Delayed Predator Prey Model Research Articles

Local Bifurcations and Optimal Theory in a Delayed Predator–Prey Model with Threshold Prey Harvesting

We investigate the effects of time delay and piecewise-linear threshold policy harvesting for a delayed predator–prey model. It is the first time that Holling response function of type III and the present threshold policy harvesting are associated with time delay. The trajectories of our delayed system are bounded; the stability of each equilibrium is analyzed with and without delay; there are local bifurcations as saddle-node bifurcation and Hopf bifurcation; optimal harvesting is also investigated. Numerical simulations are provided in order to illustrate each result.

Bifurcation and Stability in a Delayed Predator–Prey Model with Mixed Functional Responses

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003; Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie–Gower functional response as well as that of Beddington–DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E* and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E*. At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values.

Oscillations for a delayed predator–prey model with Hassell–Varley-type functional response

In this paper, a delayed predator–prey model with Hassell–Varley-type functional response is investigated. By choosing the delay as a bifurcation parameter and analyzing the locations on the complex plane of the roots of the associated characteristic equation, the existence of a bifurcation parameter point is determined. It is found that a Hopf bifurcation occurs when the parameter τ passes through a series of critical values. The direction and the stability of Hopf bifurcation periodic solutions are determined by using the normal form theory and the center manifold theorem due to Faria and Maglhalaes (1995). In addition, using a global Hopf bifurcation result of Wu (1998) for functional differential equations, we show the global existence of periodic solutions. Some numerical simulations are presented to substantiate the analytical results. Finally, some biological explanations and the main conclusions are included.

Hopf bifurcation and global stability of a delayed predator–prey model with prey harvesting

In this paper, we consider a delayed prey–predator model with modified Leslie–Gower term and Michaelis–Menten type prey harvesting. Regarding time delay as a bifurcation parameter, and using the normal form and center manifold theorem of functional differential equations and partial differential equations, we obtain the existence, stability and direction of periodic solutions of functional differential equations and partial differential equations, respectively. Numerical simulations are carried out to depict our theoretical analysis.

Bifurcation Analysis and Optimal Harvesting of a Delayed Predator–Prey Model

A delay predator–prey model is formulated with continuous threshold prey harvesting and Holling response function of type III. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The positive invariance of the non-negative orthant is proved and the uniform boundedness of the trajectories. Stability of equilibria is investigated and the existence of some local bifurcations is established: saddle-node bifurcation, Hopf bifurcation. We use optimal control theory to provide the correct approach to natural resource management. Results are also obtained for optimal harvesting. Numerical simulations are given to illustrate the results.

Effects of additional food in a delayed predator–prey model

We examine the effects of supplying additional food to predator in a gestation delay induced predator–prey system with habitat complexity. Additional food works in favor of predator growth in our model. Presence of additional food reduces the predatory attack rate to prey in the model. Supplying additional food we can control predator population. Taking time delay as bifurcation parameter the stability of the coexisting equilibrium point is analyzed. Hopf bifurcation analysis is done with respect to time delay in presence of additional food. The direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold theorem. The qualitative dynamical behavior of the model is simulated using experimental parameter values. It is observed that fluctuations of the population size can be controlled either by supplying additional food suitably or by increasing the degree of habitat complexity. It is pointed out that Hopf bifurcation occurs in the system when the delay crosses some critical value. This critical value of delay strongly depends on quality and quantity of supplied additional food. Therefore, the variation of predator population significantly effects the dynamics of the model. Model results are compared with experimental results and biological implications of the analytical findings are discussed in the conclusion section.

Dynamic of a delayed predator–prey model with birth pulse and impulsive harvesting in a polluted environment

In this paper, we propose a delayed predator–prey model with birth pulse and impulsive harvesting in a polluted environment. Existence conditions of the predator-extinction periodic solution are derived by developing the discrete dynamical system, which is determined by the stroboscopic map. Further, we discuss the global attractivity of predator-extinction periodic solution and permanence of the system, and obtain the threshold conditions. The results provide a dependable theoretical strategies to protect population from extinction in a polluted environment. Finally, the numerical simulations are presented for verifying the theoretical conclusions.

Global dynamics of a delayed predator–prey model with stage structure for the predator and the prey

A delayed predator–prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each feasible equilibrium of the system is discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory for infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using suitable Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator–extinction equilibrium and the coexistence equilibrium do not exist, and that the predator–extinction equilibrium is globally stable when the coexistence equilibrium does not exist. Further, sufficient conditions are obtained for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.

Optimal harvesting policy of a stochastic predator–prey model with time delay

This paper concerns the optimal harvesting of a stochastic delay predator–prey model. Sufficient and necessary conditions for the existence of an optimal control are established. The optimal harvesting effort and the maximum value of the cost function are obtained as well. Some numerical tests are given to illustrate the main results.

