Dynamics in a periodic two-species predator–prey system with pure delays

Abstract

A class of non-autonomous two-species Lotka–Volterra predator–prey system with pure discrete time delays is discussed. Some sufficient conditions on the boundedness, permanence, extinction, positive periodic solution and global attractivity of the system are established by means of the comparison method, coincidence degree theory and Liapunov functional.

Introduction

In the real world, there are many types of interactions between two species. Predator–prey relations are among the most common ecological interactions. Remarkably, the whole field of mathematical ecology began with the studies of population dynamics subject to the predator–prey interaction, that is, with the classical works by Lotka [1] and Volterra [2]. Traditional two-species non-autonomous Lotka–Volterra predator–prey systems take the form

$$\begin{array}{cc}\hfill \frac{d{x}_1(t)}{dt}& =x_1(t)[r_1(t)-a_{11}(t)x_1(t)-a_{12}(t)x_2(t)],\hfill \\ \hfill \frac{d{x}_2(t)}{dt}& =x_2(t)[-r_2(t)+a_{21}(t)x_1(t)-a_{22}(t)x_2(t)].\hfill \end{array}$$

(1)

Recently, the properties of system (1) are discussed by many scholars [3–7]. Some sufficient conditions are obtained for the persistence, permanence and extinction of the species, the existence and uniqueness of periodic solutions or almost periodic solutions, and the global stability of solutions for system (1).

However, in the real world, the growth rate of a natural species will not often respond immediately to changes in its own population or that of an interacting species, but will rather do so after a time lag [8]. Time delays have a great destabilizing influence on the species population; this result was put forwarded by May [9].

Therefore, we should introduce time delay into model foundation, which will have more resemblance to the real ecosystem.

In this paper, we investigate the following two-species Lotka–Volterra type predator–prey systems with pure discrete time delays

$$\begin{array}{cc}\hfill {\dot{x}}_1(t)& =x_1(t)[r_1(t)-a_{11}(t)x_1(t-{\mathit{\tau }}_1)-a_{12}(t)x_2(t-{\mathit{\tau }}_1)],\hfill \\ \hfill {\dot{x}}_2(t)& =x_2(t)[-r_2(t)+a_{21}(t)x_1(t-{\mathit{\tau }}_2)-a_{22}(t)x_2(t-{\mathit{\tau }}_2)].\hfill \end{array}$$

(2)

Our main purpose is to establish some sufficient conditions on the boundedness, permanence, extinction, positive periodic solution and global attractivity of the system (2).

The organization of this paper is as follows. In the next section, we will present some basic assumptions and main lemmas. In "Main results", we will consider conditions for the boundedness, permanence, extinction, positive periodic solution and global attractivity of the system. In the final section, as an application, one special case of the system is considered.

Preliminaries

In system (2), we have that $x_1(t)$ is the prey population density and $x_2(t)$ is the predator population density, $r_1(t)$ and $a_{11}(t)$ are the intrinsic growth rate and density-dependent coefficient of the prey, respectively, $r_2(t)$ is the intrinsic growth rate of the predator, $a_{12}(t)$ is the capturing rate of the predator and $a_{21}(t)$ is the rate of conversion of nutrients into the reproduction of the predator. Throughout this paper, for system (2) we introduce the following hypotheses.

($H_1$) ${\mathit{\tau }}_i>0(i=1,2)$ are positive constants, $r_i(t)(i=1,2)$ are continuous $\mathit{\omega }$-periodic functions with ${\int }_0^{\mathit{\omega }}{r}_i(t)dt>0$. $a_{ij}(t)(i,j=1,2)$ are continuous positive $\mathit{\omega }$-periodic functions.

From the viewpoint of mathematical biology, in this paper for system (2), we only consider the solution with the following initial condition:

$$\begin{array}{c}\hfill x_i(t)={\mathit{\varphi }}_i(t)\text{for}\text{all}t\in [-\mathit{\tau },0),i=1,2,\end{array}$$

(3)

where ${\mathit{\varphi }}_i(t)(i=1,2)$ are nonnegative continuous functions defined on $[-\mathit{\tau },0)$ satisfying ${\mathit{\varphi }}_i(0)>0(i=1,2)$ and $\mathit{\tau }=max\{{\mathit{\tau }}_1,{\mathit{\tau }}_2\}$.

In this paper, for any $\mathit{\omega }$-periodic continuous function $f(t)$, we denote

$$\begin{array}{c}\hfill f^L=\underset{t\in [0,\mathit{\omega }]}{min}f(t),f^M=\underset{t\in [0,\mathit{\omega }]}{max}f(t),\overline{f}=\frac{1}{\mathit{\omega }}{\int }_0^{\mathit{\omega }}f(t)dt.\end{array}$$

To obtain the existence of positive $\mathit{\omega }$-periodic solutions of system (2), we will use the continuation theorem. For the reader’s convenience, we will introduce the continuation theorem in the following. First we define some definitions.

Let $X$ and $Z$ be two normed vector spaces. Let $L:\text{Dom}L\subset X\to Z$ be a linear operator and $N:X\to Z$ be a continuous operator. The operator $L$ is called a Fredholm operator of index zero, if $dim$Ker$L=$codimIm$L<\infty$ and Im$L$ is a closed set in $Z$. If $L$ is a Fredholm operator of index zero, then there exist continuous projectors $P:X\to X$ and $Q:Z\to Z$ such that Im$P=$ Ker$L$ and Im$L=$ Ker$Q=$ Im$(I-Q)$. It follows that $L|$Dom$L\cap$Ker$P:$ Dom$L\cap$Ker$P\to$ Im$L$ is invertible and its inverse is denoted by $K_P$ and denoted by $J:ImQ\to KerL$ an isomorphism of $ImQ$ onto $KerL$. Let $\mathrm{\Omega }$ be a bounded open subset of $X$, we say that the operator $N$ is $L$-compact on $\overline{\mathrm{\Omega }}$, where $\overline{\mathrm{\Omega }}$ denotes the closure of $\mathrm{\Omega }$ in $X$, if $QN(\overline{\mathrm{\Omega }})$ is bounded and $K_P(I-Q)N:\overline{\mathrm{\Omega }}\to X$ is compact. Such definitions can be found in [10–12].

