1 Introduction

Ecosystems are characterized by the interaction between different species and natural environment. One of the important types of interaction, which has effect on population dynamics, is predation. Thus, predator-prey models have been the focus of ecological science since the early days of this discipline [1]. Since the great work of Lotka (in 1925) and Volterra (in 1926), modeling predator-prey interaction has been one of the central themes in mathematical ecology [2, 3].

Owing to severe competition, natural enemy, or deterioration of the patch environment, the migration phenomena of biological species can often occur between heterogeneous spatial environments and patches. More recently, increasing attention has been paid to the dynamics of a large number of mathematical models with diffusion, and many nice results have been obtained. The persistence and extinction for ordinary differential equation and delayed differential equation models were investigated in [4,5,6]. Global stability of periodic solution for the model with diffusion was studied in [7,8,9,10,11,12]. Particularly, the predator-prey system with the prey dispersal was also studied in [13,14,15,16,17]. Regretfully, in all of the above population dispersing systems, they always assumed that the dispersal occurs at every time. For example, Zhang and Teng investigated the following periodic predator-prey Lotka–Volterra type system with prey dispersal in n patches in [14]:

$$ \textstyle\begin{cases} \dot{x}_{1}(t)= x_{1}(t) [a_{1}(t)-b_{1}(t)x_{1}(t)-c(t)y(t) ] \\ \hphantom{\dot{x}_{1}(t)=}{}+ \sum_{j=1}^{n}d_{1j}(t)(x_{j}(t)-x_{1}(t)), \\ \dot{x}_{i}(t)= x_{i}(t) [a_{i}(t)-b_{i}(t)x_{i}(t) ]+\sum_{j=1}^{n}d_{ij}(t)(x _{j}(t)-x_{i}(t)), \\ \dot{y}(t)= y(t)[-e(t)+f(t)x_{1}(t)], \quad i=2,3,\ldots ,n, \end{cases} $$
(1.1)

where \(e(t)\) denotes the death rate of the predator, \(d_{ij}(t)\) (\(i, j \in I, i\neq j\)) represents the dispersal rate of the prey species from the ith patch to the jth patch. Sufficient conditions on the boundedness, permanence, and existence of a positive periodic solution for system (1.1) are established.

Actually, many man-made factors (e.g., drought, hunting, harvesting, breeding, fire, etc.) always lead to rapid increase or decrease of population number at some transitory time slots. These short-term perturbations were often assumed to be in the form of impulses. For example, birds often migrate between patches in winter to find suitable environments. Impulsive differential equations [18] have attracted the interest of researchers, and many important studies have been performed [19,20,21,22,23].

It is well known that time delay is quite common for a natural population. Therefore, it is necessary to take the effect of time delay into account in forming a biologically meaningful mathematical model. Recently, many impulsive predator-prey models with dispersion and time delay have been investigated in [24,25,26,27,28]. For example in [24], Li and Zhang proposed and studied the following delayed predator-prey system with impulsive diffusion:

$$ \textstyle\begin{cases} \dot{x}_{1}(t)= x_{1}(t) [r_{1}-a_{1}x_{1}(t) ]-\beta x_{1}(t)y(t), \\ \dot{x}_{2}(t)= -r_{2}x_{2}(t),& t\ne nT, \\ \dot{y}(t)= y(t)[-d_{1}+k\beta x_{1}(t-\tau _{1})-a_{2}y(t-\tau _{2})], \\ \Delta x_{1}(t)=d_{21}x_{2}(t)-d_{12}x_{1}(t), \\ \Delta x_{2}(t)=d_{12}x_{1}(t)-d_{21}x_{2}(t)],& t= nT, \\ \Delta y(t)=0, \end{cases} $$
(1.2)

with the initial conditions

$$\begin{aligned}& x_{1}(s)=\phi _{1}(s),\qquad x_{2}(s)=\phi _{2}(s), \\& y(s)=\phi _{3}(s),\qquad \tau =\max \{\tau _{1},\tau _{2}\}, \\& \phi =\bigl(\phi _{1}(s),\phi _{2}(s),\phi _{3}(s)\bigr)^{T}\in C\bigl([-\tau ,0],R^{3} _{+}\bigr),\qquad \phi _{i}(0)>0,\quad i=1,2,3. \end{aligned}$$

In system (1.2), they assumed that the ecosystem was composed of two isolated patches and the breeding area was damaged in patch 2. By using comparison theorem of impulsive differential equation and some analysis techniques, they got the global attractivity of predator-extinction periodic solution and permanence of the system.

