that

$$lim\underset{t\to +\mathrm{\infty }}{inf}x(t)=lim\underset{t\to +\mathrm{\infty }}{inf}{[u(t)]}^{-1}\ge \frac{a-{\xi }^{u}A}{b{e}^{{\xi }^{u}A}}.$$

This completes the proof. □

Lemma 3.4 Assume that $a\theta >{\xi }^l$ and for $x(t)>0$, we have

$$\{\begin{array}{l}\dot{x}(t)\le x(t)[a-b_{0}x(t)-b_{1}x(t-\tau (t))],t\ne t_k,\\ \mathrm{\Delta }x(t_k)\le h_{k}x(t_k),k\in \mathbb{Z},\end{array}$$

(3.6)

where

$$a>0,b_0,b_1\ge 0,b_0+b_1>0.$$

Then there exists a positive constant M such that

$$lim\underset{t\to +\mathrm{\infty }}{sup}x(t)\le \frac{e^{{\xi }^l}(a\theta -{\xi }^l)}{B\theta }:=M,$$

where $B=b_0+{inf}_{t\in \mathbb{R}}{b}_1{\prod }_{t_k\in [t-\tau (t),t)}{(1+h_k)}^{-1}{e}^{-a\tau (t)}$.

Proof From system (3.6), we have

$$\{\begin{array}{l}\dot{x}(t)\le ax(t),t\ne t_k,\\ \mathrm{\Delta }x(t_k)\le h_{k}x(t_k),k\in \mathbb{Z},\end{array}$$

is equivalent to

$$\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{dt}}[x(t)e^{-at}]\le 0,t\ne t_k,\\ \mathrm{\Delta }x(t_k)\le h_{k}x(t_k),k\in {\mathbb{Z}}^{+}.\end{array}$$

(3.7)

For some $t\in [0,+\mathrm{\infty })$ and $t\ne t_k$, $k\in {\mathbb{Z}}^{+}$, consider interval $[t-\tau (t),t)$. Assume that $t_1<t_2<\cdots <t_j$ are the impulse points in $[t-\tau (t),t)$. Integrating the first inequality of system (3.7) from $t-\tau (t)$ to $t_1$ leads to

$$x(t_1)e^{-a{t}_1}\le x(t-\tau (t))e^{-a(t-\tau (t))}.$$

Integrating the first inequality of system (3.7) from $t_1$ to $t_2$ leads to

$$x(t_2)e^{-a{t}_2}\le x\left(t_1^{+}\right)e^{-a{t}_1}\le (1+h_1)x(t_1)e^{-a{t}_1}\le (1+h_1)x(t-\tau (t))e^{-a(t-\tau (t))}.$$

Repeating the above process, integrating the first inequality of system (3.7) from $t_j$ to t leads to

$$x(t)e^{-at}\le x\left(t_j^{+}\right)e^{-a{t}_j}\le (1+h_j)x(t_j)e^{-a{t}_j}\le \prod _{t_k\in [t-\tau (t),t)}(1+h_k)x(t-\tau (t))e^{-a(t-\tau (t))}.$$

Then

$$x(t-\tau (t))\ge \prod _{t_k\in [t-\tau (t),t)}{(1+h_k)}^{-1}{e}^{-a\tau (t)}x(t).$$

(3.8)

Substituting (3.8) into system (3.7) leads to

$$\{\begin{array}{l}\dot{x}(t)\le x(t)[a-Bx(t)],t\ne t_k,\\ \mathrm{\Delta }x(t_k)\le h_{k}x(t_k),k\in \mathbb{Z}.\end{array}$$

Consider the auxiliary system

$$\{\begin{array}{l}\dot{z}(t)=z(t)[a-Bz(t)],t\ne t_k,\\ z(t_k^{+})=(1+h_k)z(t_k),k\in \mathbb{Z},\\ z(0^{+})=x(0^{+}).\end{array}$$

(3.9)

By Lemma 3.1, $x(t)\le z(t)$, where $z(t)$ is the solution of system (3.9). By Lemma 3.2, we have from (3.9)

$$lim\underset{t\to +\mathrm{\infty }}{sup}x(t)\le lim\underset{t\to +\mathrm{\infty }}{sup}z(t)\le \frac{e^{{\xi }^l}(a\theta -{\xi }^l)}{B\theta }.$$

This completes the proof. □

Lemma 3.5 Assume that $a>{\xi }^{u}A$, for $x(t)>0$ and $lim{sup}_{t\to +\mathrm{\infty }}x(t)\le M$, we have

$$\{\begin{array}{l}\dot{x}(t)\ge x(t)[a-b_{0}x(t)-b_{1}x(t-\tau (t))],t\ne t_k,\\ \mathrm{\Delta }x(t_k)=h_{k}x(t_k),\end{array}$$

(3.10)

where

$$a>K+{\xi }^{u}A,b_0,b_1\ge 0,b:=b_0+b_1>0,k\in \mathbb{Z}.$$

Then there exists a positive constant N such that

$$lim\underset{t\to +\mathrm{\infty }}{inf}x(t)\ge \frac{a-{\xi }^{u}A}{D{e}^{{\xi }^{u}A}}:=N,$$

where

$$D:=b_0+\underset{t\in \mathbb{R}}{sup}{b}_1\prod _{t_k\in [t-\tau (t),t)}{(1+h_k)}^{-1}{e}^{-[a-bM]\tau (t)}.$$

Proof According to the assumption, for $\mathrm{\forall }{ϵ}_1>0$, there exists $T_1>0$ such that

$$x(t)\le M+{ϵ}_1\text{for }t\ge T_1.$$

From system (3.10), we have

$$\{\begin{array}{l}\dot{x}(t)\ge [a-b(M+{ϵ}_1)]x(t):=L_{{ϵ}_1}x(t),t\ne t_k,t\ge T_1,\\ \mathrm{\Delta }x(t_k)=h_{k}x(t_k)+d_k,k\in \mathbb{Z},\end{array}$$

is equivalent to

$$\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{dt}}[x(t)e^{-L_{{ϵ}_1}t}]\ge 0,t\ne t_k,t\ge T_1,\\ \mathrm{\Delta }x(t_k)=h_{k}x(t_k)+d_k,k\in \mathbb{Z}.\end{array}$$

(3.11)

Similar to the arguments in (3.8), we obtain

$$x(t-\tau (t))\le \prod _{t_k\in [t-\tau (t),t)}{(1+h_k)}^{-1}{e}^{-L_{{ϵ}_1}\tau (t)}x(t).$$

(3.12)

Let

$$D_{{ϵ}_1}:=b_0+\underset{t\in \mathbb{R}}{sup}{b}_1\prod _{t_k\in [t-\tau (t),t)}{(1+h_k)}^{-1}{e}^{-[a-b(M+{ϵ}_1)]\tau (t)}.$$

Substituting (3.12) into system (3.10) leads to

$$\{\begin{array}{l}\dot{x}(t)\ge x(t)[a-D_{{ϵ}_1}x(t)],t\ne t_k,t\ge T_1,\\ \mathrm{\Delta }x(t_k)=h_{k}x(t_k),k\in \mathbb{Z}.\end{array}$$