Now, we present some useful lemmas.

Lemma 1

Set $R_{+}^2=\{(x_1,x_2):x_i>0,i=1,2\}$ is positively invariant for system (2).

The proof of Lemma 1 is simple, and here we omit it.

Lemma 2

[(see,[13])] Consider the following equation: $\dot{u}(t)=u(t)(d_1-d_{2}u(t))$, where $d_2>0$, we have If $d_1>0,then{lim}_{t\to +\infty }u(t)=d_1/d_2$. If $d_1<0,then{lim}_{t\to +\infty }u(t)=0$.

Lemma 3

(see [4]) Let $L$ be a Fredholm operator of index zero and let $N$ be $L$-compact on $\overline{\mathrm{\Omega }}$. If

• (a) for each $\mathit{\lambda }\in (0,1)$ and $x\in \mathit{\partial }\mathrm{\Omega }\cap$Dom$L$, $Lx\ne \mathit{\lambda }Nx$;

• (b) for each $x\in \mathit{\partial }\mathrm{\Omega }\cap$Ker$L$, $QNx\ne 0$;

• (c) deg$\{JQN,\mathrm{\Omega }\cap$Ker$L,0\}\ne 0$,

then the operator equation $Lx=Nx$ has at least one solution lying in Dom$L\cap \overline{\mathrm{\Omega }}$.

Main results

In this section, we will obtain some sufficient conditions for the boundedness, existence of periodic solution, global attractivity, permanence, and extinction of system (2).

Theorem 1

Suppose that assumption (H1) holds, then there exist positive constants $M_i(i=1,2)$ such that

$$\begin{array}{c}\hfill x_i(t)\le M_i,\end{array}$$

for any positive solution $x_i(t)$ of system (2).

Proof

Let $(x_1(t),x_2(t))$ be a solution of system (2). Firstly, it follows from the first equation of system (2) that for $t>\mathit{\tau }$, we have

$$\begin{array}{cc}\hfill \frac{d{x}_1(t)}{dt}& =x_1(t)[r_1(t)-a_{11}(t)x_1(t-{\mathit{\tau }}_1)-a_{12}(t)x_2(t-{\mathit{\tau }}_1)]\hfill \\ \hfill & \le x_1(t)[r_1^M-a_{11}^L{e}^{-r_1^M{\mathit{\tau }}_1}{x}_1(t)]\text{for}t>\mathit{\tau }.\hfill \end{array}$$

We consider the following auxiliary equation

$$\begin{array}{c}\hfill \frac{du(t)}{dt}=u(t)[r_1^M-a_{11}^L{e}^{-r_1^M{\mathit{\tau }}_1}u(t)].\end{array}$$

By Lemma 2, we derive

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}u(t)=\frac{r_1^M{e}^{r_1^M{\mathit{\tau }}_1}}{a_{11}^L}=:M_1.\end{array}$$

By comparison, there exists a $T_1>\mathit{\tau }$ such that $x_1(t)\le M_1$ for $t\ge T_1$.

Next, from the second equation of system (2) for $t>T_1$, we have

$$\begin{array}{cc}\hfill \frac{d{x}_2(t)}{dt}& \le x_2(t)[a_{21}^M{M}_1-a_{22}^L{x}_2(t-{\mathit{\tau }}_2)]\hfill \\ \hfill & \le x_2(t)[a_{21}^M{M}_1-a_{22}^L{e}^{-a_{21}^M{M}_1{\mathit{\tau }}_2}{x}_2(t)]\text{for}t>T_1.\hfill \end{array}$$

We consider the following auxiliary equation

$$\begin{array}{c}\hfill \frac{du(t)}{dt}=u(t)[a_{21}^M{M}_1-a_{22}^L{e}^{-a_{21}^M{M}_1{\mathit{\tau }}_2}u(t)].\end{array}$$

By Lemma 2, we derive

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}u(t)=\frac{a_{21}^M{M}_1{e}^{a_{21}^M{M}_1{\mathit{\tau }}_2}}{a_{22}^L}=:M_2.\end{array}$$

By comparison, there exists a $T_2>\mathit{\tau }$ such that $x_2(t)\le M_2$ for $t\ge T_2$. This completes the proof. $\square$

Theorem 2

Suppose that assumption (H1) holds and ${\displaystyle {\overline{r}}_1-{(\frac{a_{11}}{a_{21}})}^M{\overline{r}}_2>0}$. Then system (2) has at least one positive $\mathit{\omega }-$ periodic solution.

Proof

Let

$$\begin{array}{c}\hfill x_1(t)=exp\{u_1(t)\}\text{and}{x}_2(t)=exp\{u_2(t)\}.\end{array}$$

Then system (2) is rewritten in the following system

$$\begin{array}{cc}\hfill {\dot{u}}_1(t)& =r_1(t)-a_{11}(t)exp\{u_1(t-{\mathit{\tau }}_1)\}-a_{12}(t)exp\{u_2(t-{\mathit{\tau }}_1)\},\hfill \\ \hfill {\dot{u}}_2(t)& =-r_2(t)+a_{21}(t)exp\{u_1(t-{\mathit{\tau }}_2)\}-a_{22}(t)exp\{u_2(t-{\mathit{\tau }}_2)\}.\hfill \end{array}$$

(4)

To apply Lemma 3 to system (4), we introduce the normed vector spaces $X$ and $Z$ as follows. Let $C(R,R^2)$ denote the space of all continuous functions $u(t)=(u_1(t),u_2(t)):R\to R^2$. We take

$$\begin{array}{c}\hfill X=Z=\{u(t)\in C(R,R^2):u(t)\text{an}\mathit{\omega }-\mathrm{periodic}\mathrm{function}\}\end{array}$$

with norm

$$\begin{array}{c}\hfill \Vert u\Vert =\underset{t\in [0,\mathit{\omega }]}{max}|u_1(t)|+\underset{t\in [0,\mathit{\omega }]}{max}|u_2(t)|.\end{array}$$

