Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system

A diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered. Hopf and steady state bifurcation analysis are carried out in details. In particular we show the existence of multiple spatially non-homogeneous periodic orbits while the system parameters are all spatially homogeneous. Our results and global bifurcation theory also suggest the existence of loops of spatially non-homogeneous periodic orbits and steady state solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation.     

Steady-state and Hopf bifurcations in the Langford ODE and PDE systems

This paper is concerned with the Langford ODE and PDE systems. For the Langford ODE system, the existence of steady-state solutions is firstly obtained by Lyapunov–Schmidt method, and the stability and bifurcation direction of periodic solutions are established. Then for the Langford PDE system, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalue. Finally, by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions for the PDE system are investigated.     

具有扩散的广义Brusselator系统的Hopf分歧

在齐次Neumann边界条件下,考虑广义Brusselator系统．首先讨论常微分系统Hopf分歧的存在性,得到渐近稳定的周期解．其次讨论具有扩散的偏微分系统,在扩散系数满足一定的条件下,得到超临界的Hopf分歧,并利用规范形理论和中心流形定理给出空间齐次周期解的渐近稳定性．最后,借助Matlab软件进行数值模拟,证明了定理的结论．同时,正平衡态解和空间非齐次周期解的描绘补充了理论分析结果．     

Stability and Hopf bifurcation in a class of nonlocal delay differential equation with the zero‐flux boundary condition

The aim of this paper is to study the stability and Hopf bifurcation in a general class of differential equation with nonlocal delayed feedback that models the population dynamics of a two age structured spices. The existence of Hopf bifurcation is firstly established after delicately analyzing the eigenvalue problem of the linearized nonlocal equation. The direction of the Hopf bifurcation and stability of the bifurcated periodic solutions are then investigated by means of center manifold reduction. Subsequently, we apply our main results to explore the spatial‐temporal patterns of the nonlocal Mackey‐Glass equation. We obtain both spatially homogeneous and inhomogeneous periodic solutions and numerically show that the former is stable while the latter is unstable. We also show that the inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations and increasing of the immature mobility constant will shorten the transient oscillation time.     

Stability and Hopf bifurcation of a delayed predator-prey system with nonlocal competition and herd behaviour

In this paper, we investigate the stability and Hopf bifurcation of a diffusive predator-prey system with herd behaviour. The model is described by introducing both time delay and nonlocal prey intraspecific competition. Compared to the model without time delay, or without nonlocal competition, thanks to the together action of time delay and nonlocal competition, we prove that the first critical value of Hopf bifurcation may be homogenous or non-homogeneous. We also show that a double-Hopf bifurcation occurs at the intersection point of the homogenous and non-homogeneous Hopf bifurcation curves. Furthermore, by the computation of normal forms for the system near equilibria, we investigate the stability and direction of Hopf bifurcation. Numerical simulations also show that the spatially homogeneous and non-homogeneous periodic patters.     

Passage to diffusion chaos in a model of an ecological system

We carry out analytical and numerical analysis of a model of an ecological system described by a system of nonlinear partial differential equations of reaction-diffusion type. We find conditions for the bifurcation of periodic spatially homogeneous and inhomogeneous solutions from the thermodynamic branch of the system. We show that the passage to diffusion chaos in the model occurs, in complete agreement with the universal Feigenbaum-Sharkovskii-Magnitskii bifurcation theory, via a subharmonic cascade of bifurcations of stable limit cycles.     

Space–time chaos in a system of reaction–diffusion equations

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.     

Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models

This paper is concerned with two-species spatial homogeneous and inhomogeneous predator-prey models with Beddington-DeAngelis functional response. For the spatial homogeneous model, the asymptotic behavior of the interior equilibrium and the existence of Hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with no-flux boundary conditions, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Finally, to verify our theoretical results, some numerical simulations are also included.     

Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey model

The interactions of diffusion-driven Turing instability and delay-induced Hopf bifurcation always give rise to rich spatiotemporal dynamics. In this paper, we first derive the algorithm for the normal forms associated with the Turing-Hopf bifurcation in the reaction-diffusion system with delay, which can be used to investigate the spatiotemporal dynamical classification near the Turing-Hopf bifurcation point in the parameter plane. Then, we consider a diffusive predator-prey model with weak Allee effect and delay. Through investigating the dynamics of the corresponding normal form of Turing-Hopf bifurcation induced by diffusion and delay, the spatiotemporal dynamics near this bifurcation point can be divided into six categories. Especially, stable spatially homogeneous/inhomogeneous periodic solutions and steady states, coexistence of two stable spatially inhomogeneous periodic solutions, coexistence of two stable spaially inhomogeneous steady states and the transition from one kind of spatiotemporal patterns to another are found.     

Diffusion-driven stability and bifurcation in a predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type functional response subject to Neumann boundary conditions is considered. Hopf and steady-state bifurcation analysis are carried out in detail. First, the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated by analysing the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the centre manifold reduction for partial functional differential equations and then steady-state bifurcation is studied. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model

In this paper, we study the Hopf bifurcation phenomenon of a one-dimensional Schnakenberg reaction-diffusion model subject to the Neumann boundary condition. Our results reveal that both spatially homogeneous periodic solutions and spatially heterogeneous periodic solution exist. Moreover, we conclude that the spatially homogeneous periodic solutions are locally asymptotically stable and the spatially heterogeneous periodic solutions are unstable. In addition, we give specific examples to illustrate the phenomenon that coincides with our theoretical results.     