Consider the auxiliary system

$$\{\begin{array}{l}\dot{z}(t)=z(t)[a-D_{{ϵ}_1}z(t)],t\ne t_k,t\ge T_1,\\ z(t_k^{+})=(1+h_k)z(t_k),k\in \mathbb{Z},\\ z(T_1^{+})=x(T_1^{+}).\end{array}$$

(3.13)

By Remark 3.1, $x(t)\ge z(t)$, where $z(t)$ is the solution of system (3.13). By Lemma 3.3, we have from (3.13)

$$lim\underset{t\to +\mathrm{\infty }}{inf}x(t)\ge lim\underset{t\to +\mathrm{\infty }}{inf}z(t)\ge \frac{a-{\xi }^{u}A}{D{e}^{{\xi }^{u}A}}.$$

This completes the proof. □

Let

$$\begin{array}{c}{r}^u:=\underset{1\le i\le n}{max}{r}_i^u,a^l:=\underset{1\le i\le n}{min}{a}_i^l,h_k^u:=\underset{1\le i\le n}{max}{h}_{ik},k\in \mathbb{Z},\hfill \\ {\xi }^l:=ln\underset{k\in \mathbb{Z}}{inf}\frac{1}{1+h_k^u},{\xi }_{n+1}^l:=ln\underset{k\in \mathbb{Z}}{inf}\frac{1}{1+h_{n+1,k}}.\hfill \end{array}$$

Proposition 3.1 Every solution $x(t)={(x_1(t),x_2(t),\dots ,x_n(t),y(t))}^T$ of system (1.2) satisfies

$$\begin{array}{c}lim\underset{t\to \mathrm{\infty }}{sup}{x}_i(t)\le M_i:=\frac{e^{{\xi }^l}(r^u\theta -{\xi }^l)}{a^l\theta },\hfill \\ lim\underset{t\to \mathrm{\infty }}{sup}y(t)\le M_{n+1}:=\frac{e^{{\xi }_{n+1}^l}(r_y^u\theta -{\xi }_{n+1}^l)}{B_{n+1}\theta },\hfill \end{array}$$

if the following condition holds:

(H₆) $r^u\theta >{\xi }^l$, $r_y^u\theta >{\xi }_{n+1}^l$, $j=1,2,\dots ,n$,

where $B_{n+1}:=a_{n+1}^l+{inf}_{t\in \mathbb{R}}{b}_{n+1}^l{\prod }_{t_k\in [t-{\tau }_{n+1}(t),t)}{(1+h_{n+1,k})}^{-1}{e}^{-r_{n+1}^u{\tau }_{n+1}(t)}$, $r_y^u:=\frac{f^u{M}_1}{1+{\alpha }^l{M}_1}$.

Proof Define $V(t)=max\{x_1(t),x_2(t),\dots ,x_n(t)\}$ for $t\ge 0$. For any $t^0\ge 0$ and $t^0\ne t_k$, $k\in \mathbb{Z}$, there must exist $i\in \{1,2,\dots ,n\}$ and $\delta >t^0$ small enough such that $V(t^0)=x_i(t^0)$ and $x_j(s)\le x_i(s)$ for $\mathrm{\forall }s\in [t^0,\delta )$, $j\ne i$, $i,j\in \{1,2,\dots ,n\}$. Calculating the upper right derivative of $V(t_0)$ from the positive solution for system (1.2), we have

$$D^{+}V\left(t^0\right)={\dot{x}}_i\left(t^0\right)\le x_i\left(t^0\right)[r_i^u-a_i^l{x}_i\left(t^0\right)]\le V\left(t^0\right)[r^u-a^{l}V\left(t^0\right)].$$

By the arbitrariness of $t^0$, we have

$$D^{+}V(t)\le V(t)[r^u-a^{l}V(t)],\mathrm{\forall }t\ne t_k,k\in \mathbb{Z}.$$

(3.14)

Observe that $x_i(t_k^{+})=(1+h_{ik})x_i(t_k)$ and $1+h_{ik}>0$, $k\in \mathbb{Z}$. For arbitrary impulse point $t_k$, there exists $i_0\in \{1,2,\dots ,n\}$ such that $V(t_k)=max\{x_1(t_k),x_2(t_k),\dots ,x_n(t_k)\}=x_{i_0}(t_k)$, that is,

$$V\left(t_k^{+}\right)=x_{i_0}\left(t_k^{+}\right)=(1+h_{i_{0}k})x_{i_0}(t_k)\le (1+h_k^u)V(t_k),k\in \mathbb{Z}.$$

(3.15)

By Lemma 3.4, we obtain from (3.14)-(3.15)

$$lim\underset{t\to \mathrm{\infty }}{sup}{x}_i(t)\le lim\underset{t\to \mathrm{\infty }}{sup}V(t)\le M_i,i=1,2,\dots ,n.$$

For any positive constant ${ϵ}_2>0$, there exists $T_2>0$ such that

$$x_i(t)\le M_i+{ϵ}_2\text{for }t\ge T_2,i=1,2,\dots ,n.$$

In view of system (1.2), it follows that

$$\{\begin{array}{l}\dot{y}(t)\le y(t)[\frac{f^u(M_1+{ϵ}_2)}{1+{\alpha }^l(M_1+{ϵ}_2)}-a_{n+1}^{l}y(t)-b_{n+1}^{l}y(t-{\tau }_{n+1}(t))],t\ne t_k,\\ \mathrm{\Delta }y(t_k)=h_{n+1,k}y(t_k),k\in \mathbb{Z},\end{array}$$

which implies from Lemma 3.4 that

$$lim\underset{t\to \mathrm{\infty }}{sup}y(t)\le M_{n+1}.$$

This completes the proof. □

Define

$${\xi }_i^u:=ln\underset{k\in \mathbb{Z}}{sup}\frac{1}{1+h_{ik}},i=1,2,\dots ,n,n+1.$$

Proposition 3.2 Assume that the following condition (H₇) holds:

$$\begin{array}{c}{p}_1:=r_1^l-\sum _{i=2}^n{D}_{i1}^u-{\int }_{-{\sigma }_1}^0{k}_1^u(s)\mathrm{d}s{M}_1-c^u{M}_{n+1}\ge {\xi }_1^{u}A,\hfill \\ p_i:=r_i^l-\sum _{j=1}^n{D}_{ji}^u-{\int }_{-{\sigma }_i}^0{k}_i^u(s)\mathrm{d}s{M}_i\ge {\xi }_i^{u}A,i=2,\dots ,n,\hfill \\ p_{n+1}:=-r_{n+1}^u+\frac{f^l{N}_1}{1+{\alpha }^u{N}_1}-{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}^u(s)\mathrm{d}s{M}_{n+1}\ge {\xi }_{n+1}^{u}A,\hfill \end{array}$$

then every solution $x(t)={(x_1(t),x_2(t),\dots ,x_n(t),y(t))}^T$ of system (1.2) satisfies

$$\begin{array}{c}lim\underset{t\to +\mathrm{\infty }}{inf}{x}_i(t)\ge N_i:=\frac{p_i-{\xi }_i^{u}A}{Q_i{e}^{{\xi }_i^{u}A}},\hfill \\ lim\underset{t\to +\mathrm{\infty }}{inf}y(t)\ge N_{n+1}:=\frac{p_{n+1}-{\xi }_{n+1}^{u}A}{Q_{n+1}{e}^{{\xi }_{n+1}^{u}A}},\hfill \end{array}$$

where

$$Q_i:=a_i^u+\underset{t\in \mathbb{R}}{sup}{b}_i^u\prod _{t_k\in [t-{\tau }_i(t),t)}{(1+h_{ik})}^{-1}{e}^{-[p_i-(a_i^u+b_i^u)M_i]{\tau }_i(t)},i=1,2,\dots ,n+1.$$