Many single species models with impulsive diffusion and dispersal delay have been investigated, too. In [20], the authors studied a single species model with symmetric bidirectional impulsive diffusion and dispersal delay:

$$ \textstyle\begin{cases} \textstyle\begin{array}{@{}rcl} \dot{N}_{1}(t)= r_{1}N_{1}(t)\ln \frac{k_{1}}{N_{1}(t)}, \\ \dot{N}_{2}(t)= r_{2}N_{2}(t)\ln \frac{k_{2}}{N_{2}(t)}, \end{array}\displaystyle & t\ne nT, \\ \textstyle\begin{array}{@{}rcl} \Delta N_{1}(t) =d_{1}[N_{2}(t-\tau _{0})-N_{1}(t)], \\ \Delta N_{2}(t) =d_{2}[N_{1}(t-\tau _{0})-N_{2}(t)], \end{array}\displaystyle & t= nT, \end{cases} $$
(1.3)

where \(r_{i}\) (\(i=1,2\)) stands for the intrinsic growth rate of the population \(N_{i}\), and \(d_{i}\) represents the dispersal rate in the ith patch. \(\tau _{0}\) is the time delay, that is, a period of time of species \(N_{i}\) disperse between patches (\(\tau _{0}< T\)). Sufficient criteria were obtained for the permanence, existence, uniqueness, and global stability of positive periodic solutions by using discrete dynamical system theory.

Motivated by the above analysis, in this paper, based on system (1.3), we consider a predator-prey model with prey symmetric bidirectional impulsive diffusion and dispersal delay between two patches:

$$ \textstyle\begin{cases} \dot{N}_{1}(t)= r_{1}N_{1}(t)\ln \frac{k_{1}}{N_{1}(t)}, \\ \dot{N}_{2}(t)= N_{2}(t) [r_{2}\ln \frac{k_{2}}{N_{2}(t)}-c_{1}y(t) ],& t\ne nT, \\ \dot{y}(t)= y(t)[-r_{3}+c_{2}N_{2}(t-\tau _{1})-c_{3}y(t-\tau _{2})], \\ \Delta N_{1}(t)=d_{1}[N_{2}(t-\tau _{0})-N_{1}(t)], \\ \Delta N_{2}(t)=d_{2}[N_{1}(t-\tau _{0})-N_{2}(t)],&t= nT, n=1,2,\ldots , \\ \Delta y(t)=0, \end{cases} $$
(1.4)

with the initial conditions

$$\begin{aligned}& N_{1}(s)=\phi _{1}(s),\qquad N_{2}(s)=\phi _{2}(s), \\& y(s)=\phi _{3}(s),\qquad \tau =\max \{\tau _{0},\tau _{1},\tau _{2}\}, \\& \phi =\bigl(\phi _{1}(s),\phi _{2}(s),\phi _{3}(s)\bigr)^{T}\in C\bigl([-\tau ,0],R^{3} _{+}\bigr), \qquad \phi _{i}(0)>0,\quad i=1,2,3, \end{aligned}$$

where \(N_{i}(t)\) (\(i = 1, 2\)) denotes the density of the prey species in the ith patch at time t; \(y(t)\) denotes the density of the predator species at time t. Predator species is confined to the second patch while the prey species can disperse between two patches. \(\tau _{0}\) is a positive constant (\(\tau _{0}< T\)), which represents the time for the species to disperse between patches. \(\tau _{1}\geqslant 0\) is a constant delay due to the gestation of the predator. The term \(-c_{3}y(t-\tau _{2})\) is the negative feedback of predator crowding. We will use methods similar to those of [24] to analyze our predator-prey model with prey symmetric bidirectional impulsive diffusion and dispersal delay.

2 Preliminaries

Firstly, for simplicity and convenience, we let \(x_{1}=\frac{N_{1}}{k _{1}}\), \(x_{2}=\frac{N_{2}}{k_{2}}\), \(k=\frac{k_{2}}{k_{1}}\), then system (1.4) can be written as follows:

$$ \textstyle\begin{cases} \dot{x_{1}}(t)= r_{1}x_{1}(t)\ln \frac{1}{x_{1}(t)}, \\ \dot{x_{2}}(t)= x_{2}(t)[r_{2}\ln \frac{1}{x_{2}(t)}-c_{1}y(t)],& t\ne nT, \\ \dot{y}(t)= y(t)[-r_{3}+k_{2}c_{2}x_{2}(t-\tau _{1})-c_{3}y(t-\tau _{2})], \\ \Delta x_{1}(t)= d_{1}[kx_{2}(t-\tau _{0})-x_{1}(t)], \\ \Delta x_{2}(t)= d_{2}[\frac{1}{k}x_{1}(t-\tau _{0})-x_{2}(t)],&t= nT, n=1,2,\ldots . \\ \Delta y(t)=0, \end{cases} $$
(2.1)

Next, we discuss the dynamical behaviors of the following single species model:

$$ \textstyle\begin{cases} \textstyle\begin{array}{@{}rcl} \dot{v}_{1}(t)= r_{1}v_{1}(t)\ln \frac{1}{v_{1}(t)}, \\ \dot{v}_{2}(t)= r_{2}v_{2}(t)\ln \frac{1}{v_{2}(t)}, \end{array}\displaystyle & t\ne nT, \\ \textstyle\begin{array}{@{}lll} \Delta v_{1}(t)= d_{1}[kv_{2}(t-\tau _{0})-v_{1}(t)], \\ \Delta v_{2}(t)= d_{2}[\frac{1}{k}v_{1}(t-\tau _{0})-v_{2}(t)], \end{array}\displaystyle & t= nT. \end{cases} $$
(2.2)

We introduce the following assumptions for system (2.2):

(\(H_{1}\)):

\(0< d_{1}+d_{2}<1\),

(\(H_{2}\)):

\(b_{1}+b_{2}+d_{1}\leqslant 1\),

(\(H_{3}\)):

\(1-b_{i}\leqslant (1-b_{i}e^{r_{i}\tau _{0}})e^{(r_{1}+r _{2})\tau _{0}}\), \(i=1,2\),

where \(b_{i}=e^{-r_{i}T}\).