It is obvious that $X$ and $Z$ are the Banach spaces. We define a linear operator $L:$ Dom$L\subset X\to Z$ and a continuous operator $N:X\to Z$ as follows.

$$\begin{array}{c}\hfill Lu(t)=\dot{u}(t)\end{array}$$

and

$$\begin{array}{c}\hfill Nu(t)=(N{u}_1(t),N{u}_2(t)).\end{array}$$

(5)

where

$$\begin{array}{cc}\hfill N{u}_1(t)& =r_1(t)-a_{11}(t)exp\{u_1(t-{\mathit{\tau }}_1)\}-a_{12}(t)exp\{u_2(t-{\mathit{\tau }}_1)\},\hfill \\ \hfill N{u}_2(t)& =-r_2(t)+a_{21}(t)exp\{u_1(t-{\mathit{\tau }}_2)\}-a_{22}(t)exp\{u_2(t-{\mathit{\tau }}_2)\}.\hfill \end{array}$$

Further, we define continuous projectors $P:X\to X$ and $Q:Z\to Z$ as follows.

$$\begin{array}{c}\hfill Pu(t)=\frac{1}{\mathit{\omega }}{\int }_0^{\mathit{\omega }}u(t)dt,Qv(t)=\frac{1}{\mathit{\omega }}{\int }_0^{\mathit{\omega }}v(t)dt.\end{array}$$

We easily see $\text{Im}L=\{v\in Z:{\int }_0^{\mathit{\omega }}v(t)dt=0\}$ and $\text{Ker}L=R^2$. It is obvious that Im$L$ is closed in $Z$ and dimKer$L=2$. Since for any $v\in Z$ there are unique $v_1\in R^n$ and $v_2\in \text{Im}L$ with

$$\begin{array}{c}\hfill v_1=\frac{1}{\mathit{\omega }}{\int }_0^{\mathit{\omega }}v(t)dt,v_2(t)=v(t)-v_1,\end{array}$$

such that $v(t)=v_1+v_2(t)$, we have codimIm$L=2$. Therefore, $L$ is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to $L$) $K_p:$ Im$L\to$ Ker$P\cap$ Dom$L$ is given in the following form:

$$\begin{array}{c}\hfill K_{p}v(t)={\int }_0^{t}v(s)ds-\frac{1}{\mathit{\omega }}{\int }_0^{\mathit{\omega }}{\int }_0^{t}v(s)dsdt.\end{array}$$

For convenience, we denote $F(t)=(F_1(t),F_2(t))$ as follows

$$\begin{array}{cc}\hfill F_1(t)& =r_1(t)-a_{11}(t)exp\{u_1(t-{\mathit{\tau }}_1)\}-a_{12}(t)exp\{u_2(t-{\mathit{\tau }}_1)\},\hfill \\ \hfill F_2(t)& =-r_2(t)+a_{21}(t)exp\{u_1(t-{\mathit{\tau }}_2)\}-a_{22}(t)exp\{u_2(t-{\mathit{\tau }}_2)\}.\hfill \end{array}$$

(6)

Thus, we have

$$\begin{array}{c}\hfill QNu(t)=\frac{1}{\mathit{\omega }}{\int }_0^{\mathit{\omega }}F(t)dt\end{array}$$

(7)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill K_p(I-Q)Nu(t)& =K_{p}INu(t)-K_{p}QNu(t)\hfill \\ \hfill & {\displaystyle ={\int }_0^{t}F(s)ds-\frac{1}{\mathit{\omega }}{\int }_0^{\mathit{\omega }}{\int }_0^{t}F(s)dsdt}\hfill \\ \hfill & {\displaystyle +(\frac{1}{2}-\frac{t}{\mathit{\omega }}){\int }_0^{\mathit{\omega }}F(s)ds.}\hfill \end{array}\end{array}$$

(8)

From formulas (7) and (8), we easily see that $QN$ and $K_p(I-Q)N$ are continuous operators. Furthermore, it can be verified that $\overline{K_p(I-Q)N(\overline{\mathrm{\Omega }})}$ is compact for any open-bounded set $\mathrm{\Omega }\subset X$ by using Arzela–Ascoli theorem and $QN(\overline{\mathrm{\Omega }})$ is bounded. Therefore, $N$ is $L$-compact on $\overline{\mathrm{\Omega }}$ for any open-bounded subset $\mathrm{\Omega }\subset X$.

Now, we reach the position to search for an appropriate open-bounded subset $\mathrm{\Omega }$ for the application of the continuation theorem (Lemma 3) to system (2).

Corresponding to the operator equation $Lu(t)=\mathit{\lambda }Nu(t)$ with parameter $\mathit{\lambda }\in (0,1)$, we have

$$\begin{array}{c}\hfill {\dot{u}}_i(t)=\mathit{\lambda }{F}_i(t),i=1,2,\end{array}$$

(9)

where $F_i(t)(i=1,2)$ are given in Eq.6.

Assume that $u(t)=(u_1(t),u_2(t))\in X$ is a solution of system (9) for some parameter $\mathit{\lambda }\in (0,1)$. By integrating system (9) over the interval $[0,\mathit{\omega }]$, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{\mathit{\omega }}[r_1(t)-a_{11}(t)exp\{u_1(t-{\mathit{\tau }}_1)\}-a_{12}(t)exp\{u_2(t-{\mathit{\tau }}_1)\}]dt=0,}\hfill \\ {\displaystyle {\int }_0^{\mathit{\omega }}[-r_2(t)+a_{21}(t)exp\{u_1(t-{\mathit{\tau }}_2)\}-a_{22}(t)exp\{u_2(t-{\mathit{\tau }}_2)\}]dt=0.}\hfill \end{array}\end{array}$$

(10)