Proof For $\mathrm{\forall }{ϵ}_3>0$, there exists $T_3>0$ such that

$$x_i(t)\le M_i+{ϵ}_3,y(t)\le M_{n+1}+{ϵ}_3\text{for }t\ge T_3,i=1,2,\dots ,n.$$

From system (1.2), for $t\ge T_3$, we have

$$\{\begin{array}{l}{\dot{x}}_1(t)\ge x_1(t)[r_1^l-{\sum }_{i=2}^n{D}_{i1}^u-a_1^u{x}_1(t)-b_1^u{x}_1(t-{\tau }_1(t))\\ \phantom{{\dot{x}}_1(t)\ge }-{\int }_{-{\sigma }_1}^0{k}_1^u(s)\mathrm{d}s(M_1+{ϵ}_2)-c^u(M_{n+1}+{ϵ}_2)],\\ {\dot{x}}_i(t)\ge x_i(t)[r_i^l-{\sum }_{j=1}^n{D}_{ji}^u-a_i^u{x}_i(t)-b_i^u{x}_i(t-{\tau }_i(t))\\ \phantom{{\dot{x}}_i(t)\ge }-{\int }_{-{\sigma }_i}^0{k}_i^u(s)\mathrm{d}s(M_i+{ϵ}_2)],i=2,3,\dots ,n,t\ne t_k,\\ \mathrm{\Delta }{x}_i(t_k)=h_{ik}{x}_i(t_k),k\in \mathbb{Z}.\end{array}$$

By Lemma 3.5 and the arbitrariness of ${ϵ}_3$, we have

$$lim\underset{t\to +\mathrm{\infty }}{inf}{x}_i(t)\ge N_i,i=1,2,\dots ,n.$$

Then for $\mathrm{\forall }{ϵ}_4>0$, there exists $T_4>0$ such that

$$x_1(t)\ge N_1-{ϵ}_4,y(t)\le M_{n+1}+{ϵ}_4\text{for }t\ge T_4.$$

From system (1.2), for $t\ge T_4$, we have

$$\{\begin{array}{l}\dot{y}(t)\ge y(t)[-r_{n+1}^u+\frac{f(t)(N_1-{ϵ}_4)}{1+\alpha (t)(N_1-{ϵ}_4)}-a_{n+1}^{u}y(t)-b_{n+1}^{u}y(t-{\tau }_{n+1}(t))\\ \phantom{\dot{y}(t)\ge }-{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}^u(s)\mathrm{d}s(M_{n+1}+{ϵ}_4)],t\ne t_k,\\ \mathrm{\Delta }y(t_k)=h_{n+1,k}y(t_k),k\in \mathbb{Z}.\end{array}$$

By Lemma 3.5 and the arbitrariness of ${ϵ}_4$, we have

$$lim\underset{t\to +\mathrm{\infty }}{inf}y(t)\ge N_{n+1}.$$

This completes the proof. □

Remark 3.2 When $h_{ik}(i=1,2,\dots ,n+1,k\in \mathbb{Z})\equiv 0$ in system (1.2), then Propositions 3.1 and 3.2 improve the corresponding results in [16]. So Propositions 3.1 and 3.2 extend and improve the corresponding results in [16].

Remark 3.3 In view of Propositions 3.1 and 3.2, the distance θ between impulse points, the values of impulse coefficients $h_{ik}$ ($i=1,2,\dots ,n+1$, $k\in \mathbb{Z}$) and the number A of the impulse points in each interval of length 1 have negative effect on the uniform persistence of system (1.2).

By Propositions 3.1 and 3.2, we have:

Theorem 3.1 Assume that (H₁)-(H₇) hold, then system (1.2) is uniformly persistent.

Remark 3.4 Theorem 3.1 gives the sufficient conditions for the uniform persistence of system (1.2). Therefore, Theorem 3.1 provides a possible method to study the permanence of the models with almost periodic impulsive perturbations in biological populations.

4 Global asymptotical stability

The main result of this section concerns the global asymptotical stability of positive solution of system (1.2).

Theorem 4.1 Assume that (H₁)-(H₇) hold. Suppose further that

(H₈) there exist positive constants ${\lambda }_i$ such that

$$\begin{array}{c}\underset{t\in \mathbb{R}}{inf}[{\lambda }_1{a}_1(t)-\frac{{\lambda }_1{b}_1({\delta }_1^{-1}(t))}{1-{\dot{\tau }}_1({\delta }_1^{-1}(t))}-{\lambda }_1{\int }_{-{\sigma }_1}^0{k}_1(t-s,s)\mathrm{d}s\hfill \\ -\frac{\alpha (t)c(t)M_{n+1}}{{[1+\alpha (t)N_1]}^2}-\sum _{j=1}^n\frac{{\lambda }_1{D}_{j1}(t)}{N_1}-\frac{{\lambda }_{n+1}f(t)}{1+\alpha (t)N_1}]>0,\hfill \\ \underset{t\in \mathbb{R}}{inf}[{\lambda }_i{a}_i(t)-\frac{{\lambda }_i{b}_i({\delta }_i^{-1}(t))}{1-{\dot{\tau }}_i({\delta }_i^{-1}(t))}-{\lambda }_i{\int }_{-{\sigma }_i}^0{k}_i(t-s,s)\mathrm{d}s\hfill \\ -\sum _{j=1}^n\frac{{\lambda }_j{D}_{ij}(t)}{N_j}-\frac{{\lambda }_{n+1}f(t)}{1+\alpha (t)N_1}]>0,\hfill \\ \underset{t\in \mathbb{R}}{inf}[{\lambda }_{n+1}{a}_{n+1}(t)-\frac{{\lambda }_{n+1}{b}_{n+1}({\delta }_{n+1}^{-1}(t))}{1-{\dot{\tau }}_{n+1}({\delta }_{n+1}^{-1}(t))}\hfill \\ -{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t-s,s)\mathrm{d}s-\frac{c(t)}{1+\alpha (t)N_1}]>0,\hfill \end{array}$$

where ${\delta }_j^{-1}$ is an inverse function of ${\tau }_j$, $i=2,\dots ,n$, $j=1,2,\dots ,n+1$.

Then system (1.2) is globally asymptotically stable.

Proof Suppose that $X(t)={(x_1(t),\dots ,x_n(t),y(t))}^T$ and $X^{\ast }(t)={(x_1^{\ast }(t),\dots ,x_n^{\ast }(t),y^{\ast }(t))}^T$ are any two solutions of system (1.2).