Lemma 2.1

([20])

Suppose that assumptions (\(H_{1}\))–(\(H_{3}\)) hold, then system (2.2) has a unique globally attractive positive T-periodic solution \((v^{*}_{1}(t),v^{*}_{2}(t))\), that is,

$$ \lim_{t\to \infty }\bigl(v_{1}(t),v_{2}(t)\bigr)= \bigl(v^{*}_{1}(t),v^{*}_{2}(t)\bigr). $$

Next, we consider the following system:

$$ \textstyle\begin{cases} \textstyle\begin{array}{@{}l} \dot{v}_{1\alpha }(t)= r_{1}v_{1\alpha }(t)\ln \frac{1}{v_{1\alpha }(t)},\\ \dot{v}_{2\alpha }(t)= v_{2\alpha }(t) [r_{2}\ln \frac{1}{v_{2\alpha }(t)}-\alpha ],\end{array}\displaystyle & t\ne nT,\\ \textstyle\begin{array}{@{}l} \Delta v_{1\alpha }(t)= d_{1}[kv_{2\alpha }(t-\tau _{0})-v_{1\alpha }(t)], \\ \Delta v_{2\alpha }(t)= d_{2} [\frac{1}{k}v_{1\alpha }(t-\tau _{0})-v_{2\alpha }(t) ], \end{array}\displaystyle & t= nT, \end{cases} $$
(2.3)

where α is a positive constant.

Let \(u_{1}(t)=v_{1\alpha }(t)\), \(u_{2}(t)=e^{\frac{\alpha }{r_{2}}}v_{2\alpha }(t)\), then system (2.3) is transformed into the following form:

$$ \textstyle\begin{cases} \textstyle\begin{array}{@{}l} \dot{u}_{1}(t)= r_{1}u_{1}(t)\ln \frac{1}{u_{1}(t)},\\ \dot{u}_{2}(t)= r_{2}u_{2}(t)\ln \frac{1}{u_{2}(t)},\end{array}\displaystyle & t\ne nT,\\ \textstyle\begin{array}{@{}l} \Delta u_{1}(t)= d_{1}[k^{*}u_{2}(t-\tau _{0})-u_{1}(t)], \\ \Delta u_{2}(t)= d_{2}[\frac{1}{k^{*}}u_{1}(t-\tau _{0})-u_{2}(t)], \end{array}\displaystyle & t= nT, \end{cases} $$
(2.4)

where \(k^{*}=ke^{-\frac{\alpha }{r_{2}}}\).

Therefore system (2.3) has the following result as system (2.2).

Lemma 2.2

Suppose that assumptions (\(H_{1}\))–(\(H_{3}\)) hold, then system (2.3) has a unique globally attractive positive T-periodic solution \((v^{*}_{1\alpha }(t),v^{*}_{2\alpha }(t))\), that is,

$$ \lim_{t\to \infty }\bigl(v_{1\alpha }(t),v_{2\alpha }(t)\bigr)= \bigl(v^{*}_{1\alpha } (t),v^{*}_{2\alpha }(t)\bigr). $$

Definition 2.1

For any positive solution \((x_{1}(t),x_{2}(t), y(t))\) of system (2.1), if there are positive constants m and M such that

$$ m\leqslant x_{i}(t)\leqslant M,\qquad m\leqslant y(t)\leqslant M, \quad i=1,2, \text{as } t\rightarrow \infty , $$

then system (2.1) is said to be permanent.

Lemma 2.3

([29])

Assume that for \(y(t)>0\), \(t\geqslant 0\), it holds that

$$ \dot{y}(t)\leqslant y(t) \bigl(a-by(t-\tau )\bigr) $$
(2.5)

with initial conditions, \(y(s)=\phi (s)\geqslant 0\) for \(s \in [-\tau ,0]\), where a, b are positive constants. Then

$$ \limsup_{t\to +\infty }y(t)\leqslant \frac{ae^{a\tau }}{b}. $$
(2.6)

3 Main results

Theorem 3.1

Suppose that assumptions (\(H_{1}\))–(\(H_{3}\)) hold. If

(\(H_{4}\)):

\(k_{2} c_{2}\min_{t\in [0,T]}v^{*}_{2}(t)>r_{3}\),

then system (2.1) is permanent.

