Stability analysis of a twopatch predator–prey model with two dispersal delays
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Abstract
This paper deals with a predator–prey model with both species in the delayeddispersal case in a twopatch environment. The purpose of this paper is to study the effect of two dispersal delays on the stability of three equilibria. It turns out that the stability of the trivial equilibrium and the boundary equilibrium is delayindependent. However, the stability of the coexistence equilibrium is delaydependent. Numerical simulations are performed to demonstrate the obtained results.
Keywords
Dispersal Delay Stability Bifurcation Predator–prey modelMSC
92B05 34K20 34K18 34K101 Introduction
The relationship of predator and prey is prevalent in nature and hence is one of the most important themes in ecological and mathematical models. Since the Lotka–Volterra predator–prey model was formulated, various predator–prey models have been studied by incorporating additional ecological concepts into the classical Lotka–Volterra model, such as functional responses, dispersal and time delay. In predator–prey models, dispersal will represent migration of either the prey population, the predator population, or both [1, 2].
Population dispersal is very common in ecology. Species migrate from one patch to another patch, due to some kinds of factor in the initial patch. For instance, a prey species will choose to move on the basis of resource availability and predation risk, while predators tend to migrate to the better patch to gain more prey. In nature, lack of food, competition, sex, age, lack of security (mainly for the prey), climatic conditions, season, overpopulation in a patch—these factors make species move from a patch to another [3]. For example, in aquatic environments, many zooplankton species exhibit vertical movements each day due to light and food. During the day time, some species migrate downwards into the darkness to reduce the predation risk by fish, while at night time, these species move upward to consume the phytoplankton [4]. There has been great interest in the study of mathematical models of populations with species dispersal among patches, such as a single population dispersal [5, 6, 7, 8], and the dispersal of both prey and predator among patches [9, 10, 11].
It is worth pointing out that most of the research work in population models with patchy structures assumed the dispersal to be instantaneous. In fact, species movement between patches takes some time. Recently, Zhang et al. considered a predator–prey metapopulation model with travel time delay and showed that such delay can stabilize and destabilize the system [8].
To the best of our knowledge, there is little research on joining considerations of migration and dispersal delay in biological models. It is challenging to add time delays to predator–prey models for the mathematical analysis. Delay may change the stability of dynamics and Hopf bifurcation may occur. Motivated by the above predator–prey model, we will integrate dispersal of both species and dispersal delays into a twopatch Rosenzweig–MacArthur predator–prey model. We shall investigate how the dispersal and dispersal delays interact to affect the stability of the predator–prey metapopulation model.
The rest of the paper is organized as follows. The formulation of mathematical model is presented in Sect. 2. The stability analysis of our model at three equilibria are given in Sect. 3. Then numerical simulations based on the analysis are reported in Sect. 4. Finally, we conclude the paper by a short discussion.
2 Model formulation
If \(d=0\) and \(m=0\), the single patch model has three equilibria: \((0,0)\), \((\kappa,0)\) and \((h^{*},p^{*})\) with \(h^{*}=\frac{\mu}{1\mu}\) and \(p^{*}=(1\frac{h^{*}}{\kappa })(1+h^{*})\). Its dynamics is described by the following results [12].
Theorem 2.1
 (1)
The trivial equilibrium\((0,0)\)is unstable.
 (2)
The predatorextinction equilibrium\((\kappa,0)\)is stable when\(\mu>\frac{\kappa}{1+\kappa}\).
 (3)
The coexistence equilibrium\((h^{*},p^{*})\)exists if and only if\(0<\mu<\frac{\kappa}{1+\kappa}\).
 (4)
The coexistence equilibrium is globally stable when\(\frac{\kappa1}{\kappa+1}<\mu<\frac{\kappa}{1+\kappa}\).
 (5)
There is a unique globally stable limit cycle if\(0<\mu <\frac{\kappa1}{\kappa+1}\).
3 Stability analysis of the equilibria
It is easy to calculate that \(\det(J)=\det(J_{1} + J_{2})\cdot\det(J_{1}  J_{2})\). The characteristic equation determines the local stability of equilibria. The equilibrium is stable if and only if all the characteristic roots have negative real part. In the following, we will analyze the stability of our model (2.2) at three equilibria, respectively.
3.1 The stability analysis of \(E_{0}\)
Based on the technique of [13, 14, 15], the stability change at the equilibrium can only happen when characteristic roots appear on or cross the imaginary axis as τ increases. Here we assume that \(\varepsilon\neq2d\), then we look for a pair of purely imaginary roots of the characteristic equations (3.1).
Similarly, for the second equation of (3.1), there is a pair of purely imaginary roots \(\pm i\omega_{0}\) with \(\omega_{0}=\sqrt {d^{2}(\varepsilond)^{2}}\).
For the third equation and the last equation of (3.1), by using the same method, we have \(\omega^{2}=m^{2}(\mu+m)^{2}<0\). Thus, there are no purely imaginary roots for the last two equations of (3.1).
Lemma 3.1
Proof
For the second equation of (3.1), the conclusion of the lemma also holds. The proof is complete. □
By Lemma 3.1, we know that \(\frac{d\operatorname{Re}(\lambda)}{d\tau _{1}} _{\lambda=i\omega_{0}}>0\). This indicates that, as \(\tau_{1}\) increases, for the first and second equation of (3.1), the characteristic roots cross the imaginary axis through \(\pm i\omega _{0}\) at \(\tau=\tau_{1}\) from left to right and the number of characteristic roots with positive real parts is increased by 2.
On the other hand, for the third and fourth equations of (3.1), there are no purely imaginary roots. Note the instability of \(E_{0}\) for \(\tau_{1}=\tau_{2}=0\), thus \(E_{0}\) remains unstable for \(\tau_{1}>0\) and \(\tau_{2}>0\).
3.2 The stability analysis of \(E_{1}\)
When \(\tau_{i}=0\), the roots of the characteristic equations (3.3) are −ε, \(\varepsilon2d\), \(\frac{\kappa }{1+\kappa }\mu\), \(\frac{\kappa}{1+\kappa}\mu2m\). Here we have two cases to consider: (1) \(\mu>\frac{\kappa}{1+\kappa}\), all roots are negative, so equilibrium \(E_{1}\) is stable; (2) \(0<\mu<\frac{\kappa}{1+\kappa}\), the roots at least have a positive root, \(E_{1}\) is unstable.
Substituting \(\lambda=i\omega\) with \(\omega>0\) into these characteristic equations (3.3), and separating the real and imaginary parts, we can calculate that if \(\mu>\frac{\kappa}{1+\kappa}\), \(\cos\omega\tau_{1}=1+\frac{\varepsilon}{d}>1\), \(\cos \omega\tau _{2}=1\frac{\frac{\kappa}{1+\kappa}\mu}{m}>1\), which means there is no solutions of (3.3) can appear on the imaginary axis for any \(\tau_{i}\). Therefore, \(E_{1}\) is locally asymptotically stable when \(\mu>\frac {\kappa}{1+\kappa}\).
For case (2), if \(0<\mu<\frac{\kappa}{1+\kappa}\), by the expression of \(\cos\omega\tau_{1}\), the first and second equations of (3.3) have no imaginary roots. However, the third and fourth equations admit purely imaginary roots \(\pm i \omega\) with \(\omega=\sqrt{m^{2} (m(\frac{\kappa }{1+\kappa }\mu) )^{2}}\) if \(\frac{\kappa}{1+\kappa}2m<\mu<\frac{\kappa}{1+\kappa}\); the two equations have no imaginary roots while if \(0<\mu<\frac {\kappa }{1+\kappa}2m\). It is easy to show that \(\operatorname{sign} (\frac{d(\operatorname{Re}\lambda)}{d\tau _{2}} )_{\lambda=i\omega}=\operatorname{sign}(\omega^{2})>0\), Note that equilibrium \(E_{1}\) is unstable with \(\tau_{1}=\tau_{2}=0\) in case (2). So when \(0<\mu<\frac{\kappa}{1+\kappa}\), \(E_{1}\) remains unstable as \(\tau _{1}\) and \(\tau_{2}\) increase.
3.3 The stability analysis of \(E_{2}\)
When \(m=0\), this implies that prey disperses only. Based on the results of [8], we summarize the related conclusions in the following theorem.
Theorem 3.2
 Case (i)