By Theorem 3.1 and (H₈), for ${ϵ}_5>0$ small enough, there exist $T_5>0$ and $\mathrm{\Theta }>0$ such that

$$\begin{array}{c}0<N_i-{ϵ}_5\le x_i(t)\le M_i+{ϵ}_5,0<N_{n+1}-{ϵ}_5\le y(t)\le M_{n+1}+{ϵ}_5\text{for }t\ge T_5,\hfill \\ \underset{t\in \mathbb{R}}{inf}[{\lambda }_1{a}_1(t)-\frac{{\lambda }_1{b}_1({\delta }_1^{-1}(t))}{1-{\dot{\tau }}_1({\delta }_1^{-1}(t))}-{\lambda }_1{\int }_{-{\sigma }_1}^0{k}_1(t-s,s)\mathrm{d}s\hfill \\ -\frac{\alpha (t)c(t)(M_{n+1}+{ϵ}_5)}{{[1+\alpha (t)(N_1-{ϵ}_5)]}^2}-\sum _{j=1}^n\frac{{\lambda }_1{D}_{j1}(t)}{N_1-{ϵ}_5}-\frac{{\lambda }_{n+1}f(t)}{[1+\alpha (t)(N_1-{ϵ}_5)]}]>\mathrm{\Theta },\hfill \\ \underset{t\in \mathbb{R}}{inf}[{\lambda }_i{a}_i(t)-\frac{{\lambda }_i{b}_i({\delta }_i^{-1}(t))}{1-{\dot{\tau }}_i({\delta }_i^{-1}(t))}-{\lambda }_i{\int }_{-{\sigma }_i}^0{k}_i(t-s,s)\mathrm{d}s\hfill \\ -\sum _{j=1}^n\frac{{\lambda }_j{D}_{ij}(t)}{N_j-{ϵ}_5}-\frac{{\lambda }_{n+1}f(t)}{[1+\alpha (t)(N_1-{ϵ}_5)]}]>\mathrm{\Theta },\hfill \\ \underset{t\in \mathbb{R}}{inf}[{\lambda }_{n+1}{a}_{n+1}(t)-\frac{{\lambda }_{n+1}{b}_{n+1}({\delta }_{n+1}^{-1}(t))}{1-{\dot{\tau }}_{n+1}({\delta }_{n+1}^{-1}(t))}-{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t-s,s)\mathrm{d}s\hfill \\ -\frac{c(t)}{[1+\alpha (t)(N_1-{ϵ}_5)]}]>\mathrm{\Theta },\hfill \end{array}$$

where $i=2,3,\dots ,n$.

Construct a Lyapunov functional as follows:

$$V(t)=V_1(t)+V_2(t)+V_3(t),\mathrm{\forall }t\ge T_5,$$

where

$$\begin{array}{c}{V}_1(t)=\sum _{i=1}^n{\lambda }_i|ln{x}_i(t)-ln{x}_i^{\ast }(t)|+{\lambda }_{n+1}|lny(t)-ln{y}^{\ast }(t)|,\hfill \\ V_2(t)=\sum _{i=1}^n{\int }_{t-{\tau }_i(t)}^t\frac{{\lambda }_i{b}_i({\delta }_i^{-1}(s))}{1-{\dot{\tau }}_i({\delta }_i^{-1}(s))}|x_i(s)-x_i^{\ast }(s)|\mathrm{d}s\hfill \\ \phantom{V_2(t)=}+{\int }_{t-{\tau }_{n+1}(t)}^t\frac{{\lambda }_{n+1}{b}_{n+1}({\delta }_{n+1}^{-1}(s))}{1-{\dot{\tau }}_{n+1}({\delta }_{n+1}^{-1}(s))}|y(s)-y^{\ast }(s)|\mathrm{d}s,\hfill \\ V_3(t)=\sum _{i=1}^n{\lambda }_i{\int }_{-{\sigma }_i}^0{\int }_{t+s}^t{k}_i(l-s,s)|x_i(l)-x_i^{\ast }(l)|\mathrm{d}l\mathrm{d}s\hfill \\ \phantom{V_3(t)=}+{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{\int }_{t+s}^t{k}_{n+1}(l-s,s)|y(l)-y^{\ast }(l)|\mathrm{d}l\mathrm{d}s.\hfill \end{array}$$

For $t\ne t_k$, $k\in \mathbb{Z}$, calculating the upper right derivative of $V_1(t)$ along the solution of system (1.2), it follows that

$$\begin{array}{rcl}{D}^{+}{V}_1(t)& =& \sum _{i=1}^n{\lambda }_i[\frac{{\dot{x}}_i(t)}{x_i(t)}-\frac{{\dot{x}}_i^{\ast }(t)}{x_i^{\ast }(t)}]sgn(x_i(t)-x_i^{\ast }(t))\\ +{\lambda }_{n+1}[\frac{\dot{y}(t)}{y(t)}-\frac{{\dot{y}}^{\ast }(t)}{y^{\ast }(t)}]sgn(y(t)-y^{\ast }(t))\\ \le & \sum _{i=1}^n{\lambda }_i[-a_i(t)|x_i(t)-x_i^{\ast }(t)|+b_i(t)|x_i(t-{\tau }_i(t))-x_i^{\ast }(t-{\tau }_i(t))|\\ +{\int }_{-{\sigma }_i}^0{k}_i(t,s)|x_i(t+s)-x_i^{\ast }(t+s)|\mathrm{d}s]\\ +{\lambda }_{1}sgn(x_1(t)-x_1^{\ast }(t))\sum _{j=2}^n{D}_{j1}(t)\frac{[x_j(t)x_1^{\ast }(t)-x_1(t)x_j^{\ast }(t)]}{x_1(t)x_1^{\ast }(t)}\\ +\sum _{i=2}^n{\lambda }_{i}sgn(x_i(t)-x_i^{\ast }(t))\sum _{j=1}^n{D}_{ji}(t)\frac{[x_j(t)x_i^{\ast }(t)-x_i(t)x_j^{\ast }(t)]}{x_i(t)x_i^{\ast }(t)}\\ +{\lambda }_{1}sgn(x_1(t)-x_1^{\ast }(t))[-\frac{c(t)y(t)}{1+\alpha (t)x_1(t)}+\frac{c(t)y^{\ast }(t)}{1+\alpha (t)x_1^{\ast }(t)}]\\ +{\lambda }_{n+1}|\frac{f(t)x_1(t)}{1+\alpha (t)x_1(t)}-\frac{f(t)x_1^{\ast }(t)}{1+\alpha (t)x_1^{\ast }(t)}|\\ -{\lambda }_{n+1}{a}_{n+1}(t)|y(t)-y^{\ast }(t)|+{\lambda }_{n+1}{b}_{n+1}(t)|y(t-{\tau }_{n+1}(t))-y^{\ast }(t-{\tau }_{n+1}(t))|\\ +{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t,s)|y(t+s)-y^{\ast }(t+s)|\mathrm{d}s\\ \le & -\sum _{i=1}^n{\lambda }_i{a}_i(t)|x_i(t)-x_i^{\ast }(t)|+\sum _{i=1}^n{\lambda }_i{b}_i(t)|x_i(t-{\tau }_i(t))-x_i^{\ast }(t-{\tau }_i(t))|\\ +\sum _{i=1}^n{\lambda }_i{\int }_{-{\sigma }_i}^0{k}_i(t,s)|x_i(t+s)-x_i^{\ast }(t+s)|\mathrm{d}s\\ +\sum _{j=1}^n\frac{{\lambda }_1{D}_{j1}(t)}{N_1-{ϵ}_5}|x_1(t)-x_1^{\ast }(t)|+\sum _{i=2}^n\sum _{j=1}^n\frac{{\lambda }_j{D}_{ij}(t)}{N_j-{ϵ}_5}|x_i(t)-x_i^{\ast }(t)|\\ +\frac{\alpha (t)c(t)(M_{n+1}+{ϵ}_5)}{{[1+\alpha (t)(N_1-{ϵ}_5)]}^2}|x_1(t)-x_1^{\ast }(t)|+\frac{c(t)}{[1+\alpha (t)(N_1-{ϵ}_5)]}|y(t)-y^{\ast }(t)|\\ +\frac{{\lambda }_{n+1}f(t)}{[1+\alpha (t)(N_1-{ϵ}_5)]}|x_1(t)-x_1^{\ast }(t)|\\ -{\lambda }_{n+1}{a}_{n+1}(t)|y(t)-y^{\ast }(t)|+{\lambda }_{n+1}{b}_{n+1}(t)|y(t-{\tau }_{n+1}(t))-y^{\ast }(t-{\tau }_{n+1}(t))|\\ +{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t,s)|y(t+s)-y^{\ast }(t+s)|\mathrm{d}s.\end{array}$$