Proof

We first prove the ultimate boundedness of all positive solutions of system (2.1). Let \((x_{1}(t),x_{2}(t),y(t))\) be any positive solution of system (2.1). Then we obtain

$$ \textstyle\begin{cases} \textstyle\begin{array}{@{}l} \dot{x}_{1}(t)= r_{1}x_{1}(t)\ln \frac{1}{x_{1}(t)}, \\ \dot{x}_{2}(t)\leqslant r_{2}x_{2}(t)\ln \frac{1}{x_{2}(t)}, \end{array}\displaystyle & t\ne nT, \\ \textstyle\begin{array}{@{}l} \Delta x_{1}(t)= d_{1}[kx_{2}(t-\tau _{0})-x_{1}(t)], \\ \Delta x_{2}(t)= d_{2}[\frac{1}{k}x_{1}(t-\tau _{0})-x_{2}(t)], \end{array}\displaystyle & t= nT, \end{cases} $$
(3.1)

for all \(t>\tau _{0}\). Consider the auxiliary system (2.2). From Lemma 2.1 and the comparison theorem of impulsive differential equations, we have that, for any constant \(\varepsilon >0\) small enough, there is \(T_{0}>0\) such that

$$ x_{i}(t)\leqslant v_{i}(t)< v^{*}_{i}(t)+ \varepsilon \leqslant \max_{t\in [0,T]}v^{*}_{i}(t)+ \varepsilon \triangleq M_{i}, \quad i=1,2, $$
(3.2)

for all \(t\geqslant T_{0}\). Hence, from the third equation of (2.1) and (3.2), we have

$$ \dot{y}(t)\leqslant y(t)\bigl[-r_{3}+k_{2} c_{2}M_{2}-c_{3}y(t-\tau _{2})\bigr], \quad t\geqslant T_{0}+\tau . $$

By Lemma 2.3, we can obtain

$$ y(t)\leqslant \frac{-r_{3}+k_{2}c_{2}M_{2}}{c_{3}}e^{(-r_{3}+c_{2}k _{2}M_{2})\tau _{2}}\triangleq M_{3}, \quad t\geqslant T_{0}+\tau , $$
(3.3)

where \(-r_{3}+k_{2}c_{2}M_{2}>0\) can be easily obtained by (\(H_{4}\)). Take \(M=\max \{M_{1},M_{2},M_{3}\}\), then \(x_{i}(t)\leqslant M\), \(y(t) \leqslant M\), \(i=1,2\), \(t\geqslant T_{0}+\tau \).

The proof of the permanence of species x is simple. In fact, let \((x_{1}(t),x_{2}(t),y(t))\) be any positive solution of system (2.1), then from systems (2.1) and (3.3) we obtain

$$ \textstyle\begin{cases} \textstyle\begin{array}{@{}l} \dot{x}_{1}(t)= r_{1}x_{1}(t)\ln \frac{1}{x_{1}(t)}, \\ \dot{x}_{2}(t)\geqslant x_{2}(t) [r_{2}\ln \frac{1}{x_{2}(t)}-\alpha ], \end{array}\displaystyle & t\ne nT, \\ \textstyle\begin{array}{@{}l} \Delta x_{1}(t)= d_{1}[kx_{2}(t-\tau _{0})-x_{1}(t)], \\ \Delta x_{2}(t)= d_{2}[\frac{1}{k}x_{1}(t-\tau _{0})-x_{2}(t)], \end{array}\displaystyle & t= nT, \end{cases} $$
(3.4)

where \(\alpha =c_{1}M_{3}\). Consider the auxiliary system (2.3). From Lemma 2.2 and the comparison theorem of impulsive differential equations, we obtain that, for above \(\varepsilon >0\), there exist \(T_{1}\geqslant T_{0}+\tau \) such that

$$ x_{i}(t)\geqslant v_{i\alpha }(t)> v^{*}_{i\alpha }(t)- \varepsilon \geqslant \min_{t\in [0,T]}v^{*}_{i \alpha }(t)- \varepsilon \triangleq m_{i},\quad i=1,2. $$
(3.5)

This shows that species \(x_{i}\) (\(i=1,2\)) are permanent in system (2.1).

Now, in system (2.1) we prove the permanence of species y. From assumption (\(H_{4}\)), we take a constant \(\varepsilon _{0}>0\) small enough such that

$$ \delta \triangleq k_{2} c_{2}\Bigl(\min _{t\in [0,T]}v^{*}_{2}(t)- \varepsilon _{0}\Bigr)-c_{3}\varepsilon _{0}-r_{3}>0. $$
(3.6)

