Dynamical behaviors of a predatorprey system with prey impulsive diffusion and dispersal delay between two patches
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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.
Keywords
Predatorprey system Impulsive diffusion Dispersal delay Permanence Global attractivityMSC
34A37 34C55 34C25 34D231 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].
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].
2 Preliminaries
 (\(H_{1}\))

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

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

\(1b_{i}\leqslant (1b_{i}e^{r_{i}\tau _{0}})e^{(r_{1}+r _{2})\tau _{0}}\), \(i=1,2\),
Lemma 2.1
([20])
Therefore system (2.3) has the following result as system (2.2).
Lemma 2.2
Definition 2.1
Lemma 2.3
([29])
3 Main results
Theorem 3.1
 (\(H_{4}\))

\(k_{2} c_{2}\min_{t\in [0,T]}v^{*}_{2}(t)>r_{3}\),
Proof
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 \(y(s_{k})=y(t_{k})=\varepsilon _{1}\).
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 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
 (\(H_{5}\))

\(k_{2} c_{2}\max_{t\in [0,T]}v^{*}_{2}(t)\leqslant r_{3}\),
Proof
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.
Notes
Acknowledgements
The authors would like to thank the anonymous reviewers for their constructive comments and suggestions.
Availability of data and materials
Not applicable.
Authors’ contributions
The authors contributed equally in this article. They read and approved the final manuscript.
Funding
This work was supported by the Doctoral Foundation of Heze University [Grant numbers, XY18BS12].
Competing interests
The authors declare that they have no competing interests.
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