(4.1)

Here we use the following inequality which has been proved in [16]:

$$sgn(x_i(t)-x_i^{\ast }(t))\sum _{j=1}^n{D}_{ji}(t)\frac{[x_j(t)x_i^{\ast }(t)-x_i(t)x_j^{\ast }(t)]}{x_i(t)x_i^{\ast }(t)}\le \sum _{j=1}^n\frac{D_{ji}(t)}{N_i-{ϵ}_3}|x_j(t)-x_j^{\ast }(t)|.$$

Moreover, we obtain

$$\begin{array}{c}{D}^{+}{V}_2(t)=\sum _{i=1}^n\frac{{\lambda }_i{b}_i({\delta }_i^{-1}(t))}{1-{\dot{\tau }}_i({\delta }_i^{-1}(t))}|x_i(t)-x_i^{\ast }(t)|+\frac{{\lambda }_{n+1}{b}_{n+1}({\delta }_{n+1}^{-1}(t))}{1-{\dot{\tau }}_{n+1}({\delta }_{n+1}^{-1}(t))}|y(t)-y^{\ast }(t)|\hfill \\ \phantom{D^{+}{V}_2(t)=}-\sum _{i=1}^n{\lambda }_i{b}_i(t)|x_i(t-{\tau }_i(t))-x_i^{\ast }(t-{\tau }_i(t))|\hfill \\ \phantom{D^{+}{V}_2(t)=}-{\lambda }_{n+1}{b}_{n+1}(t)|y(t-{\tau }_{n+1}(t))-y^{\ast }(t-{\tau }_{n+1}(t))|,\hfill \end{array}$$

(4.2)

$$\begin{array}{c}{D}^{+}{V}_3(t)=\sum _{i=1}^n{\lambda }_i{\int }_{-{\sigma }_i}^0{k}_i(t-s,s)|x_i(t)-x_i^{\ast }(t)|\mathrm{d}s\hfill \\ \phantom{D^{+}{V}_3(t)=}+{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t-s,s)|y(t)-y^{\ast }(t)|\mathrm{d}s\hfill \\ \phantom{D^{+}{V}_3(t)=}-\sum _{i=1}^n{\lambda }_i{\int }_{-{\sigma }_i}^0{k}_i(t,s)|x_i(t+s)-x_i^{\ast }(t+s)|\mathrm{d}s\hfill \\ \phantom{D^{+}{V}_3(t)=}-{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t,s)|y(t+s)-y^{\ast }(t+s)|\mathrm{d}s.\hfill \end{array}$$

(4.3)

From (4.1)-(4.3), one has

$$\begin{array}{rcl}{D}^{+}V(t)& \le & -[{\lambda }_1{a}_1(t)-\frac{{\lambda }_1{b}_1({\delta }_1^{-1}(t))}{1-{\dot{\tau }}_1({\delta }_1^{-1}(t))}-{\lambda }_1{\int }_{-{\sigma }_1}^0{k}_1(t-s,s)\mathrm{d}s\\ -\frac{\alpha (t)c(t)(M_{n+1}+{ϵ}_5)}{{[1+\alpha (t)(N_1-{ϵ}_5)]}^2}-\sum _{j=1}^n\frac{{\lambda }_1{D}_{j1}(t)}{N_1-{ϵ}_5}-\frac{{\lambda }_{n+1}f(t)}{[1+\alpha (t)(N_1-{ϵ}_5)]}\left]\right|x_1(t)-x_1^{\ast }(t)|\\ -\sum _{i=2}^n[{\lambda }_i{a}_i(t)-\frac{{\lambda }_i{b}_i({\delta }_i^{-1}(t))}{1-{\dot{\tau }}_i({\delta }_i^{-1}(t))}-{\lambda }_i{\int }_{-{\sigma }_i}^0{k}_i(t-s,s)\mathrm{d}s\\ -\sum _{j=1}^n\frac{{\lambda }_j{D}_{ij}(t)}{N_j-{ϵ}_5}-\frac{{\lambda }_{n+1}f(t)}{[1+\alpha (t)(N_1-{ϵ}_5)]}\left]\right|x_i(t)-x_i^{\ast }(t)|\\ -[{\lambda }_{n+1}{a}_{n+1}(t)-\frac{{\lambda }_{n+1}{b}_{n+1}({\delta }_{n+1}^{-1}(t))}{1-{\dot{\tau }}_{n+1}({\delta }_{n+1}^{-1}(t))}-{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}(t-s,s)\mathrm{d}s\\ -\frac{c(t)}{[1+\alpha (t)(N_1-{ϵ}_5)]}\left]\right|y(t)-y^{\ast }(t)|\\ \le & -\mathrm{\Theta }\left[\sum _{i=1}^n\right|x_i(t)-x_i^{\ast }(t)|+|y(t)-y^{\ast }(t)\left|\right].\end{array}$$

(4.4)

For $t=t_k$, $k\in \mathbb{Z}$, we have

$$\begin{array}{rcl}V\left(t_k^{+}\right)& =& V_1\left(t_k^{+}\right)+V_2\left(t_k^{+}\right)+V_3\left(t_k^{+}\right)\\ =& \sum _{i=1}^n{\lambda }_i|ln{x}_i\left(t_k^{+}\right)-ln{x}_i^{\ast }\left(t_k^{+}\right)|+{\lambda }_{n+1}|lny\left(t_k^{+}\right)-ln{y}^{\ast }\left(t_k^{+}\right)|+V_2(t_k)+V_3(t_k)\\ =& \sum _{i=1}^n{\lambda }_i|ln\frac{(1+h_{ik})x_i(t_k)+d_{ik}}{(1+h_{ik})x_i^{\ast }(t_k)+d_{ik}}|\\ +{\lambda }_{n+1}|ln\frac{(1+h_{n+1,k})y(t_k)+d_{n+1,k}}{(1+h_{n+1,k})y^{\ast }(t_k)+d_{n+1,k}}|+V_2(t_k)+V_3(t_k)\\ =& V_1(t_k)+V_2(t_k)+V_3(t_k)\\ =& V(t_k).\end{array}$$