For any constant \(\alpha >0\), according to assumptions (\(H_{1}\))–(\(H _{3}\)), we have that system (2.3) has a unique globally attractive positive T-periodic solution \((v^{*}_{1\alpha }(t),v^{*}_{2\alpha }(t))\). Since system (2.3) is periodic, we obtain that \((v^{*}_{1\alpha }(t),v^{*}_{2\alpha }(t))\) is globally uniformly attractive. Hence, for above \(\varepsilon _{0} \) and M, there is a constant \(T^{*}=T^{*}(\varepsilon _{0},M)>0\) such that, for any initial value \((t_{0},v_{1\alpha }(t_{0}),v_{2 \alpha }(t_{0}))\) with \(t_{0}\geqslant 0\) and \(0< v_{i\alpha }(t _{0})\leqslant M\) (\(i=1,2\)), we have

$$ \bigl\vert v_{i\alpha }(t)-v^{*}_{i\alpha }(t) \bigr\vert < \frac{\varepsilon _{0}}{2} \quad \text{for all } t\geqslant t_{0}+T^{*}. $$
(3.7)

Therefore, we further have

$$ v_{i\alpha }(t)>v^{*}_{i\alpha }(t)-\frac{\varepsilon _{0}}{2} \quad \text{for all } t\geqslant t_{0}+T^{*}. $$
(3.8)

By the continuity of solutions with respect to parameters, there is \(\alpha _{0}\in (0,\varepsilon _{0})\) such that

$$ \bigl\vert v^{*}_{i\alpha _{0}}(t)-v^{*}_{i}(t) \bigr\vert < \frac{\varepsilon _{0}}{2} \quad \text{for all } t\in R. $$
(3.9)

We further have

$$ v^{*}_{i\alpha _{0}}(t)\geqslant v^{*}_{i}(t)- \frac{\varepsilon _{0}}{2}, \quad t\geqslant 0. $$
(3.10)

Let \(\varepsilon _{1}=\min \{\frac{\alpha _{0}}{c_{1}},\varepsilon _{0} \}\). There are three cases as follows for species \(y(t)\).

Case 1. For all \(t\geqslant T_{2}\), there is a constant \(T_{2}\geqslant T_{1}\) such that \(y(t)\leqslant \varepsilon _{1}\).

Case 2. For all \(t\geqslant T_{2}\), there is a constant \(T_{2}\geqslant T_{1}\) such that \(y(t)\geqslant \varepsilon _{1}\).

Case 3. There is an interval sequence \(\{[s_{k},t_{k}]\}\) with \(T_{1}\leqslant s_{1}< t_{1}< s_{2}< t_{2}<\cdots <s_{k}<t_{k}<\cdots \) and \(\lim_{k\to \infty }s_{k}=\infty \) such that \(y(t)\leqslant \varepsilon _{1}\) for all \(t\in \bigcup_{k=1}^{\infty }[s_{k},t_{k}]\), \(y(t)\geqslant \varepsilon _{1}\) for all \(t\notin \bigcup_{k=1}^{ \infty }(s_{k},t_{k})\), and \(y(s_{k})=y(t_{k})=\varepsilon _{1}\).

For Case 1, from system (2.1), we have

$$ \textstyle\begin{cases} \textstyle\begin{array}{@{}l} \dot{x}_{1}(t)= r_{1}x_{1}(t)\ln \frac{1}{x_{1}(t)}, \\ \dot{x}_{2}(t)\geqslant x_{2}(t) [r_{2}\ln \frac{1}{x_{2}(t)}-\alpha _{0} ], \end{array}\displaystyle & t\ne nT, \\ \textstyle\begin{array}{@{}l} \Delta x_{1}(t)= d_{1}[kx_{2}(t-\tau _{0})-x_{1}(t)], \\ \Delta x_{2}(t)= d_{2}[\frac{1}{k}x_{1}(t-\tau _{0})-x_{2}(t)], \end{array}\displaystyle & t= nT. \end{cases} $$
(3.11)

Consider the auxiliary system (2.3). From Lemma 2.2, (3.8), (3.10), and the comparison theorem of impulsive differential equations, we have that

$$\begin{aligned} x_{i}(t) \geqslant& v_{i\alpha _{0}}(t)> v^{*}_{i\alpha _{0}}(t)- \frac{\varepsilon _{0}}{2} \\ \geqslant& v^{*}_{i}(t)- \varepsilon _{0} \geqslant \min_{t\in [0,T]}v^{*}_{i}(t)- \varepsilon _{0},\quad i=1,2, t\geqslant T_{1}+T^{*}. \end{aligned}$$
(3.12)

Consider the third equation of system (2.1), we further obtain

$$ \dot{y}(t)\geqslant y(t)\Bigl[-r_{3}+k_{2} c_{2} \Bigl(\min_{t\in [0,T]}v^{*} _{2}(t)-\varepsilon _{0}\Bigr)-c_{3}\varepsilon _{0}\Bigr], \quad t \geqslant T_{1}+T ^{*}+\tau . $$
(3.13)

For any \(t=T_{2}+n_{1}T\), we choose an integer \(n_{1}\geqslant 0\), where \(T_{2}=T_{1}+T^{*}+\tau \), and integrate (3.13) from \(T_{2}\) to t, then from (3.6) we have

$$\begin{aligned} y(t) \geqslant & y(T_{2})\exp \Bigl\{ \Bigl[-r_{3}+k_{2} c_{2}\Bigl(\min_{t\in [0,T]}v^{*}_{2}(t)- \varepsilon _{0}\Bigr)-c_{3}\varepsilon _{0}) \Bigr](t-T_{2})\Bigr\} \\ =& y(T_{2})e^{n_{1}T\delta }. \end{aligned}$$
(3.14)

We have \(y(t)\rightarrow \infty \) as \(n_{1}\rightarrow \infty \), which leads to a contradiction.