\(d\in(0, A/2)\): the coexistence equilibrium\(E_{2}\)remains unstable for\(\tau_{1}>0\).
 Case (ii)

\(d\in(A/2,A)\): the coexistence equilibrium\(E_{2}\)remains unstable for\(\tau_{1}>0\).
 Case (iii)

\(d\in(A, \infty)\): there may exist stability switches.
When both species are mobile, due to the presence of two different time delays in characteristic equations, it is usually difficult to analyze the transcendental equation with two delays. Actually, finding all the characteristic roots of Eq. (3.4) have negative real parts is hopeless [16]. This indicates the difficulty in investigating the distribution of the zeros of Eq. (3.4). Thus, we mainly numerically examine how the two dispersal delays affect the stability of the coexistence equilibrium in our model in the next section.
4 Numerical simulations
From the above section, we know that the stability of the trivial equilibrium \(E_{0}\) and the boundary equilibrium \(E_{1}\) is relatively simple. However, the stability analysis of the coexistence equilibrium \(E_{2}\) is complicated. Therefore, in this section, we mainly present some numerical examples of our model (2.2) and investigate that the effect of delay on the stability and instability of the coexistence equilibrium \(E_{2}\). Based on Theorem 3.2, we display the numerical simulations in each case.
Example 1
Example 2
Example 3
In order to investigate our model (2.2) in the case \(d>A>0\), two dispersal delays may induce Hopf bifurcation. Thus, we first plot the numerical solutions and the \(\tau _{1}\)bifurcation diagram of (2.2) with \(m=0\) in each cases. Then we choose values of \(\tau_{1}\) in its stable intervals and unstable intervals, respectively. We regard \(\tau_{2}\) as bifurcation parameter and display the bifurcation diagram of (2.2).
Example 4
In all, as shown in Fig. 5 and Fig. 6, dispersal delays may exhibit both stabilizing and destabilizing effects on the coexistence equilibrium.
5 Summary and discussion
This paper is concerned with a predator–prey model with two dispersal delays and the stability analysis of three equilibria. We focus attention on the effect of two dispersal delays on the dynamics of our model (2.2). Moreover, we show that, for the trivial equilibrium and the boundary equilibrium, dispersal delays have no impact on the stability and instability of two equilibria. However, the stability of the coexistence equilibrium is delaydependent. Delays can destabilize and stabilize the coexistence equilibrium. Indeed, the dispersal delays cannot only switch the stability but also induce a Hopf bifurcation. If the species’ mortality during dispersal is considered, the stability analysis becomes very difficult. Numerical simulations are carried out showing that stability switches are possible. We leave the related analysis for our future work.
Notes
Acknowledgements
The authors wish to thank the editor and the referees for reading this manuscript.
Authors’ contributions
All authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.
Funding
This work was supported by the National Natural Science Foundation of China (No. 11371313 and No. 61573016) and by the Foundation of Yuncheng University (No. YQ2017003 and No. YQ2014011). AM was partially supported by China Scholarship Council (No. 201608140214).
Competing interests
The authors declare that they have no competing interests.
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