Therefore, V is nonincreasing. Integrating (4.4) from $T_5$ to t leads to

$$V(t)+\mathrm{\Theta }{\int }_{T_5}^t\left[\sum _{i=1}^n\right|x_i(s)-x_i^{\ast }(s)|+|y(s)-y^{\ast }(s)\left|\right]\mathrm{d}s\le V(T_5)<+\mathrm{\infty },\mathrm{\forall }t\ge T_5,$$

that is,

$${\int }_{T_5}^{+\mathrm{\infty }}\left[\sum _{i=1}^n\right|x_i(s)-x_i^{\ast }(s)|+|y(s)-y^{\ast }(s)\left|\right]\mathrm{d}s<+\mathrm{\infty },$$

which implies that

$$\underset{s\to +\mathrm{\infty }}{lim}|x_i(s)-x_i^{\ast }(s)|=0,\underset{s\to +\mathrm{\infty }}{lim}|y(s)-y^{\ast }(s)|=0,i=1,2,\dots ,n.$$

Thus, system (1.2) is globally asymptotically stable. This completes the proof. □

Remark 4.1 Theorem 4.1 gives a sufficient condition for the global asymptotical stability of system (1.2). Therefore, Theorem 4.1 extends the corresponding result in [16] and provides a possible method to study the global asymptotical stability of the models with impulsive perturbations in biological populations.

5 Almost periodic solution

In this section, we investigate the existence and uniqueness of a globally asymptotically stable positive almost periodic solution of system (1.2) by using almost periodic functional hull theory of impulsive differential equations.

Let $\{s_n\}$ be any integer valued sequence such that $s_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. Taking a subsequence if necessary, we have $r_i(t+s_n)\to r_i^{\ast }(t)$, $a_i(t+s_n)\to a_i^{\ast }(t)$, $b_i(t+s_n)\to b_i^{\ast }(t)$, $c(t+s_n)\to c^{\ast }(t)$, $f(t+s_n)\to f^{\ast }(t)$, $\alpha (t+s_n)\to {\alpha }^{\ast }(t)$, $D_{ij}(t+s_n)\to D_{ij}^{\ast }(t)$, ${\tau }_i(t+s_n)\to {\tau }_i^{\ast }(t)$, $k_i(t+s_n)\to k_i^{\ast }(t,s)$, as $n\to \mathrm{\infty }$ for $t\in \mathbb{R}$, $s\in (-\mathrm{\infty },0]$, $i=1,2,\dots ,n+1$, $j=1,2,\dots ,n$. From Lemma 2.1 it follows that the set of sequences $\{t_k-s_n\}$, $k\in \mathbb{Z}$ is convergent to the sequence $\{t_k^s\}$ uniformly with respect to $k\in \mathbb{Z}$ as $n\to \mathrm{\infty }$.

By $\{k_{n_i}\}$ we denote the sequence of integers such that the subsequence $\{t_{k_{n_i}}\}$ is convergent to the sequence $\{t_k^s\}$ uniformly with respect to $k\in \mathbb{Z}$ as $i\to \mathrm{\infty }$.

From the almost periodicity of $\{h_{ik}\}$, it follows that there exists a subsequence of the sequence $\{k_{n_i}\}$ such that the sequences $\{h_{i{k}_{n_i}}\}$ are convergent uniformly to the limits denoted by $h_{ik}^s$, $i=1,2,\dots ,n+1$.

Then we get hull equations of system (1.2) as follows:

$$\{\begin{array}{l}{\dot{x}}_1(t)=x_1(t)[r_1^{\ast }(t)-a_1^{\ast }(t)x_1(t)-b_1^{\ast }(t)x_1(t-{\tau }_1^{\ast }(t))\\ \phantom{{\dot{x}}_1(t)=}-{\int }_{-{\sigma }_1}^0{k}_1^{\ast }(t,s)x_1(t+s)\mathrm{d}s-\frac{c^{\ast }(t)y(t)}{1+{\alpha }^{\ast }(t)x_1(t)}]+{\sum }_{i=2}^n{D}_{i1}^{\ast }(t)[x_i(t)-x_1(t)],\\ {\dot{x}}_i(t)=x_i(t)[r_i^{\ast }(t)-a_i^{\ast }(t)x_i(t)-b_i^{\ast }(t)x_i(t-{\tau }_i^{\ast }(t))\\ \phantom{{\dot{x}}_i(t)=}-{\int }_{-{\sigma }_i}^0{k}_i^{\ast }(t,s)x_i(t+s)\mathrm{d}s]+{\sum }_{j=1}^n{D}_{ji}^{\ast }(t)[x_j(t)-x_i(t)],i=2,3,\dots ,n,\\ \dot{y}(t)=y(t)[-r_{n+1}^{\ast }(t)+\frac{f^{\ast }(t)x_1(t)}{1+{\alpha }^{\ast }(t)x_1(t)}-a_{n+1}^{\ast }(t)y(t)-b_{n+1}^{\ast }(t)y(t-{\tau }_{n+1}^{\ast }(t))\\ \phantom{\dot{y}(t)=}-{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}^{\ast }(t,s)y(t+s)\mathrm{d}s],t\ne t_k^s,\\ \mathrm{\Delta }{x}_j(t_k^s)=h_{jk}^s{x}_j(t_k^s),j=1,2,\dots ,n,\\ \mathrm{\Delta }y(t_k^s)=h_{n+1,k}^{s}y(t_k^s),k\in \mathbb{Z}.\end{array}$$

(5.1)

By the almost periodic theory, we can conclude that if system (1.2) satisfies (H₁)-(H₈), then the hull equations (5.1) of system (1.2) also satisfy (H₁)-(H₈).

By Lemma 4.15 in [18], we can easily obtain the lemma as follows.

Lemma 5.1 If each hull equation of system (1.2) has a unique strictly positive solution, then system (1.2) has a unique strictly positive almost periodic solution.

By using Lemma 5.1, we obtain the following result.

Lemma 5.2 If system (1.2) satisfies (H₁)-(H₈), then system (1.2) admits a unique strictly positive almost periodic solution.

Proof By Lemma 5.1, in order to prove the existence of a unique strictly positive almost periodic solution of system (1.2), we only need to prove that each hull equation of system (1.2) has a unique strictly positive solution.

