Chaos and Hopf bifurcation analysis for a two species predator–prey system with prey refuge and diffusion

In this paper, a delayed predator–prey model with Holling type II functional response incorporating a constant prey refuge and diffusion is considered. By analyzing the characteristic equation of linearized system corresponding to the model, we study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Further, an example is presented to illustrate our main results. Finally, recurring to the numerical method, the influences of impulsive perturbations on the dynamics of the system are also investigated.     

Hopf bifurcation analysis for a delayed predator–prey system with diffusion effects

This paper is concerned with a delayed predator–prey diffusive system with Neumann boundary conditions. The bifurcation analysis of the model shows that Hopf bifurcation can occur by regarding the delay as the bifurcation parameter. In addition, the direction of Hopf bifurcation and the stability of bifurcated periodic solution are also discussed by employing the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, the effect of the diffusion on bifurcated periodic solution is considered.     

A bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting

In this paper, we propose a bioeconomic differential algebraic predator–prey model with Holling type II functional response and nonlinear prey harvesting. As the nonlinear prey harvesting is introduced, the proposed model displays a complex dynamics in the predator–prey plane. Taking into account of the economic factor, our predator–prey system is established by bioeconomic differential algebraic equations. The effect of economic profit on the proposed model is analyzed by viewing it as a bifurcation parameter. By jointly using the normal form of differential algebraic models and the bifurcation theory, the stability and bifurcations (singularity induced bifurcation, Hopf bifurcation) are discussed. These results obtained here reveal richer dynamics of the bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting, and suggest a guidance for harvesting in the practical word. Finally, numerical simulations are given to demonstrate the results.     

Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay

A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.     

Dynamics in a diffusive predator–prey system with a constant prey refuge and delay

In this paper, a diffusive predator–prey system with a constant prey refuge and time delay subject to Neumann boundary condition is considered. Local stability and Turing instability of the positive equilibrium are studied. The effect of time delay on the model is also obtained, including locally asymptotical stability and existence of Hopf bifurcation at the positive equilibrium. And the properties of Hopf bifurcation are determined by center manifold theorem and normal form theorem of partial functional differential equations. Some numerical simulations are carried out.     

Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey

This paper is concerned with a mathematical model dealing with a predator–prey system with disease in the prey. Mathematical analysis of the model regarding stability has been performed. The effect of delay on the above system is studied. By regarding the time delay as the bifurcation parameter, the stability of the positive equilibrium and Hopf bifurcations are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, some numerical simulations are also included.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

Dynamics of a delayed predator–prey model with predator migration

Based on the availability of prey and a simple predator–prey model, we propose a delayed predator–prey model with predator migration to describe biological control. We first study the existence and stability of equilibria. It turns out that backward bifurcation occurs with the migration rate as bifurcation parameter. The stability of the trivial equilibrium and the boundary equilibrium is delay-independent. However, the stability of the positive equilibrium may be delay-dependent. Moreover, delay can switch the stability of the positive equilibrium. When the positive equilibrium loses stability, Hopf bifurcation can occur. The direction and stability of Hopf bifurcation is derived by applying the center manifold method and the normal form theory. The main theoretical results are illustrated with numerical simulations.     

Bifurcation analysis of an autonomous epidemic predator–prey model with delay

In this paper, a class of an autonomous epidemic predator–prey model with delay is considered. Its linear stability and Hopf bifurcation are investigated. Applying the normal form theory and center manifold theory, the explicit formulas for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.     

Hopf bifurcation and center stability for a predator–prey biological economic model with prey harvesting

In this paper, we investigate Hopf bifurcation and center stability of a predator–prey biological economic model. By employing the local parameterization method, Hopf bifurcation theory and the formal series method, we obtain some testable results on these issues. The economic profit is chosen as a positive bifurcation parameter here. It shows that a phenomenon of Hopf bifurcation occurs as the economic profit increases beyond a certain threshold. Besides, we also find that the center of the biological economic model is always unstable. Finally, some numerical simulations are given to illustrate the effectiveness of our results.     