By (10) we get ,

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle {\int }_0^{\mathit{\omega }}[a_{11}(t)exp\{u_1(t-{\mathit{\tau }}_1)\}+a_{12}(t)exp\{u_2(t-{\mathit{\tau }}_1)\}]dt={\overline{r}}_1\mathit{\omega },}\hfill \\ {\displaystyle {\int }_0^{\mathit{\omega }}[a_{21}(t)exp\{u_1(t-{\mathit{\tau }}_2)\}-a_{22}(t)exp\{u_2(t-{\mathit{\tau }}_2)\}]dt={\overline{r}}_2\mathit{\omega }.}\hfill \end{array}\end{array}$$

(11)

For each $i,j=1,2$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle {\int }_0^{\mathit{\omega }}{a}_{ij}(t)exp\{u_i(t-{\mathit{\tau }}_i)\}dsdt}\\ \hfill {\displaystyle ={\int }_{-{\mathit{\tau }}_i}^{\mathit{\omega }-{\mathit{\tau }}_i}{a}_{ij}(s+{\mathit{\tau }}_i)exp\{u_i(s)\}ds}\\ \hfill {\displaystyle ={\int }_0^{\mathit{\omega }}{a}_{ij}(s+{\mathit{\tau }}_i)exp\{u_i(s)\}ds}\\ \hfill {\displaystyle ={\int }_0^{\mathit{\omega }}{a}_{ij}(t+{\mathit{\tau }}_i)exp\{u_i(t)\}dt.}\end{array}\end{array}$$

(12)

From the continuity of $u(t)=(u_1(t),u_2(t))$, there exist constants ${\mathit{\xi }}_i,{\mathit{\eta }}_i\in [0,\mathit{\omega }](i=1,2)$ such that

$$\begin{array}{c}\hfill u_i({\mathit{\xi }}_i)=\underset{t\in [0,\mathit{\omega }]}{max}{u}_i(t),u_i({\mathit{\eta }}_i)=\underset{t\in [0,\mathit{\omega }]}{min}{u}_i(t),i=1,2.\end{array}$$

(13)

From (11)–(13) and the condition of Theorem 2, we further obtain

$$\begin{array}{c}\hfill u_i({\mathit{\eta }}_i)\le ln(\frac{{\overline{r}}_1}{{\overline{A}}_i}),u_1({\mathit{\xi }}_1)\ge ln(\frac{{\overline{r}}_2}{{\overline{A}}_3})=:B_1,u_2({\mathit{\xi }}_2)\ge ln{A}_4=:B_2,\end{array}$$

(14)

where

$$\begin{array}{c}\hfill A_i(t)=a_{1i}(t+{\mathit{\tau }}_1),A_3(t)=a_{22}(t+{\mathit{\tau }}_2),A_4=\frac{{\overline{r}}_1-{(\frac{a_{11}}{a_{21}})}^M{\overline{r}}_2}{{\overline{A}}_2+{(\frac{a_{11}}{a_{21}})}^M{\overline{A}}_3},i=1,2.\end{array}$$

From (4) and (11) we have

$$\begin{array}{cc}\hfill {\int }_0^{\mathit{\omega }}|{\dot{u}}_1(t)|dt& \le {\int }_0^{\mathit{\omega }}[r_1(t)+a_{11}(t)exp\{u_1(t-{\mathit{\tau }}_1)\}+a_{12}(t)exp\{u_2(t-{\mathit{\tau }}_1)\}]dt\hfill \\ \hfill & \le 2{\overline{r}}_1\mathit{\omega }=:C_1,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\int }_0^{\mathit{\omega }}|{\dot{u}}_2(t)|dt& \le {\int }_0^{\mathit{\omega }}[r_2(t)+a_{21}(t)exp\{u_1(t-{\mathit{\tau }}_2)\}+a_{22}(t)exp\{u_2(t-{\mathit{\tau }}_2)\}]dt\hfill \\ \hfill & {\displaystyle \le \frac{2a_{21}^M{\overline{r}}_1\mathit{\omega }}{a_{11}^L}=:C_2.}\hfill \end{array}$$

(16)

By (14)–(16), we have

$$\begin{array}{c}\hfill u_i(t)\le u_i({\mathit{\eta }}_i)+{\int }_0^{\mathit{\omega }}|{\dot{u}}_i(t)|dt\le ln(\frac{{\overline{r}}_1}{A_i})+C_i=:M_{i}i=1,2,\end{array}$$

(17)

and

$$\begin{array}{c}\hfill u_i(t)\ge u_i({\mathit{\xi }}_i)-{\int }_0^{\mathit{\omega }}|{\dot{u}}_i(t)|dt\ge B_i-C_i=:N_{i}i=1,2.\end{array}$$

(18)

Therefore, from (17), (18) we have

$$\begin{array}{c}\hfill \underset{t\in [0,\mathit{\omega }]}{max}|u_i(t)|\le max\{|M_i|,|N_i|\}=:H_i,i=1,2.\end{array}$$

It can be seen that the constants $H_i(i=1,2)$ are independent of parameter $\mathit{\lambda }\in (0,1)$.

For any $u=(u_1,u_2)\in R^2$, from (5) we can obtain

$$\begin{array}{c}\hfill QNu=(QN{u}_1,QN{u}_2).\end{array}$$

where

$$\begin{array}{cc}\hfill QN{u}_1& ={\overline{r}}_1-{\overline{a}}_{11}exp\{u_1\}-{\overline{a}}_{12}exp\{u_2\},\hfill \\ \hfill QN{u}_2& =-{\overline{r}}_2+{\overline{a}}_{21}exp\{u_1\}-{\overline{a}}_{22}exp\{u_2\}.\hfill \end{array}$$

We consider the following system of algebraic equations

$$\begin{array}{c}\hfill \begin{array}{cc}& {\overline{r}}_1-{\overline{a}}_{11}{\mathit{\upsilon }}_1-{\overline{a}}_{12}{\mathit{\upsilon }}_2=0,\hfill \\ & -{\overline{r}}_2+{\overline{a}}_{21}{\mathit{\upsilon }}_1-{\overline{a}}_{22}{\mathit{\upsilon }}_2=0.\hfill \end{array}\end{array}$$