Firstly, we prove the existence of a strictly positive solution of any hull equations (5.1). According to the almost periodic hull theory of impulsive differential equations (see [9]), there exists a time sequence $\{s_n\}$ with $s_n\to \mathrm{\infty }$ as $n\to +\mathrm{\infty }$ such that $r_i(t+s_n)\to r_i^{\ast }(t)$, $a_i(t+s_n)\to a_i^{\ast }(t)$, $b_i(t+s_n)\to b_i^{\ast }(t)$, $c(t+s_n)\to c^{\ast }(t)$, $f(t+s_n)\to f^{\ast }(t)$, $\alpha (t+s_n)\to {\alpha }^{\ast }(t)$, $D_{ij}(t+s_n)\to D_{ij}^{\ast }(t)$, ${\tau }_i(t+s_n)\to {\tau }_i^{\ast }(t)$, $k_i(t+s_n)\to k_i^{\ast }(t,s)$, as $n\to \mathrm{\infty }$ for $t\in \mathbb{R}$, $t\ne t_k$, $k\in \mathbb{Z}$, $s\in (-\mathrm{\infty },0]$, $i=1,2,\dots ,n+1$, $j=1,2,\dots ,n$. There exists a subsequence $\{k_n\}$ of $\{n\}$, $k_n\to +\mathrm{\infty }$, $n\to +\mathrm{\infty }$ such that $t_{k_n}\to t_k^s$, $h_{i{k}_n}\to h_{ik}^s$, $i=1,2,\dots ,n+1$. Suppose $x(t)={(x_1(t),x_2(t),\dots ,x_n(t),y(t))}^T$ is any positive solution of hull equations (5.1). By the proof of Theorem 3.1, for $\mathrm{\forall }ϵ>0$, there exists $T_0>0$ such that

$$\begin{array}{r}{N}_i-ϵ\le x_i(t)\le M_i+ϵ,\\ N_{n+1}-ϵ\le y(t)\le M_{n+1}+ϵ,t\ge T_0,i=1,2,\dots ,n.\end{array}$$

(5.2)

Let $x_n(t)=x(t+s_n)$ for all $t\ge -s_n+T_0$, $n=1,2,\dots$ , such that

$$\{\begin{array}{l}{\dot{x}}_1(t)=x_1(t)[r_1^{\ast }(t+s_n)-a_1^{\ast }(t+s_n)x_1(t)-b_1^{\ast }(t+s_n)x_1(t-{\tau }_1^{\ast }(t+s_n))\\ \phantom{{\dot{x}}_1(t)=}-{\int }_{-{\sigma }_1}^0{k}_1^{\ast }(t+s_n,s)x_1(t+s)\mathrm{d}s-\frac{c^{\ast }(t+s_n)y(t)}{1+{\alpha }^{\ast }(t+s_n)x_1(t)}]\\ \phantom{{\dot{x}}_1(t)=}+{\sum }_{i=2}^n{D}_{i1}^{\ast }(t+s_n)[x_i(t)-x_1(t)],\\ {\dot{x}}_i(t)=x_i(t)[r_i^{\ast }(t+s_n)-a_i^{\ast }(t+s_n)x_i(t)-b_i^{\ast }(t+s_n)x_i(t-{\tau }_i^{\ast }(t+s_n))\\ \phantom{{\dot{x}}_i(t)=}-{\int }_{-{\sigma }_i}^0{k}_i^{\ast }(t+s_n,s)x_i(t+s)\mathrm{d}s]\\ \phantom{{\dot{x}}_i(t)=}+{\sum }_{j=1}^n{D}_{ji}^{\ast }(t+s_n)[x_j(t)-x_i(t)],i=2,3,\dots ,n,\\ \dot{y}(t)=y(t)[-r_{n+1}^{\ast }(t+s_n)+\frac{f^{\ast }(t+s_n)x_1(t)}{1+{\alpha }^{\ast }(t+s_n)x_1(t)}-a_{n+1}^{\ast }(t+s_n)y(t)\\ \phantom{\dot{y}(t)=}-b_{n+1}^{\ast }(t+s_n)y(t-{\tau }_{n+1}^{\ast }(t+s_n))\\ \phantom{\dot{y}(t)=}-{\int }_{-{\sigma }_{n+1}}^0{k}_{n+1}^{\ast }(t+s_n,s)y(t+s)\mathrm{d}s],t\ne t_k^s,\\ \mathrm{\Delta }{x}_j(t_k^s)=h_{jk}^s{x}_j(t_k^s),j=1,2,\dots ,n,\\ \mathrm{\Delta }y(t_k^s)=h_{n+1,k}^{s}y(t_k^s),k\in \mathbb{Z}.\end{array}$$

(5.3)

From the inequality (5.2), there exists a positive constant K which is independent of n such that $|{\dot{x}}_n|\le K$ for all $t\ge -s_n+T_0$, $n=1,2,\dots$ . Therefore, for any positive integer r sequence $\{x_n(t):n\ge r\}$ is uniformly bounded and equicontinuous on $[-s_n+T_0,\mathrm{\infty })$. According to Ascoli-Arzela theorem, one can conclude that there exists a subsequence $\{s_m\}$ of $\{s_n\}$ such that sequence $\{x_m(t)\}$ not only converges on t on ℝ, but it also converges uniformly on any compact set of ℝ as $m\to +\mathrm{\infty }$. Suppose ${lim}_{m\to +\mathrm{\infty }}{x}_m(t)=x^{\ast }(t)={(x_1^{\ast }(t),x_2^{\ast }(t),\dots ,x_n^{\ast }(t),y(t))}^T$, then we have

$$\begin{array}{c}{N}_i-ϵ\le x_i(t)\le M_i+ϵ,\hfill \\ N_{n+1}-ϵ\le y(t)\le M_{n+1}+ϵ,t\in \mathbb{R},i=1,2,\dots ,n.\hfill \end{array}$$

From differential equations (5.3) and the arbitrariness of ϵ, we can easily see that $x^{\ast }(t)$ is the solution of the hull equations (5.1) and $N_i\le x_i^{\ast }(t)\le M_i$ for all $t\in \mathbb{R}$, $i=1,2,\dots ,n$. Hence each hull equation of the almost periodic system (1.2) has at least a strictly positive solution.

Now we prove the uniqueness of the strictly positive solution of each hull equations (5.1). Suppose that the hull equations (5.1) have two arbitrary strictly positive solutions $x(t)={(x_1(t),x_2(t),\dots ,x_n(t),y(t))}^T$ and $x^{\ast }(t)={(x_1^{\ast }(t),x_2^{\ast }(t),\dots ,x_n^{\ast }(t),y^{\ast }(t))}^T$, which satisfy

$$\begin{array}{c}{N}_i-ϵ\le x_i(t),x_i^{\ast }(t)\le M_i+ϵ,\hfill \\ N_{n+1}-ϵ\le y(t),y^{\ast }(t)\le M_{n+1}+ϵ,t\in \mathbb{R},i=1,2,\dots ,n.\hfill \end{array}$$