Stability and Hopf bifurcation of a modified delay predator-prey model with stage structure

In this paper, a modified delay predator-prey model with stage structure is established, which involves the economic factor and internal competition of all the prey and predator populations. By the methods of normal form and characteristic equation, we obtain the stability of the positive equilibrium point and the sufficient condition of the existence of Hopf bifurcation. We analyze the influence of the time delay on the equation and show the occurrence of Hopf bifurcation periodic solution. The simulation gives a visual understanding for the existence and direction of Hopf bifurcation of the model.     

Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, dynamic behavior of the proposed model system with and without discrete time delay is investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; Furthermore, local stability of the model system with discrete time delay is studied. It reveals that the discrete time delay has a destabilizing effect in the population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Finally, numerical simulations are carried out to show the consistency with theoretical analysis obtained in this paper.     

Hopf bifurcation in a delayed differential–algebraic biological economic system

In this paper, we consider a differential–algebraic biological economic system with time delay where the model with Holling type II functional response incorporates a constant prey refuge and prey harvesting. By considering time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the differential–algebraic biological economic system based on the new normal form approach of the differential–algebraic system and the normal form approach and the center manifold theory. Finally, numerical simulations illustrate the effectiveness of our results.     

A delayed prey–predator model with Crowley–Martin‐type functional response including prey refuge

In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin‐type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators

This paper is concerned with a predator–prey system with Holling II functional response and hunting delay and gestation. By regarding the sum of delays as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. We obtained explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation. Using a global Hopf bifurcation result of Wu [Wu JH. Symmetric functional differential equations and neural networks with memory, Trans Amer Math Soc 1998;350:4799–4838] for functional differential equations, we may show the global existence of the periodic solutions. Finally, several numerical simulations illustrating the theoretical analysis are also given.     

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     

Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model

In this paper we explore how the two mechanisms, Turing instability and Hopf bifurcation, interact to determine the formation of spatial patterns in a ratio-dependent prey–predator model with discrete time delay. We conduct both rigorous analysis and extensive numerical simulations. Results show that four types of patterns, cold spot, labyrinthine, chaotic as well as mixture of spots and labyrinthine can be observed with and without time delay. However, in the absence of time delay, the two aforementioned mechanisms have a significant impact on the emergence of spatial patterns, whereas only Hopf bifurcation threshold is derived by considering the discrete time delay as the bifurcation parameter. Moreover, time delay promotes the emergence of spatial patterns via spatio-temporal Hopf bifurcation compared to the non-delayed counterpart, implying the destabilizing role of time delay. In addition, the destabilizing role is prominent when the magnitude of time delay and the ratio of diffusivity are comparatively large.     

Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay

In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. 26 We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.     

Persistence and global stability of Bazykin predator–prey model with Beddington–DeAngelis response function

In this article, a predator–prey model of Beddington–DeAngelis type with discrete delay is proposed and analyzed. The essential mathematical features of the proposed model are investigated in terms of local, global analysis and bifurcation theory. By analyzing the associated characteristic equation, it is found that the Hopf bifurcation occurs when the delay parameter τ crosses some critical values. In this article, the classical Bazykin’s model is modified with Beddington–DeAngelis functional response. The parametric space under which the system enters into Hopf bifurcation for both delay and non-delay cases are investigated. Global stability results are obtained by constructing suitable Lyapunov functions for both the cases. We also derive the explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions by using normal form and central manifold theory. Our analytical findings are supported by numerical simulations. Biological implication of the analytical findings are discussed in the conclusion section.     

Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays

In this paper we give a detailed Hopf bifurcation analysis of a ratio-dependent predator–prey system involving two different discrete delays. By analyzing the characteristic equation associated with the model, its linear stability is investigated. Choosing delay terms as bifurcation parameters the existence of Hopf bifurcations is demonstrated. Stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. Furthermore, some of the bifurcation properties including direction, stability and period are given. Finally, theoretical results are supported by some numerical simulations.