Stability and Hopf bifurcation of a diffusive predator‐prey model with hyperbolic mortality

The dynamics of a reaction‐diffusion predator‐prey model with hyperbolic mortality and Holling type II response effect is considered. The stability of the positive equilibrium and the existence of Hopf bifurcation are investigated by analyzing the distribution of eigenvalues without diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system which are spatially homogeneous. To verify our theoretical results, some numerical simulations are also presented. © 2015 Wiley Periodicals, Inc. Complexity 21: 34–43, 2016     

Hopf bifurcations in general systems of Brusselator type

This paper is concerned with general models of Brusselator type subject to the homogeneous Neumann boundary condition. The existence of Hopf bifurcation for the ODE and PDE models is obtained. By the center manifold theory and the normal form method, the bifurcation direction and stability of bifurcating periodic solutions are established. Moreover, some numerical simulations are shown to support the analytical results.     

Stability and Hopf bifurcation in a diffusive predator–prey system with delay effect

This paper is concerned with a delayed predator–prey system with diffusion effect. First, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the distribution of the eigenvalues. Next the direction and the stability of Hopf bifurcation are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Principal Asymptotics in the Problem on the Andronov–Hopf Bifurcation and Their Applications

New formulas are obtained for the principal asymptotics of bifurcation solutions in the problem on the Andronov–Hopf bifurcation, leading to new algorithms for studying bifurcations in the general setting. The approach proposed in the paper allows one to consider not only the classical problems about bifurcations of codimension one but also some problems concerning bifurcations of codimension two. A new approach to the analysis of bifurcations of cycles in systems with homogeneous nonlinearities is proposed. As an application, we consider the problem on the bifurcation of periodic solutions of the van der Pol equation.     

Complicated dynamics of a predator–prey system with Watt-type functional response and impulsive control strategy

Based on the classical predator–prey system with Watt-type functional response, an impulsive differential equations to model the process of periodic perturbations on the predator at different fixed time for pest control is proposed and investigated. It proves that there exists a globally asymptotically stable prey-eradication periodic solution when the impulse period is less than some critical value, and otherwise, the system can be permanent. Numerical results show that the system considered has more complicated dynamics involving quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and “crises”, etc. It will be useful for studying the dynamic complexity of ecosystems.     

Bifurcation analysis on a diffusive Holling–Tanner predator–prey model

This paper is concerned with a diffusive Holling–Tanner predator–prey model subject to homogeneous Neumann boundary condition. By choosing the ratio of intrinsic growth rates of predator to prey λ as the bifurcation parameter, we find that spatially homogeneous and non-homogeneous Hopf bifurcation occur at the positive constant steady state as λ varies. The steady state bifurcation of simple and double eigenvalues are intensively investigated. The techniques of space decomposition and the implicit function theorem are adopted to deal with the case of double eigenvalues. Our results show that this model can exhibit spatially non-homogeneous periodic and stationary patterns induced by the parameter λ. Numerical simulations are presented to illustrate our theoretical results.     

Hopf Bifurcations in a Predator-Prey Diffusion System with Beddington-DeAngelis Response

This paper is concerned with a two-species predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and subject to homogeneous Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, the asymptotic stability of the positive constant steady-state solution and the existence of local Hopf bifurcations are investigated. Also, it is shown that the appearance of the diffusion and homogeneous Neumann boundary conditions can lead to the appearance of codimension two Bagdanov-Takens bifurcation. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Finally, numerical simulations supporting the theoretical analysis are also included.     

Interactions of Turing and Hopf bifurcations in an additional food provided diffusive predator-prey model

Complex spatiotemporal dynamics of a diffusive predator-prey system involving additional food supply to predator and intra-specific competition among predator, are investigated. We establish critical conditions of the occurrence of Turing instability, which are necessary and sufficient. Furthermore, we also establish conditions of the occurrence of codimension-2 Turing-Hopf bifurcation and Turing-Turing bifurcation, by exploring interactions of Turing bifurcations and Hopf bifurcation. For Turing-Hopf bifurcation, by analyzing normal form truncated to order 3 which are derived by applying normal form method, it is shown that under proper conditions, diffusive predator-prey system generates interesting spatial, temporal and spatiotemporal patterns, including a pair of spatially inhomogeneous steady states, a spatially homogeneous periodic solution and a pair of spatially inhomogeneous periodic solutions. And numerical simulations are also shown to support theory analysis. Moreover, it is found that proper intra-specific competition among predator helps generate complex spatiotemporal dynamics. And, proper additional food supply to predator helps control the population fluctuations of predator and prey, while large quantity and high quality of additional food supply will lead to the extinction of prey and make predator change the source of food, which finally destroy the ecosystem. These investigations might help understand complex spatiotemporal dynamics of predator-prey system involving additional food supply to predator and intra-specific competition among predator, and help conserve species in an ecosystem via supplying suitable additional food.     

Hopf bifurcation in a reaction–diffusion model with Degn–Harrison reaction scheme

In this paper, we consider a chemical reaction–diffusion model with Degn–Harrison reaction scheme under homogeneous Neumann boundary conditions. The existence of Hopf bifurcation to ordinary differential equation (ODE) and partial differential equation (PDE) models are derived, respectively. Furthermore, by using the center manifold theory and the normal form method, we establish the bifurcation direction and stability of periodic solutions. Finally, some numerical simulations are shown to support the analytical results, and to reveal new phenomenon on the Hopf bifurcation.