By direct calculation we can get

$$\begin{array}{c}\hfill v_1^{\ast }=\frac{{\overline{r}}_1{\overline{a}}_{22}+{\overline{r}}_2{\overline{a}}_{12}}{{\overline{a}}_{11}{\overline{a}}_{22}+{\overline{a}}_{12}{\overline{a}}_{21}},v_2^{\ast }=\frac{{\overline{r}}_1{\overline{a}}_{21}-{\overline{r}}_2{\overline{a}}_{11}}{{\overline{a}}_{11}{\overline{a}}_{22}+{\overline{a}}_{12}{\overline{a}}_{21}}>\frac{{\overline{a}}_{21}[{\overline{r}}_1-{(\frac{a_{11}}{a_{21}})}^M{\overline{r}}_2]}{{\overline{a}}_{11}{\overline{a}}_{22}+{\overline{a}}_{12}{\overline{a}}_{21}}.\end{array}$$

From the assumption of Theorem 2, the system of algebraic equations has a unique positive solution $v^{\ast }=(v_1^{\ast },v_2^{\ast })$. Hence, the equation $QNu=0$ has a unique solution $u^{\ast }=(u_1^{\ast },u_2^{\ast })=(ln{v}_1^{\ast },ln{v}_2^{\ast },ln)\in R^2$.

Choosing constant $H>0$ large enough such that $|u_1^{\ast }|+|u_2^{\ast }|<H$ and $H>H_1+H_2$, we define a bounded open set $\mathrm{\Omega }\subset X$ as follows

$$\begin{array}{c}\hfill \mathrm{\Omega }=\{u\in X:\Vert u\Vert <H\}.\end{array}$$

It is clear that $\mathrm{\Omega }$ satisfies conditions $(a)$ and $(b)$ of Lemma 3. On the other hand, by directly calculating we can obtain

$$\begin{array}{c}\hfill \text{deg}\{JQN,\mathrm{\Omega }\cap \text{Ker}L,(0,0)\}\\ \hfill =\text{sgn}\left|\begin{array}{cc}-{\overline{a}}_{11}{K}_1& -{\overline{a}}_{12}{K}_2\\ {\overline{a}}_{21}{K}_1& -{\overline{a}}_{22}{K}_2\end{array}\right|,\end{array}$$

where $K_i=exp\{u_i\}(i=1,2)$.

From the assumption of Theorem 2, we have

$$\begin{array}{c}\hfill \left|\begin{array}{cc}-{\overline{a}}_{11}{K}_1& -{\overline{a}}_{12}{K}_2\\ {\overline{a}}_{21}{K}_1& -{\overline{a}}_{22}{K}_2\end{array}\right|\ne 0.\end{array}$$

From this, we finally have

$$\begin{array}{c}\hfill \text{deg}\{JQN,\mathrm{\Omega }\cap \text{Ker}L,(0,0)\}\ne 0.\end{array}$$

This shows that $\mathrm{\Omega }$ satisfies condition $(c)$ of Lemma 3. Therefore, system (4) has a $\mathit{\omega }$-periodic solution $u^{\ast }(t)=(u_1^{\ast }(t),u_2^{\ast }(t))\in \overline{\mathrm{\Omega }}$. Hence, system (2) has a positive $\mathit{\omega }$-periodic solution $x^{\ast }(t)=(x_1^{\ast }(t),x_2^{\ast }(t))$. $\square$

Theorem 3

Suppose that assumptions of Theorem 2 hold. Further suppose that the following ($H_2$) holds.

($H_2$) There exists a constant ${\mathit{\mu }}_i>0(i=1,2)$ such that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; inf}{A}_i(t)>0,i=1,2,\end{array}$$

where

$$\begin{array}{cc}\hfill A_1(t)& ={\mathit{\mu }}_1{a}_{11}(t)-{\mathit{\mu }}_1{\int }_{t-{\mathit{\tau }}_1}^t{a}_{11}(u+{\mathit{\tau }}_1)du[r_1(t)+(a_{11}(t)+a_{12}(t))M]\hfill \\ \hfill & -M\sum _{j=1}^2{\mathit{\mu }}_j{\mathit{\tau }}_j{a}_{jj}^M{a}_{j1}(t+{\mathit{\tau }}_j)-{\mathit{\mu }}_2{a}_{21}(t+{\mathit{\tau }}_2),\hfill \\ \hfill A_2(t)& ={\mathit{\mu }}_2{a}_{22}(t)-{\mathit{\mu }}_2{\int }_{t-{\mathit{\tau }}_2}^t{a}_{22}(u+{\mathit{\tau }}_2)du[r_2(t)+(a_{21}(t)+a_{22}(t))M]\hfill \\ \hfill & -M\sum _{j=1}^2{\mathit{\mu }}_j{\mathit{\tau }}_j{a}_{jj}^M{a}_{j2}(t+{\mathit{\tau }}_j)-{\mathit{\mu }}_1{a}_{12}(t+{\mathit{\tau }}_1),\hfill \end{array}$$

where $M=max\{M_1,M_2\}$ and $M_1,M_2$ are defined in Theorem 1. Then system (2) has a positive periodic solution which is globally attractive.

Proof

From Theorem 2, we can obtain that system (2) has a positive periodic solution.

Let $x(t)=(x_1(t),x_2(t))$ be a positive periodic solution of system (2) and $(y_1(t),y_2(t))$ be a any positive solution of system (2). From Theorem 1, choose positive constants $m_i>0$, $M_i>0$ such that

$$\begin{array}{c}\hfill m\le x_i(t)\le M,i=1,2,\end{array}$$

(19)

where $M=max\{M_1,M_2\}$ and $m=min\{m_1,m_2\}$ for all $t\ge T$. Let

$$\begin{array}{c}\hfill W_1(t)=\sum _{i=1}^2{\mathit{\mu }}_i|ln{x}_i(t)-ln{y}_i(t)|.\end{array}$$

