The joint effects of diffusion and delay on the stability of a ratiodependent predatorprey model
 Kejun Zhuang^{1, 2}Email author and
 Gao Jia^{3}
https://doi.org/10.1186/s1366201710963
© The Author(s) 2017
Received: 10 November 2016
Accepted: 19 January 2017
Published: 7 February 2017
Abstract
This paper is concerned with a diffusive and delayed predatorprey system with LeslieGower and ratiodependent Holling type III schemes subject to homogeneous Neumann boundary conditions. Preliminary analyses on the wellposedness of solutions and the dissipativeness of the system are presented with assistance of inequality technique. Then the Hopf bifurcation induced by spatial diffusion and time delay is discussed, respectively. Moreover, the bifurcation properties are obtained by computing the norm forms on the center manifold. Finally, some numerical simulations and conclusions are given to verify and illustrate the theoretical results.
Keywords
predatorprey system Hopf bifurcation reactiondiffusion system delayMSC
35K57 35B32 92D251 Introduction
In population ecology, the dynamics of species populations and the way these populations interact with the environment have raised widespread concerns [1]. It is the study of how the population sizes of species change over time and space. Among these interactions, the predatorprey interaction is the most basic one, and plenty of mathematical models have been established since the pioneering work by Lotka and Volterra [2, 3].
To better understand the relationship between predator and prey, a functional response is utilized to model the intake rate of a consumer as a function of food density, such as Holling types IIV [4, 5], BeddingtonDeAngelis type [6], HassellVarley type [7], LeslieGower type [8], CrowleyMartin type [9], and so on [10, 11]. In many situations, when predators have to search, share or compete for their resources, the socalled ratiodependent functional response is reasonable. It means that the per capita predator growth rate is a function of the ratio of prey to predator abundance. This is strongly supported by numerous fields and laboratory experiments and observations [12]. Such ratiodependent models can present rich dynamic behaviors, see [13–15]. In addition, the environmental carrying capacity of predator is proportional to the number of prey; consequently, the LeslieGower type functional response was proposed in [16, 17].
However, to the best of our knowledge, there are no results on the joint effects of diffusion and delay on the spatiotemporal dynamics of the delayed reactiondiffusion system (1). As a consequence, our major goal is to investigate the basic properties of system (1), the stability of constant steady states and the time periodic solutions generated by diffusion and delay. The rest of this paper is arranged as follows. In Section 2, we give some preliminary results on the wellposedness of solutions and the dissipativeness of the system. In Section 3, we derive sufficient conditions for the stability of nonnegative constant steady states and the existence of Hopf bifurcation. In Section 4, we compute the formulae for determining the Hopf bifurcation properties. In Section 5, we conduct some numerical simulations to illustrate our theoretical results. Finally, we give some conclusions and biological interpretations.
2 Elementary results
Here, we establish some basic properties of the solutions of system (1), specifically, the wellposedness of solutions and the dissipativeness of system (1).
For convenience, we first restate a useful lemma from [26] as follows.
Lemma 1
Theorem 1
System (1) has a unique global solution, and the solution remains nonnegative and uniformly bounded for all \(t > 0\).
Proof
Following the process in [27, 28], we can similarly get the local existence and uniqueness of solution \((u(x,t), v(x,t))\) with \(x\in\overline{\Omega}\) and \(t\in[0,T)\), where T is the maximal existence time of the solution.
In the following, we will show that any nonnegative solution of system (1) is bounded as \(t\rightarrow+\infty\) for all \(x\in\Omega\).
Theorem 2
Dissipativeness
Proof
3 The stability of positive steady states and the existence of Hopf bifurcation
 (H1):

\(\beta< 1+m \).
In the following, we shall explore the effect of spatial diffusion and time delay on the dynamic behaviors of system (1), respectively.
3.1 The effect of diffusion
 (H2):

\((1+m)(1+r) < 2 \).
Theorem 3
 (i)
If \(\beta\in(0,\beta_{0})\), then the positive steady state \(E_{\ast}\) is asymptotically stable.
 (ii)
If \(\beta\in(\beta_{0}, m+1)\), then the positive steady state \(E_{\ast}\) is unstable.
 (iii)
The periodic solutions bifurcating from \(\beta=\beta_{0}\) are spatially homogeneous, and the periodic solutions bifurcating from \(\beta=\beta_{n}\) (\(1\leq n\leq N_{1}, \beta_{N_{1}}< m+1, \beta_{N_{1}+1} \geq m+1\)) are spatially inhomogeneous.
3.2 The effect of delay
Next, we discuss the dynamic behaviors of \(E_{\ast}\) by taking time delay τ as the bifurcation parameter.
 (H3):