Similar to Theorem 4.1, we define a Lyapunov functional

$$V^{\ast }(t)=V_1^{\ast }(t)+V_2^{\ast }(t)+V_3^{\ast }(t),\mathrm{\forall }t\in \mathbb{R},$$

where

$$\begin{array}{c}{V}_1^{\ast }(t)=\sum _{i=1}^n{\lambda }_i|ln{x}_i(t)-ln{x}_i^{\ast }(t)|+{\lambda }_{n+1}|lny(t)-ln{y}^{\ast }(t)|,\hfill \\ V_2^{\ast }(t)=\sum _{i=1}^n{\int }_{t-{\tau }_i(t)}^t\frac{{\lambda }_i{b}_i^{\ast }({\delta }_i^{\ast -1}(s))}{1-{\dot{{\tau }^{\ast }}}_i({\delta }_i^{\ast -1}(s))}|x_i(s)-x_i^{\ast }(s)|\mathrm{d}s\hfill \\ \phantom{V_2^{\ast }(t)=}+{\int }_{t-{\tau }_{n+1}^{\ast }(t)}^t\frac{{\lambda }_{n+1}{b}_{n+1}^{\ast }({\delta }_{n+1}^{\ast -1}(s))}{1-{\dot{{\tau }^{\ast }}}_{n+1}({\delta }_{n+1}^{\ast -1}(s))}|y(s)-y^{\ast }(s)|\mathrm{d}s,\hfill \\ V_3^{\ast }(t)=\sum _{i=1}^n{\lambda }_i{\int }_{-{\sigma }_i}^0{\int }_{t+s}^t{k}_i^{\ast }(l-s,s)|x_i(l)-x_i^{\ast }(l)|\mathrm{d}l\mathrm{d}s\hfill \\ \phantom{V_3^{\ast }(t)=}+{\lambda }_{n+1}{\int }_{-{\sigma }_{n+1}}^0{\int }_{t+s}^t{k}_{n+1}^{\ast }(l-s,s)|y(l)-y^{\ast }(l)|\mathrm{d}l\mathrm{d}s,\hfill \end{array}$$

where ${\delta }_j^{\ast -1}$ is an inverse function of ${\tau }_j^{\ast }$, $j=1,2,\dots ,n+1$. Similar to the argument in (4.4), one has

$$D^{+}{V}^{\ast }(t)\le -\mathrm{\Theta }\left[\sum _{i=1}^n\right|x_i(t)-x_i^{\ast }(t)|+|y(t)-y^{\ast }(t)\left|\right],\mathrm{\forall }t\in \mathbb{R}.$$

Summing both sides of the above inequality from t to 0, we have

$$\mathrm{\Theta }{\int }_t^0\left[\sum _{i=1}^n\right|x_i(s)-x_i^{\ast }(s)|+|y(s)-y^{\ast }(s)\left|\right]\mathrm{d}s\le V^{\ast }(t)-V(0),\mathrm{\forall }t\le 0.$$

Note that $V^{\ast }$ is bounded. Hence we have

$${\int }_{-\mathrm{\infty }}^0\left[\sum _{i=1}^n\right|x_i(s)-x_i^{\ast }(s)|+|y(s)-y^{\ast }(s)\left|\right]\mathrm{d}s<\mathrm{\infty },$$

which implies that

$$\underset{s\to -\mathrm{\infty }}{lim}\sum _{i=1}^n|x_i(s)-x_i^{\ast }(s)|=\underset{s\to -\mathrm{\infty }}{lim}|y(s)-y^{\ast }(s)|=0,i=1,2,\dots ,n.$$

For arbitrary ${ϵ}_0>0$, there exists a positive constant L such that

$$max\left\{\right|x_i(t)-x_i^{\ast }(t)|,|y(t)-y^{\ast }(t)\left|\right\}<{ϵ}_0,\mathrm{\forall }t<-L,i=1,2,\dots ,n.$$

Hence, one has

$$\begin{array}{c}{V}_1^{\ast }(t)\le \sum _{i=1}^{n+1}\frac{{\lambda }_i{ϵ}_0}{N_i},\mathrm{\forall }t<-L,\hfill \\ V_2^{\ast }(t)\le \sum _{i=1}^{n+1}{\tau }_i^u\frac{{\lambda }_i{b}_i^u}{1-{sup}_{t\in \mathbb{R}}{\dot{\tau }}_i(t)}{ϵ}_0,\mathrm{\forall }t<-L,\hfill \\ V_3^{\ast }(t)\le \sum _{i=1}^{n+1}{\lambda }_i{\int }_{-{\sigma }_i}^0(-s)k_i^u(s)\mathrm{d}s{ϵ}_0,\mathrm{\forall }t<-L,\hfill \end{array}$$

which imply that there exists a positive constant ρ such that

$$V^{\ast }(t)<\rho {ϵ}_0,\mathrm{\forall }t<-L.$$

So

$$\underset{t\to -\mathrm{\infty }}{lim}{V}^{\ast }(t)=0.$$

Note that $V^{\ast }(t)$ is a nonincreasing function on ℝ, and then $V^{\ast }(t)\equiv 0$. That is,

$$x_i(t)=x_i^{\ast }(t),y(t)=y^{\ast }(t),\mathrm{\forall }t\in \mathbb{R},i=1,2,\dots ,n.$$

Therefore, each hull equation of system (1.2) has a unique strictly positive solution.

In view of the above discussion, any hull equation of system (1.2) has a unique strictly positive solution. By Lemma 5.1, system (1.2) has a unique strictly positive almost periodic solution. The proof is completed. □

By Theorem 4.1 and Lemma 5.2, we obtain the following.

Theorem 5.1 Suppose that (H₁)-(H₈) hold, then system (1.2) admits a unique strictly positive almost periodic solution, which is globally asymptotically stable.

Remark 5.1 Theorem 5.1 gives sufficient condition for the global asymptotical stability of a unique positive almost periodic solution of system (1.2). Therefore, Theorem 5.1 extends the corresponding result in [16] and provides a possible method to study the existence, uniqueness, and stability of positive almost periodic solution of the models with impulsive perturbations in biological populations.

6 An example and numerical simulations

Example 6.1 Consider the following Lotka-Volterra type predator-prey dispersal system with impulsive effects:

$$\{\begin{array}{l}{\dot{x}}_1(t)=x_1(t)[10-(5+sin(\sqrt{2}t))x_1(t)-0.1x_1(t-1)-\frac{0.02y(t)}{1+x_1(t)}]\\ \phantom{{\dot{x}}_1(t)=}+0.3[x_2(t)-x_1(t)],\\ {\dot{x}}_2(t)=x_2(t)[8+cos(\sqrt{3}t)-4x_2(t)-{\int }_{-0.1}^0{x}_2(t+s)\mathrm{d}s]\\ \phantom{{\dot{x}}_2(t)=}+0.1cos(\sqrt{5}t)[x_1(t)-x_2(t)],\\ \dot{y}(t)=y(t)[-0.01|cos(\sqrt{5}t)|+\frac{2x_1(t)}{1+x_1(t)}-2y(t)],t\ne t_k,\\ \mathrm{\Delta }{x}_i(t_k)=-0.4x_i(t_k),i=1,2,\\ \mathrm{\Delta }y(t_k)=-0.5y(t_k),\{t_k:k\in \mathbb{Z}\}\subset \{10k:k\in \mathbb{Z}\}.\end{array}$$

(6.1)

Then system (6.1) is uniformly persistent and has a unique globally asymptotically stable almost periodic solution.

Proof Corresponding to system (1.2), we have $r^u\theta -{\xi }^l=90-0.5>0$ and $r_y^u\theta -{\xi }_3^l=1.6\times 10-0.7>$