Spatiotemporal dynamics in a predator-prey model with a functional response increasing in both predator and prey densities

In this paper, we studied a diffusive predator-prey model with a functional response increasing in both predator and prey densities. The Turing instability and local stability are studied by analyzing the eigenvalue spectrum. Delay induced Hopf bifurcation is investigated by using time delay as bifurcation parameter. Some conditions for determining the property of Hopf bifurcation are obtained by utilizing the normal form method and center manifold reduction for partial functional differential equation.     

Stability and Hopf bifurcation in a predator–prey model with stage structure for the predator

A predator–prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results.     

Stability and Hopf bifurcations in a delayed Leslie-Gower predator-prey system

In this paper, we consider the following delayed Leslie-Gower predator-prey system

(∗)     

Hopf bifurcation in a two-competitor, one-prey system with time delay

In this paper, we consider a delayed two-competitor/one-prey system in which both two competitors exhibit Holling II functional response. By choosing the time delay as a bifurcation parameter, it is found that the Hopf bifurcation occurs when the delay passes through a certain critical value. Numerical simulations are performed to illustrate the analytical results.     

Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response

We consider a delayed predator-prey system with Beddington-DeAngelis functional response. The stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are performed to illustrate the obtained results.     

Stability and bifurcation in a non-kolmogorov type prey-predator system with time delay

The purpose of this paper is to study a non-Kolmogrov type prey-predator system. First, we investigate the linear stability of the model by analyzing the associated characteristic equation of the linearized system. Second, we show that the system exhibits the Hopf bifurcation. The stability and direction of the Hopf bifurcation are determined by applying the norm form theory and center manifold theorem. Finally, numerical simulations are performed to illustrate the obtained results.     

Hopf bifurcation and stability for a neural network model with mixed delays

A generalized model of the two-neuron network with mixed delays is studied. The main purpose of this paper is to explore the linear stability of the trivial solution and Hopf bifurcation of a two-neuron network with continuous and discrete delays. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solutions are also studied. Finally, the numerical simulations are given to illustrate the theoretical analysis.     

含两时滞捕食-被捕食系统的稳定性及分歧

本文研究一类含两相异时滞的捕食－被捕食系统的稳定性及分歧。首先，我们讨论两相异时滞对系统唯一正平衡点的稳定性的影响，通过对系数与时滞有关的特征方程的分析，建立了一种稳定性判别性。其次，将一个时滞看成分歧参数，而另一个看作固定参数，我们证明了该系统具有HOPF分歧特性。最后，我们讨论了分歧解的稳定性。     

Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays

In this paper, we investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay τ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results.     

Existence of periodic travelling waves solutions in predator prey model with diffusion

This paper deals with the qualitative analysis of the travelling waves solutions of a reaction diffusion model that refers to the competition between the predator and prey with modified Leslie–Gower and Holling type II schemes. The well posedeness of the problem is proved. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of the forth degree exponential polynomial characteristic equation. We also prove the existence of a Hopf bifurcation which leads to periodic oscillating travelling waves by considering the diffusion coefficient as a parameter of bifurcation. Numerical simulations are given to illustrate the analytical study.     

一类带有竞争和自反时滞的网站竞争系统的Hopf分支

不同于以往研究网站竞争系统时,仅考虑带有自反时滞或竞争时滞的情况,本文研究了一类同时带有竞争时滞和自反时滞的网站竞争系统,并以时滞作为分支参数,通过分析正平衡点处的特征方程,研究了正平衡点的稳定性,证明了Hopf分支的存在性,得到了发生Hopf分支时的临界的时滞值,最后通过数值模拟进一步验证了所得结论.     

Stability switch and Hopf bifurcation for a diffusive prey-predator system with delay

The increasing time delay usually destabilizes any dynamical system. In this paper we give an example that in some special cases the opposite effect can be experienced if the time delay is sufficiently great. We investigate the effect of both the parameter in the time delay kernel and diffusion coefficient on the stability of the positive steady state for a diffusive prey-predator system with delay. We obtain the condition of the occurrence of the stability switches of the positive steady state.     