We now consider Case 3. For any \(t\geqslant T_{1}\), when \(t\in \bigcup_{k=1}^{\infty }[s_{k},t_{k}]\), then \(t\in [s_{k},t_{k}]\) for some k. Assume \(t_{k}-s_{k}\leqslant T^{*}\). Since for any \(t\in [s_{k},t_{k}]\)

$$ \dot{y}(t)\geqslant y(t) (-r_{3}-c_{3}\varepsilon _{0}), $$
(3.15)

then we obtain

$$\begin{aligned} y(t) \geqslant & y(s_{k})\exp \bigl\{ -(r_{3}+c_{3} \varepsilon _{0})T^{*}\bigr\} \\ =&\varepsilon _{1}\exp \bigl\{ -(r_{3}+c_{3} \varepsilon _{0})T^{*}\bigr\} \\ \triangleq & m^{*}. \end{aligned}$$
(3.16)

Assume \(t_{k}-s_{k}\geqslant T^{*}\). For any \(t\in [s_{k},t_{k}]\), if \(t\leqslant s_{k}+T^{*}\), then according to the above discussion on the case of \(t_{k}-s_{k}\leqslant T^{*}\), we obtain inequality (3.16). Particularly, we have \(y(s_{k}+T^{*})\geqslant m^{*}\). Since \(y(t)\leqslant \varepsilon _{1}\) for all \(t\in [s_{k},t_{k}]\), then according to the discussion on Case 1, we have inequality (3.13). For any \(t\in [s_{k}+T^{*},t_{k}]\), we choose an integer \(n_{2}\geqslant 0\) such that \(t\in [s_{k}+T^{*}+n_{2}T,s_{k}+T^{*}+(n_{2}+1)T)\). Then integrating (3.13) from \(s_{k}+T^{*}\) to t, we obtain

$$\begin{aligned} y(t) \geqslant & y\bigl(s_{k}+T^{*}\bigr)\exp \biggl\{ \int _{s_{k}+T^{*}}^{t}\Bigl[-r_{3}+k_{2} c_{2}\Bigl( \min_{t\in [0,T]}v^{*}_{2}(t)- \varepsilon _{0}\Bigr)-c_{3}\varepsilon _{0}) \Bigr]\,dt \biggr\} \\ \geqslant & m^{*}\exp \biggl\{ \int _{s_{k}+T^{*}}^{s_{k}+T^{*}+n_{2}T}\Bigl[-r_{3}+k_{2} c_{2}\Bigl(\min_{t\in [0,T]}v^{*}_{2}(t)- \varepsilon _{0}\Bigr)-c_{3}\varepsilon _{0}) \Bigr]\,dt \\ &{} + \int _{s_{k}+T^{*}+n_{2}T}^{t}\Bigl[-r_{3}+k_{2} c_{2}\Bigl(\min_{t\in [0,T]}v ^{*}_{2}(t)- \varepsilon _{0}\Bigr)-c_{3}\varepsilon _{0}) \Bigr]\,dt \biggr\} \\ \geqslant & m^{*}\exp \biggl\{ \int _{s_{k}+T^{*}+n_{2}T}^{t}\Bigl[-r_{3}+k_{2} c_{2}\Bigl( \min_{t\in [0,T]}v^{*}_{2}(t)- \varepsilon _{0}\Bigr)-c_{3}\varepsilon _{0}) \Bigr]\,dt \biggr\} \\ \geqslant & m^{*}\exp \bigl\{ -(r_{3}+c_{3} \varepsilon _{0})T\bigr\} \\ =& \varepsilon _{1}\exp \bigl\{ -(r_{3}+c_{3} \varepsilon _{0}) \bigl(T+T^{*}\bigr)\bigr\} \\ \triangleq & m_{3}. \end{aligned}$$
(3.17)

From the above discussion, we obtain

$$ y(t)\geqslant m_{3} \quad \text{for all } t\in \bigcup _{k=1}^{\infty }[s_{k},t_{k}]. $$
(3.18)

For any \(t\notin \bigcup_{k=1}^{\infty }(s_{k},t_{k})\), we obviously have

$$ y(t)\geqslant \varepsilon _{1}>m_{3} \quad \text{for all } t\geqslant T_{1}. $$
(3.19)

Hence, for Case 3 we finally have

$$ y(t)\geqslant m_{3} \quad \text{for all } t\geqslant T_{1}. $$
(3.20)