Calculating the upper right derivation of $W_1(t)$ along system (2) for all $t\ge T$, we have

$$\begin{array}{cc}\hfill D^{+}{W}_1(t)=& {\mathit{\mu }}_1\text{sign}(x_1(t)-y_1(t))[-a_{11}(t)(x_1(t-{\mathit{\tau }}_1)-y_1(t-{\mathit{\tau }}_1))\hfill \\ \hfill & -a_{12}(t)(x_2(t-{\mathit{\tau }}_1)-y_2(t-{\mathit{\tau }}_1))]+{\mathit{\mu }}_2\text{sign}(x_2(t)-y_2(t))\hfill \\ \hfill & \times [a_{21}(t)(x_1(t-{\mathit{\tau }}_2)-y_1(t-{\mathit{\tau }}_2))-a_{22}(t)(x_2(t-{\mathit{\tau }}_2)-y_2(t-{\mathit{\tau }}_2))]\hfill \\ \hfill =& {\mathit{\mu }}_1\text{sign}(x_1(t)-y_1(t))[-a_{11}(t)(x_1(t)-y_1(t))-a_{12}(t)(x_2(t-{\mathit{\tau }}_1)-y_2(t-{\mathit{\tau }}_1))\hfill \\ \hfill & +a_{11}(t){\int }_{t-{\mathit{\tau }}_1}^t({\dot{x}}_1(u)-{\dot{y}}_1(u))du]+{\mathit{\mu }}_2\text{sign}(x_2(t)-y_2(t))[-a_{22}(t)(x_2(t)-y_2(t))\hfill \\ \hfill & +a_{21}(t)(x_1(t-{\mathit{\tau }}_2)-y_1(t-{\mathit{\tau }}_2))+a_{22}(t){\int }_{t-{\mathit{\tau }}_2}^t({\dot{x}}_2(u)-{\dot{y}}_2(u))du]\hfill \\ \hfill =& {\mathit{\mu }}_1\text{sign}(x_1(t)-y_1(t))[-a_{11}(t)(x_1(t)-y_1(t))-a_{12}(t)(x_2(t-{\mathit{\tau }}_1)-y_2(t-{\mathit{\tau }}_1))\hfill \\ \hfill & +a_{11}(t){\int }_{t-{\mathit{\tau }}_1}^t(x_1(u)[r_1(u)-a_{11}(u)x_1(u-{\mathit{\tau }}_1)-a_{12}(u)x_2(u-{\mathit{\tau }}_1)]\hfill \\ \hfill & -y_1(u)[r_1(u)-a_{11}(u)y_1(u-{\mathit{\tau }}_1)-a_{12}(u)y_2(u-{\mathit{\tau }}_1)])du]\hfill \\ \hfill & +{\mathit{\mu }}_2\text{sign}(x_2(t)-y_2(t))[-a_{22}(t)(x_2(t)-y_2(t))+a_{21}(t)(x_1(t-{\mathit{\tau }}_2)-y_1(t-{\mathit{\tau }}_2))\hfill \\ \hfill & +a_{22}(t){\int }_{t-{\mathit{\tau }}_2}^t(x_2(u)[r_2(u)+a_{21}(u)x_1(u-{\mathit{\tau }}_2)-a_{22}(u)x_2(u-{\mathit{\tau }}_2)]\hfill \\ \hfill & -y_2(u)[r_2(u)+a_{21}(u)y_1(u-{\mathit{\tau }}_2)-a_{22}(u)y_2(u-{\mathit{\tau }}_2)])du]\hfill \\ \hfill =& {\mathit{\mu }}_1\text{sign}(x_1(t)-y_1(t))[-a_{11}(t)(x_1(t)-y_1(t))-a_{12}(t)(x_2(t-{\mathit{\tau }}_1)-y_2(t-{\mathit{\tau }}_1))\hfill \\ \hfill & +a_{11}(t){\int }_{t-{\mathit{\tau }}_1}^t((x_1(u)-y_1(u))[r_1(u)-a_{11}(u)y_1(u-{\mathit{\tau }}_1)-a_{12}(u)y_2(u-{\mathit{\tau }}_1)]\hfill \\ \hfill & +x_1(u)[-a_{11}(u)(x_1(u-{\mathit{\tau }}_1)-y_1(u-{\mathit{\tau }}_1))-a_{12}(u)(x_2(u-{\mathit{\tau }}_1)-y_2(u-{\mathit{\tau }}_1))])du]\hfill \\ \hfill & +{\mathit{\mu }}_2\text{sign}(x_2(t)-y_2(t))[-a_{22}(t)(x_2(t)-y_2(t))+a_{21}(t)(x_1(t-{\mathit{\tau }}_2)-y_1(t-{\mathit{\tau }}_2))\hfill \\ \hfill & +a_{22}(t){\int }_{t-{\mathit{\tau }}_2}^t((x_2(u)-y_2(u))[r_2(u)+a_{21}(u)y_1(u-{\mathit{\tau }}_2)-a_{22}(u)y_2(u-{\mathit{\tau }}_2)]\hfill \\ \hfill & +x_2(u)[a_{21}(u)(x_1(u-{\mathit{\tau }}_2)-y_1(u-{\mathit{\tau }}_2))-a_{22}(u)(x_2(u-{\mathit{\tau }}_2)-y_2(u-{\mathit{\tau }}_2))])du]\hfill \\ \hfill \le & -\sum _{i=1}^2{\mathit{\mu }}_i{a}_{ii}(t)|x_i(t)-y_i(t)|+{\mathit{\mu }}_1{a}_{12}(t)|x_2(t-{\mathit{\tau }}_1)-y_2(t-{\mathit{\tau }}_1)|\hfill \\ \hfill & +{\mathit{\mu }}_2{a}_{21}(t)|x_1(t-{\mathit{\tau }}_2)-y_1(t-{\mathit{\tau }}_2)|+{\mathit{\mu }}_1{a}_{11}(t){\int }_{t-{\mathit{\tau }}_1}^t(|x_1(u)-y_1(u)|[r_1(u)\hfill \\ \hfill & +a_{11}(u)y_1(u-{\mathit{\tau }}_1)+a_{12}(u)y_2(u-{\mathit{\tau }}_1)]+x_1(u)[a_{11}(u)|x_1(u-{\mathit{\tau }}_1)-y_1(u-{\mathit{\tau }}_1)|\hfill \\ \hfill & +a_{12}(u)|x_2(u-{\mathit{\tau }}_1)-y_2(u-{\mathit{\tau }}_1)|])du\hfill \\ \hfill & +{\mathit{\mu }}_2{a}_{22}(t){\int }_{t-{\mathit{\tau }}_2}^t(|x_2(u)-y_2(u)|[r_2(u)+a_{21}(u)y_1(u-{\mathit{\tau }}_2)+a_{22}(u)y_2(u-{\mathit{\tau }}_2)]\hfill \\ \hfill & +x_2(u)[a_{21}(u)|x_1(u-{\mathit{\tau }}_2)-y_1(u-{\mathit{\tau }}_2)|+a_{22}(u)|x_2(u-{\mathit{\tau }}_2)-y_2(u-{\mathit{\tau }}_2)|])du.\hfill \end{array}$$

