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Dynamical behaviors of a predatorprey system with prey impulsive diffusion and dispersal delay between two patches
Advances in Difference Equations volumeÂ 2019, ArticleÂ number:Â 191 (2019)
Abstract
In this paper, we consider a predatorprey model with prey impulsive diffusion and dispersal delay. By utilizing the dynamical properties of a singlespecies model with diffusion and dispersal delay between two patches and the comparison principle of impulsive differential equations, we establish the sufficient conditions on the global attractivity of predatorextinction periodic solution and the permanence of species for the model.
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, predatorprey 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 predatorprey 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 predatorprey 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 predatorprey Lotkaâ€“Volterra type system with prey dispersal in n patches in [14]:
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 manmade 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 shortterm 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 predatorprey 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 predatorprey system with impulsive diffusion:
with the initial conditions
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 predatorextinction 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:
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 predatorprey model with prey symmetric bidirectional impulsive diffusion and dispersal delay between two patches:
with the initial conditions
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 predatorprey 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:
Next, we discuss the dynamical behaviors of the following single species model:
We introduce the following assumptions for system (2.2):
 (\(H_{1}\)):

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

\(b_{1}+b_{2}+d_{1}\leqslant 1\),
 (\(H_{3}\)):

\(1b_{i}\leqslant (1b_{i}e^{r_{i}\tau _{0}})e^{(r_{1}+r _{2})\tau _{0}}\), \(i=1,2\),
where \(b_{i}=e^{r_{i}T}\).
Lemma 2.1
([20])
Suppose that assumptions (\(H_{1}\))â€“(\(H_{3}\)) hold, then system (2.2) has a unique globally attractive positive Tperiodic solution \((v^{*}_{1}(t),v^{*}_{2}(t))\), that is,
Next, we consider the following system:
where Î± 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:
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 (\(H_{1}\))â€“(\(H_{3}\)) hold, then system (2.3) has a unique globally attractive positive Tperiodic solution \((v^{*}_{1\alpha }(t),v^{*}_{2\alpha }(t))\), that is,
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
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
with initial conditions, \(y(s)=\phi (s)\geqslant 0\) for \(s \in [\tau ,0]\), where a, b are positive constants. Then
3 Main results
Theorem 3.1
Suppose that assumptions (\(H_{1}\))â€“(\(H_{3}\)) 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
for all \(t>\tau _{0}\). Consider the auxiliary system (2.2). From LemmaÂ 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
for all \(t\geqslant T_{0}\). Hence, from the third equation of (2.1) and (3.2), we have
By LemmaÂ 2.3, we can obtain
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
where \(\alpha =c_{1}M_{3}\). Consider the auxiliary system (2.3). From LemmaÂ 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
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
For any constant \(\alpha >0\), according to assumptions (\(H_{1}\))â€“(\(H _{3}\)), we have that system (2.3) has a unique globally attractive positive Tperiodic 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
Therefore, we further have
By the continuity of solutions with respect to parameters, there is \(\alpha _{0}\in (0,\varepsilon _{0})\) such that
We further have
Let \(\varepsilon _{1}=\min \{\frac{\alpha _{0}}{c_{1}},\varepsilon _{0} \}\). There are three cases as follows for species \(y(t)\).
CaseÂ 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 CaseÂ 1, from system (2.1), we have
Consider the auxiliary system (2.3). From LemmaÂ 2.2, (3.8), (3.10), and the comparison theorem of impulsive differential equations, we have that
Consider the third equation of system (2.1), we further obtain
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
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 someÂ k. Assume \(t_{k}s_{k}\leqslant T^{*}\). Since for any \(t\in [s_{k},t_{k}]\)
then we obtain
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 CaseÂ 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
From the above discussion, we obtain
For any \(t\notin \bigcup_{k=1}^{\infty }(s_{k},t_{k})\), we obviously have
Hence, for Case 3 we finally have
Lastly, we consider Case 2. Since \(y(t)\geqslant \varepsilon _{1}\) for any \(t\geqslant T_{2}\), we obtain
Therefore, we finally have
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.â€ƒâ–¡
For system (2.1), if we let \(y(t)\equiv 0\), then system (2.1) degenerates into system (2.2). From LemmaÂ 2.1 we know that system (2.2) has a unique globally attractive positive Tperiodic solution \((v^{*}_{1}(t),v^{*}_{2}(t))\). Therefore, system (2.1) has a nonnegative Tperiodic solution \((v^{*}_{1}(t),v^{*}_{2}(t),0)\).
Next, we present conditions to ensure the global attractivity of a nonnegative Tperiodic solution \((v^{*}_{1}(t),v^{*}_{2}(t),0)\) of system (2.1).
Theorem 3.2
Suppose that assumptions (\(H_{1}\))â€“(\(H_{3}\)) 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 predatorextinction periodic solution, which is globally attractive.
Proof
From TheoremÂ 3.1, for any \(\varepsilon >0\) small enough, we have
According to assumption \((H_{5})\), for any \(\eta _{1}>0\), there is \(\eta _{0}\in (\varepsilon ,\eta _{1})\) such that
From the third equation of system (2.1) and (3.23), we have
Assume \(y(t)\geqslant \eta _{1}\) for all \(t>T_{0}\). From (3.25) we obtain
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
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
which is a contradiction. So, we finally have
Since \(\eta _{1}\) is arbitrary and Î» is a constant, from (3.29) we have
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
where \(\alpha _{1}=c_{1}\varepsilon _{2}\). Consider the auxiliary system (2.3). From LemmaÂ 2.2 and the comparison theorem of impulsive differential equations, we obtain that, for above Îµ, there is \(T_{4}>0\) such that
Combining (3.23), (3.30), and (3.32), we have
That is, system (2.1) admits a predatorextinction periodic solution, which is globally attractive. The proof of TheoremÂ 3.2 is completed.â€ƒâ–¡
Remark 3.1
In this paper, we have proposed a predatorprey 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 predatorextinction 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 predatorprey 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.
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The authors would like to thank the anonymous reviewers for their constructive comments and suggestions.
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This work was supported by the Doctoral Foundation of Heze University [Grant numbers, XY18BS12].
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Wan, H., Jiang, H. Dynamical behaviors of a predatorprey system with prey impulsive diffusion and dispersal delay between two patches. Adv Differ Equ 2019, 191 (2019). https://doi.org/10.1186/s1366201921322
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DOI: https://doi.org/10.1186/s1366201921322