Asymptotic stability of positive equilibrium solution for a delayed prey–predator diffusion system

This article is concerned with a delayed Lotka–Volterra two-species prey–predator diffusion system with a single discrete delay and homogeneous Dirichlet boundary conditions. By applying the implicit function theorem, the asymptotic expressions of positive equilibrium solutions are obtained. And then, the asymptotic stability of positive equilibrium solutions is investigated by linearizing the system at the positive equilibrium solutions and analyzing the associated eigenvalue problem. It is demonstrated that the positive equilibrium solutions are asymptotically stable when the delay is less than a certain critical value and unstable when the delay is greater than this critical value. In addition, it is also found that the system under consideration can undergo a Hopf bifurcation when the delay crosses through a sequence of critical values. Finally, to verify our theoretical predictions, some numerical simulations are also included.     

Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.     

Stability and bifurcation of a Cohen-Grossberg neural network with discrete delays

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. The qualitative analysis is given for the system and it is found that the system undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. Moreover, by applying the normal form theory and the center manifold theorem, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained. Numerical simulations are given to illustrate the obtained results.     

一类具有时滞的病菌-免疫力模型的定性研究

考虑病菌的一种信息交流机制,建立一类病菌与免疫系统竞争的时滞传染病模型.分析正平衡点的存在性、渐近稳定性、Hopf分歧的存在性.特别的,研究了无病平衡点E0 在奇异条件(R0=1)下的稳定性.数值模拟验证了所得理论结果.     

Bifurcation analysis of a predator–prey model with memory-based diffusion

In this paper, we investigate the predator–prey model equipped with Fickian diffusion and memory-based diffusion of predators. The stability and bifurcation analysis explores the impacts of the memory-based diffusion and the averaged memory period on the dynamics near the positive steady state. Specifically, when the memory-based diffusion coefficient is less than a critical value, we show that the stability of the positive steady state can be destabilized as the average memory period increases, which leads to the occurrence of Hopf bifurcations. Moreover, we also analyze the bifurcation properties using the central manifold theorem and normal form theory. This allows us to prove the existence of stable spatially inhomogeneous periodic solutions arising from Hopf bifurcation. In addition, the sufficient and necessary conditions for the occurrence of stability switches are also provided.     

Impact on the predator population while lethal disease spreads in the prey

An eco‐epidemiological model is proposed to interpret the impact of an infectious disease on ecosystem, where prey is infected by lethal bacteria and the consumption of infected prey contributes to the negative growth of predator population. Of course, the required time delay is incorporated into the model. The biodiversity conservation is concerned. Stability, persistence, and bifurcation are carried out. Finally, numerical simulations are performed to support the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Cholera model incorporating media coverage with multiple delays

In order to investigate the impact of awareness programs and time delays on the cholera outbreaks, we propose a cholera epidemic model, incorporating awareness programs by media as a separate class and two time‐delay factors. The bifurcation theory is applied to explore the variety of dynamics of this model for various combinations of the delays when R₀>1. Moreover, we analyze the direction, stability, and period of the bifurcating periodic solutions arising through Hopf bifurcation by using the normal form concept and the center manifold theory. Finally, we present numerical simulations to verify the main theoretical results.     

The effect of prey refuge and time delay on a diffusive predator‐prey system with hyperbolic mortality

In this article, we investigate the effect of prey refuge and time delay on a diffusive predator‐prey system with Holling II functional response and hyperbolic mortality subject to Neumann boundary condition. More precisely, we study Turing instability of positive equilibrium by using refuge as parameter, instability and Hopf bifurcation induced by time delay. In addition, by the theory of normal form and center manifold, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution. © 2016 Wiley Periodicals, Inc. Complexity 21: 446–459, 2016