(20)

Define

$$\begin{array}{c}\hfill W_2(t)={\mathit{\mu }}_1{V}_1(t)+{\mathit{\mu }}_2{V}_2(t),\end{array}$$

where

$$\begin{array}{cc}\hfill V_1(t)=& {\int }_{t-{\mathit{\tau }}_1}^t{\int }_u^t{a}_{11}(u+{\mathit{\tau }}_1)([r_1(s)+a_{11}(s)y_1(s-{\mathit{\tau }}_1)+a_{12}(s)y_2(s-{\mathit{\tau }}_1)]|x_1(s)-y_1(s)|\hfill \\ \hfill & +x_1(s)[a_{11}(s)|x_1(s-{\mathit{\tau }}_1)-y_1(s-{\mathit{\tau }}_1)|+a_{12}(s)|x_2(s-{\mathit{\tau }}_1)-y_2(s-{\mathit{\tau }}_1)|)dsdu,\hfill \\ \hfill V_2(t)=& {\int }_{t-{\mathit{\tau }}_2}^t{\int }_u^t{a}_{22}(u+{\mathit{\tau }}_2)([r_2(s)+a_{21}(s)y_1(s-{\mathit{\tau }}_2)+a_{22}(s)y_2(s-{\mathit{\tau }}_2)]|x_2(s)-y_2(s)|\hfill \\ \hfill & +x_2(s)[a_{21}(s)|x_1(s-{\mathit{\tau }}_2)-y_1(s-{\mathit{\tau }}_2)|+a_{22}(s)|x_2(s-{\mathit{\tau }}_2)-y_2(s-{\mathit{\tau }}_2)|)dsdu,\hfill \end{array}$$

Calculating the upper right derivative, from (20) we have

$$\begin{array}{cc}\hfill \sum _{i=1}^2{D}^{+}{W}_i(t)\le & -\sum _{i=1}^2{\mathit{\mu }}_i{a}_{ii}(t)|x_i(t)-y_i(t)|+{\mathit{\mu }}_1{a}_{12}(t)|x_2(t-{\mathit{\tau }}_1)-y_2(t-{\mathit{\tau }}_1)|\hfill \\ \hfill & +{\mathit{\mu }}_2{a}_{21}(t)|x_1(t-{\mathit{\tau }}_2)-y_1(t-{\mathit{\tau }}_2)|+{\mathit{\mu }}_1{\int }_{t-{\mathit{\tau }}_1}^t{a}_{11}(u+{\mathit{\tau }}_1)du[r_1(t)\hfill \\ \hfill & +(a_{11}(t)+a_{12}(t))M]|x_1(t)-y_1(t)|+{\mathit{\mu }}_1{\mathit{\tau }}_1{a}_{11}^{M}M\sum _{i=1}^2{a}_{1i}(t)\hfill \\ \hfill & \times |x_i(t-{\mathit{\tau }}_1)-y_i(t-{\mathit{\tau }}_1)|+{\mathit{\mu }}_2{\int }_{t-{\mathit{\tau }}_2}^t{a}_{22}(u+{\mathit{\tau }}_2)du[r_2(t)\hfill \\ \hfill & +(a_{21}(t)+a_{22}(t))M]|x_2(t)-y_2(t)|+{\mathit{\mu }}_2{\mathit{\tau }}_2{a}_{22}^{M}M\sum _{i=1}^2{a}_{2i}(t)\hfill \\ \hfill & \times \sum _{i=1}^2{a}_{2i}(t)|x_i(t-{\mathit{\tau }}_2)-y_i(t-{\mathit{\tau }}_2)|.\hfill \end{array}$$

(21)

Define

$$\begin{array}{c}\hfill W_3(t)={\mathit{\mu }}_1{V}_3(t)+{\mathit{\mu }}_2{V}_4(t),\end{array}$$

where

$$\begin{array}{cc}\hfill V_3(t)=& {\mathit{\mu }}_1{\mathit{\tau }}_1{a}_{11}^{M}M\sum _{i=1}^2{\int }_{t-{\mathit{\tau }}_1}^t{a}_{1i}(u+{\mathit{\tau }}_1)|x_i(u)-y_i(u)|du+{\int }_{t-{\mathit{\tau }}_1}^t{a}_{12}(u+{\mathit{\tau }}_1)|x_2(u)-y_2(u)|du,\hfill \\ \hfill V_4(t)=& {\mathit{\mu }}_2{\mathit{\tau }}_2{a}_{22}^{M}M\sum _{i=1}^2{\int }_{t-{\mathit{\tau }}_1}^t{a}_{2i}(u+{\mathit{\tau }}_2)|x_i(u)-y_i(u)|du+{\int }_{t-{\mathit{\tau }}_2}^t{a}_{21}(u+{\mathit{\tau }}_2)|x_1(u)-y_1(u)|du.\hfill \end{array}$$

Further, we define a Liapunov function as follows

$$\begin{array}{c}\hfill V(t)=\sum _{i=1}^3{W}_i(t).\end{array}$$

Calculating the upper right derivation of $V(t)$, from (20) and (21) we finally can obtain for all $t\ge T$

$$\begin{array}{c}\hfill D^{+}V(t)\le -\sum _{i=1}^2{A}_i(t)|x_i(t)-y_i(t)|.\end{array}$$