Lastly, we consider Case 2. Since \(y(t)\geqslant \varepsilon _{1}\) for any \(t\geqslant T_{2}\), we obtain

$$ y(t)\geqslant m_{3} \quad \text{for all } t\geqslant T_{2}. $$
(3.21)

Therefore, we finally have

$$ y(t)\geqslant m_{3} \quad \text{for all } t\geqslant T_{2}. $$
(3.22)

Take \(m=\min \{m_{1},m_{2},m_{3}\}\), then \(x_{i}(t)\geqslant m\) (\(i=1,2\)), \(y(t)\geqslant m\) hold as \(t\rightarrow +\infty \). This completes the proof. □

For system (2.1), if we let \(y(t)\equiv 0\), then system (2.1) degenerates into system (2.2). From Lemma 2.1 we know that system (2.2) has a unique globally attractive positive T-periodic solution \((v^{*}_{1}(t),v^{*}_{2}(t))\). Therefore, system (2.1) has a nonnegative T-periodic solution \((v^{*}_{1}(t),v^{*}_{2}(t),0)\).

Next, we present conditions to ensure the global attractivity of a nonnegative T-periodic solution \((v^{*}_{1}(t),v^{*}_{2}(t),0)\) of system (2.1).

Theorem 3.2

Suppose that assumptions (\(H_{1}\))–(\(H_{3}\)) hold. If

(\(H_{5}\)):

\(k_{2} c_{2}\max_{t\in [0,T]}v^{*}_{2}(t)\leqslant r_{3}\),

then system (2.1) admits a predator-extinction periodic solution, which is globally attractive.

Proof

From Theorem 3.1, for any \(\varepsilon >0\) small enough, we have

$$ x_{i}(t)\leqslant v_{i}(t)< v^{*}_{i}(t)+ \varepsilon \leqslant \max_{t\in [0,T]}v^{*}_{i}(t)+ \varepsilon ,\quad i=1,2, t\geqslant T_{0}. $$
(3.23)

According to assumption \((H_{5})\), for any \(\eta _{1}>0\), there is \(\eta _{0}\in (\varepsilon ,\eta _{1})\) such that

$$ \sigma \triangleq k_{2} c_{2}\Bigl(\max _{t\in [0,T]}v^{*}_{2}(t)+\eta _{0} \Bigr)-c _{3}\eta _{1}-r_{3}< 0. $$
(3.24)

From the third equation of system (2.1) and (3.23), we have

$$ \dot{y}(t)\leqslant y(t)\Bigl[-r_{3}+k_{2} c_{2} \Bigl(\max_{t\in [0,T]}v^{*} _{2}(t)+\eta _{0}\Bigr)-c_{3}y(t-\tau _{2})\Bigr], \quad t \geqslant T_{0}+\tau . $$
(3.25)

Assume \(y(t)\geqslant \eta _{1}\) for all \(t>T_{0}\). From (3.25) we obtain

$$ \dot{y}(t)\leqslant y(t)\Bigl[-r_{3}+k_{2} c_{2} \Bigl(\max_{t\in [0,T]}v^{*} _{2}(t)+\eta _{0}\Bigr)-c_{3}\eta _{1}\Bigr],\quad t\geqslant T_{0}+\tau . $$
(3.26)

For any \(t\geqslant T_{0}+\tau \), we choose an integer \(n_{3}\geqslant 0\) such that \(t\in [n_{3}T+T_{0}+\tau ,(n_{3}+1)T+T_{0}+\tau )\). Then integrating (3.26) from \(T_{0}+\tau \) to t, we have

$$\begin{aligned} y(t) \leqslant & y(T_{0}+\tau )\exp \biggl\{ \int _{T_{0}+\tau }^{t}\Bigl[-r_{3}+k_{2}c_{2} \Bigl( \max_{t\in [0,T]}v^{*}_{2}(t)+\eta _{0}\Bigr)-c_{3}\eta _{1})\Bigr]\,dt \biggr\} \\ \leqslant & y(T_{0}+\tau )\exp \biggl\{ \int _{T_{0}+\tau }^{n_{3}T+T_{0}+\tau }\Bigl[-r _{3}+k_{2}c_{2} \Bigl(\max_{t\in [0,T]}v^{*}_{2}(t)+\eta _{0}\Bigr)-c_{3}\eta _{1})\Bigr]\,dt \\ &{} + \int _{n_{3}T+T_{0}+\tau }^{(n_{3}+1)T+T_{0}+\tau }\Bigl[-r_{3}+k_{2}c_{2} \Bigl( \max_{t\in [0,T]}v^{*}_{2}(t)+\eta _{0}\Bigr)-c_{3}\eta _{1})\Bigr]\,dt \biggr\} \\ \leqslant & y(T_{0}+\tau )\exp \{n_{3}T\sigma +\lambda T \}, \end{aligned}$$
(3.27)