\(\beta<\min\{ m+1,(1+m)^{2}/2 \}\).
From above, we can establish the existence of Hopf bifurcation induced by time delay.
Theorem 4
 (i)
For \(\tau\in[0,\tau_{0})\), the positive steady state \((u_{\ast},v_{\ast})\) is asymptotically stable.
 (ii)
or \(\tau>\tau_{0}\), the positive steady state \((u_{\ast},v_{\ast})\) is unstable. Furthermore, \(\tau=\tau_{j}^{(n)}\) (\(j=0,1,2,\ldots\) ; \(n=0,1,2,\ldots,\min\{ N_{2}, N_{3} \}\)) are Hopf bifurcation values.
4 Bifurcation properties
In this section, we mainly analyze the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions obtained in Theorem 4. The methods here are based on the center manifold theorem and normal form theory for partial functional differential equations in [29, 30].
In general, we use \(\tau^{\ast}\) to denote an arbitrary value of \(\tau _{j}^{(n)}\) with \(j\in\mathbb{N}_{0}\) and \(n\in\{0,1,2,\ldots,\min\{{N_{2}, N_{3}}\}\}\). And we also use \(\pm i\omega^{\ast}\) to denote the corresponding simply purely imaginary roots \(\pm i\omega_{n}\).
Then the center subspace of system (1) is \(P=\operatorname{span}\{ q(\theta ),\overline{q(\theta)} \}\), and the adjoint subspace is \(P^{\ast}=\operatorname{span}\{ q^{\ast}(s),\overline{q^{\ast}(s)} \}\).
Theorem 5
 (i)
\(\ell_{2}\) determines the bifurcation direction: if \(\ell _{2}>0\), then the bifurcation is supercritical and the periodic solution exists for \(\tau>\tau_{0}\); if \(\ell_{2}<0\), then the bifurcation is subcritical and the periodic solution exists for \(\tau<\tau_{0}\).
 (ii)
\(\iota_{2}\) determines the stability of bifurcating periodic solutions: the periodic solutions are orbitally asymptotically stable if \(\iota_{2}<0\), or unstable if \(\iota_{2}>0\).
 (iii)
\(\chi_{2}\) determines the period of the bifurcating periodic solutions: the period is monotonically increasing at time delay τ when \(\chi_{2}>0\), or is monotonically decreasing at time delay τ when \(\chi_{2}<0\).
5 Numerical simulations
In this section, to illustrate the analytic results, we will conduct some numerical examples by the aid of MATLAB.
6 Conclusions
In this paper, we have considered a ratiodependent predatorprey system with spatial diffusion and time delay and have investigated the joint effects of spatial diffusion and time delay. The wellposedness of solutions and the dissipativeness of the system have been established based on inequality techniques. Hopf bifurcation conditions have also been derived by choosing different bifurcation parameters respectively. It is observed that a periodic phenomenon appears when the bifurcation parameter passes through some critical value.
It is shown that the parameter β, which reflects the specific predation rate or the interaction strength between two species, can make the equilibrium solution \(E_{\ast}=(u_{\ast}, v_{\ast})\) asymptotically stable or unstable without time delay. From Theorem 3, we can control the parameter β sufficiently small to achieve the stabilization. For example, some measures can be adopted to decrease the value of β, such as founding a refuge for prey species or increasing the interference with interaction between two species. On the other hand, the numerical examples indicate that a spatially homogeneous periodic solution will exist when the parameter β is larger than the Hopf bifurcation value. Nevertheless, when β is far away from the critical value, the two species will be extinct. It reflects that overhunting or denudation may seriously destroy the ecological environment.
Our results also show that time delay has a vital impact on the dynamics of system (1). The second group of parameters we choose in Section 5 also satisfy the conditions of Theorem 2.7 in [18]. That is to say, the equilibrium solution \(E_{\ast}\) is globally asymptotically stable when time delay is equal to zero. However, the asymptotic behavior is not able to always keep stable when time delay varies. If time delay τ is sufficiently small, then the equilibrium solution \(E_{\ast}\) is still asymptotically stable. When τ is slightly larger than a certain critical value, the equilibrium solution \(E_{\ast}\) is no longer stable and a spatially periodic solution may appear. Furthermore, if we choose other diffusion coefficients, we can find spatially inhomogeneous periodic solutions, which are not included in [19, 20], without time delay.
Well, due to the local existence of Hopf bifurcation, the periodic solutions only exist in a small neighborhood of bifurcation value. It is interesting and significant to further explore the global continuation of local Hopf bifurcation, which can ensure the existence of periodic solutions when the parameter is much larger or less than the bifurcation value. We will continue this research in the near future. Still, the methods and results in the present paper have supplemented the ones in [18–20] and can also be applied to other reactiondiffusion systems without or with time delay. We hope that our work could be useful to study the effects of spatial diffusion and time delay on the population dynamics.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11301001 and 11171220). It is also supported by the Key Project for Excellent Young Talents Fund Program of Higher Education Institutions of Anhui Province (gxyqZD2016100) and the Anhui Provincial Natural Science Foundation (1508085MA09 and 1508085QA13).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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