(22)

From assumption ($H_2$), there exists a constant $\mathit{\alpha }>0$ and $T^{\ast }\ge T$ such that for all $t\ge T^{\ast }$ we have

$$\begin{array}{c}\hfill A_i(t)\ge \mathit{\alpha }>0,i=1,2.\end{array}$$

(23)

Integrating from $T^{\ast }$ to $t$ on both sides of (22) and by (23) produces

$$\begin{array}{c}\hfill V(t)+\mathit{\alpha }{\int }_{T^{\ast }}^t(\sum _{i=1}^2|x_i(s)-y_i(s)|)ds\le V(T^{\ast }),\end{array}$$

(24)

then

$$\begin{array}{c}\hfill {\int }_0^t(\sum _{i=1}^2|x_i(s)-y_i(s)|)ds\le \frac{V(T^{\ast })}{\mathit{\alpha }},t\ge T^{\ast }.\end{array}$$

(25)

By the definition of $V(t)$ and (24) we have

$$\begin{array}{c}\hfill \sum _{i=1}^2{\mathit{\mu }}_i|ln{x}_i(t)-ln{y}_i(t)|\le V(t)\le V(T^{\ast }),t\ge 0.\end{array}$$

(26)

Therefore, for $i=1,2$ we have

$$\begin{array}{c}\hfill {\mathit{\mu }}_i|ln{x}_i(t)-ln{y}_i(t)|\le V(T^{\ast }),t\ge 0.\end{array}$$

(27)

which, together with (19), lead to

$$\begin{array}{c}\hfill m_{i}exp\{-V(T^{\ast })/{\mathit{\mu }}_i\}\le y_i(t)\le M_{i}exp\{V(T^{\ast })/{\mathit{\mu }}_i\},i=1,2.\end{array}$$

(28)

From the boundedness of $x_i(t)$ and (25), it follows that $y_i(t)(i=1,2)$ are bounded for $t\ge 0$. From the boundedness of $x_i(t)$ and $y_i(t)$ we know that the derivatives ${\dot{x}}_i(t)$ and ${\dot{y}}_i(t)$ are bounded. Furthermore, we can obtain that $x_i(t)-y_i(t)$ and their derivatives remain bounded on $[0,+\infty )$. Therefore ${\sum }_{i=1}^2|x_i(t)-y_i(t)|$ is uniformly continuous on $[0,+\infty )$. By Barbalat’s theorem it follows that:

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}\sum _{i=1}^2|x_i(t)-y_i(t)|=0.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}(x_i(t)-y_i(t))=0,i=1,2.\end{array}$$

This completes the proof of Theorem 3. $\square$

From the global attractivity of bounded positive solutions, we have the following result.

Corollary 1

Suppose that the conditions of Theorem 3 hold, then system (2) is permanent.

As a direct corollary of Lemma 2, we have

Corollary 2

Suppose that $a_{21}^M{M}_1<r_2^L$, then the predator species of system (2) goes to extinction.

Application

In this section, we will apply the results in Sect. 3 to the following predator–prey system with pure delays

$$\begin{array}{cc}\hfill {\dot{x}}_1(t)& =x_1(t)[r_1(t)-a_{11}(t)x_1(t-{\mathit{\tau }}_1)-a_{12}(t)x_2(t-{\mathit{\tau }}_1)],\hfill \\ \hfill {\dot{x}}_2(t)& =x_2(t)[r_2(t)+a_{21}(t)x_1(t-{\mathit{\tau }}_2)-a_{22}(t)x_2(t-{\mathit{\tau }}_2)].\hfill \end{array}$$

(29)

Corollary 3

Suppose that assumption (H1) holds, then there exist positive constants $N_i(i=1,2)$ such that

$$\begin{array}{c}\hfill x_i(t)\le N_i,\end{array}$$

for any positive solution $x_i(t)$ of system (29).

Corollary 4

Suppose that assumption (H1) holds and ${\displaystyle {\overline{r}}_1-{(\frac{a_{12}}{a_{22}})}^M{\overline{r}}_2>0}$. Then system (29) has at least one positive $\mathit{\omega }-$ periodic solution.

Corollary 5

Suppose that assumptions of Corollary 4 hold. Further suppose that the following ($H_3$) holds.

($H_3$) There exists a constant ${\mathit{\nu }}_i>0(i=1,2)$ such that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; inf}{B}_i(t)>0,i=1,2,\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_1(t)=& {\displaystyle {\mathit{\nu }}_1{a}_{11}(t)-{\mathit{\nu }}_1{\int }_{t-{\mathit{\tau }}_1}^t{a}_{11}(u+{\mathit{\tau }}_1)du[r_1(t)+(a_{11}(t)+a_{12}(t))N]}\hfill \\ \hfill & {\displaystyle -N{\sum }_{j=1}^2{\mathit{\nu }}_j{\mathit{\tau }}_j{a}_{jj}^M{a}_{j1}(t+{\mathit{\tau }}_j)-{\mathit{\nu }}_2{a}_{21}(t+{\mathit{\tau }}_2),}\hfill \\ \hfill B_2(t)=& {\displaystyle {\mathit{\nu }}_2{a}_{22}(t)-{\mathit{\nu }}_2{\int }_{t-{\mathit{\tau }}_2}^t{a}_{22}(u+{\mathit{\tau }}_2)du[r_2(t)+(a_{21}(t)+a_{22}(t))N]}\hfill \\ \hfill & {\displaystyle -N{\sum }_{j=1}^2{\mathit{\nu }}_j{\mathit{\tau }}_j{a}_{jj}^M{a}_{j2}(t+{\mathit{\tau }}_j)-{\mathit{\nu }}_1{a}_{12}(t+{\mathit{\tau }}_1),}\hfill \end{array}\end{array}$$

where $N=max\{N_1,N_2\}$. Then system (29) has a positive periodic solution which is globally attractive.

Corollary 6

Suppose that the conditions of Theorem 3 hold, then system (29) is permanent.