where \(\lambda =k_{2}c_{2}(\max_{t\in [0,T]}v^{*}_{2}(t)+\eta _{0})\). Since \(n_{3}\rightarrow \infty \) and \(\sigma <0\), then \(y(t)\rightarrow 0\) as \(t\rightarrow \infty \). This leads to a contradiction. Hence, there is \(t_{1}>T_{0}\) such that \(y(t)\leqslant \eta _{1}\). Since \(y(t)\) is continuous for all \(t\geqslant 0\), if further exists \(t_{3}>t_{1}\) such that \(y(t_{3})>\eta _{1}e^{\lambda T}\), then there is \(t_{2}\in (t_{1},t_{3})\) such that \(y(t_{2})=\eta _{1}\) and \(y(t)>\eta _{1}\) for any \(t\in (t_{2},t_{3}]\). When \(t\in [t_{2},t_{3}]\), we can easy find that inequality (3.26) holds. Further, we choose an integer \(n_{4}\geqslant 0\) such that \(t_{3}\in [t_{2}+n_{4}T,t_{2}+(n_{4}+1)T)\). Integrating (3.26) from \(t_{2}\) to \(t_{3}\), we obtain

$$\begin{aligned} y(t) \leqslant & y(t_{2})\exp \biggl\{ \int _{t_{2}}^{t_{3}}\Bigl[-r_{3}+k_{2}c_{2} \Bigl( \max_{t\in [0,T]}v^{*}_{2}(t)+\eta _{0}\Bigr)-c_{3}\eta _{1})\Bigr]\,dt \biggr\} \\ =& y(t_{2})\exp \biggl\{ \int _{t_{2}}^{t_{2}+n_{4}T}\Bigl[-r_{3}+k_{2}c_{2} \Bigl( \max_{t\in [0,T]}v^{*}_{2}(t)+\eta _{0}\Bigr)-c_{3}\eta _{1})\Bigr]\,dt \\ & {} + \int _{t_{2}+n_{4}T}^{t_{3}}\Bigl[-r_{3}+k_{2}c_{2} \Bigl(\max_{t\in [0,T]}v^{*} _{2}(t)+\eta _{0}\Bigr)-c_{3}\eta _{1})\Bigr]\,dt \biggr\} \\ \leqslant & \eta _{1}e^{\lambda T}, \end{aligned}$$
(3.28)

which is a contradiction. So, we finally have

$$ y(t)\leqslant \eta _{1}e^{\lambda T}\quad \text{for any } t>T_{0}. $$
(3.29)

Since \(\eta _{1}\) is arbitrary and λ is a constant, from (3.29) we have

$$ \lim_{t\to \infty }y(t)=0. $$
(3.30)

Therefore, for any \(\varepsilon _{2}\geqslant 0\) small enough, there is \(T_{3}>T_{0}\) such that \(0< y(t)<\varepsilon _{2}\), \(t>T_{3}\).

For the second equation of system (2.1), we have

$$ \textstyle\begin{cases} \textstyle\begin{array}{@{}l} \dot{x}_{1}(t)= r_{1}x_{1}(t)\ln \frac{1}{x_{1}(t)}, \\ \dot{x}_{2}(t)\geqslant x_{2}(t) [r_{2}\ln \frac{1}{x_{2}(t)}-\alpha _{1} ], \end{array}\displaystyle & t\ne nT, \\ \textstyle\begin{array}{@{}l} \Delta x_{1}(t)= d_{1}[kx_{2}(t-\tau _{0})-x_{1}(t)], \\ \Delta x_{2}(t)= d_{2}[\frac{1}{k}x_{1}(t-\tau _{0})-x_{2}(t)], \end{array}\displaystyle & t= nT, \end{cases} $$
(3.31)

where \(\alpha _{1}=c_{1}\varepsilon _{2}\). Consider the auxiliary system (2.3). From Lemma 2.2 and the comparison theorem of impulsive differential equations, we obtain that, for above ε, there is \(T_{4}>0\) such that

$$ x_{i}(t)\geqslant v_{i\alpha _{1}}(t)> v^{*}_{i\alpha _{1}}(t)- \frac{\varepsilon _{0}}{2}\geqslant v^{*}_{i}(t)- \varepsilon _{0},\quad i=1,2, t\geqslant T_{4}. $$
(3.32)

Combining (3.23), (3.30), and (3.32), we have

$$ x_{i}(t)\rightarrow v_{i}^{*}(t),\quad y(t) \rightarrow 0, \quad i=1,2, t\rightarrow \infty . $$
(3.33)

That is, system (2.1) admits a predator-extinction periodic solution, which is globally attractive. The proof of Theorem 3.2 is completed. □

Remark 3.1

In this paper, we have proposed a predator-prey model with prey impulsive diffusion and dispersal delay. By using the comparison theorem of impulsive differential equation and other analysis methods, we have established a set of easily verifiable sufficient conditions on the global attractivity of the predator-extinction periodic solution and the permanence of species. The highlight of this paper is that we considered the prey with impulsive diffusion and dispersal delay. However, we only discussed the case of the predator-prey model with prey impulsive diffusion in two patches. For this model with prey impulsive diffusion in multiple patches, the results that can be obtained are still important and interesting open problems.