A density dependent delayed predator–prey model with Beddington–DeAngelis type function response incorporating a prey refuge

This paper describes a predator–prey model incorporating a prey refuge. The feeding rate of consumers (predators) per consumer (i.e. functional response) is considered to be of Beddington–DeAngelis type. The Beddington–DeAngelis functional response is similar to the Holling-type II functional response but contains an extra term describing mutual interference by predators. We investigate the role of prey refuge and degree of mutual interference among predators in the dynamics of system. The dynamics of the system is discussed mainly from the point of view of permanence and stability. We obtain conditions that affect the persistence of the system. Local and global asymptotic stability of various equilibrium solutions is explored to understand the dynamics of the model system. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional. The dynamical behaviour of the delayed system is further analyzed through incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. The analytical results found in the paper are illustrated with the help of numerical examples.

Persistence and extinction of a stochastic delay predator-prey model under regime switching

The paper is concerned with a stochastic delay predator-prey model under regime switching. Sufficient conditions for extinction and non-persistence in the mean of the system are established. The threshold between persistence and extinction is also obtained for each population. Some numerical simulations are introduced to support our main results.

Global dynamics of a delayed predator–prey model with stage structure and holling type II functional response

The global properties of a delayed predator-prey model with Holling type II functional response and stage structure for both the predator and the prey are investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the model is discussed. Further, it is proved that the model undergoes a Hopf bifurcation at the coexistence equilibrium. By means of the persistence theory on infinite dimensional systems, it is shown that the model is permanent if the coexistence equilibrium exists. By using Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium is globally asymptotically stable when the predator-extinction equilibrium is not feasible, the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main results.

Multiple periodic solutions of a delayed predator–prey model with non-monotonic functional response and stage structure

The paper studies a periodic and delayed predator–prey system with non-monotonic functional responses and stage structure. In the system, both the predator and prey are divided into immature individuals and mature individuals by two fixed ages. It is assumed that the immature predators cannot attack preys, and the case of the mature predators attacking the immature preys is also ignored. Based on Mawhin's coincidence degree, sufficient conditions are obtained for the existence of two positive periodic solutions of the system. An example is presented to illustrate the feasibility of the main results.

On a stochastic delayed predator–prey model with Lévy jumps

This note is concerned with a stochastic delayed predator–prey system with Lévy jumps. Under some simple conditions, sufficient and necessary conditions for stability in time average and extinction of each species are obtained.

Global asymptotic stability of a stochastic delayed predator–prey model with Beddington–DeAngelis functional response

This paper is concerned with a stochastic predator–prey system with Beddington–DeAngelis functional response and time delay. Sufficient conditions for global asymptotic stability of the positive equilibrium point are established. The result demonstrates that the delay is harmless for global stability of the stochastic system in some cases. Some figures are given to validate the analytical results.

Modelling and analysis of a delayed predator–prey model with disease in the predator

In this paper, we study a predator–prey model with a transmissible disease spreading in the predator population and a time delay representing the gestation period of the predator. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium and the coexistence equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are derived for the global stability of the predator-extinction equilibrium and the disease-free equilibrium and the global attractiveness of the coexistence equilibrium of the system, respectively. Numerical simulations are carried out to support the theoretical analysis.

Permanence of a stage-structured predator–prey system

A stage-structured predator–prey system (stage structure for both predator and prey) with discrete delay is studied in this paper. By introducing a new lemma and applying the standard comparison theorem, a set of novel criteria which ensure the permanence of the system is obtained. Our result supplements the main results of Chen, Xie and Li [Partial survival and extinction of a delayed predator–prey model with stage structure, Applied Mathematics and Computation, 219 (8) (2012) 4157–4162].

Analysis of Stochastic Delay Predator-Prey System with Impulsive Toxicant Input in Polluted Environments

A stochastic delay predator-prey model in a polluted environment with impulsive toxicant input is proposed and studied. The thresholds between stability in time average and extinction of each population are obtained. Some recent results are extended and improved greatly. Several simulation figures are introduced to support the conclusions.

Stability and bifurcation of a stage-structured predator–prey model with both discrete and distributed delays

This paper concerns with a new delayed predator–prey model with stage structure on prey, in which the immature prey and the mature prey are preyed by predator and the delay is the length of the immature stage. Mathematical analysis of the model equations is given with regard to invariance of non-negativity, boundedness of solutions, permanence and global stability and nature of equilibria. Our work shows that the stage structure on the prey is one of the important factors that affect the extinction of the predator, and the predation on immature prey is a cause of periodic oscillation of population and can make the behaviors of the system more complex. The predation on the immature and mature prey brings both positive and negative effects on the permanence of the predator, if ignore the predation on immature prey in the system, the stage-structure on prey brings only negative effect on the permanence of the predator.

Partial survival and extinction of a delayed predator–prey model with stage structure

In this paper, we revisit the non-persistent property of a stage-structured predator–prey system which was proposed by Ma et al. (2008) [Z.H. Ma, Z.Z. Li, S.F. Wang, T.Li, F.P. Zhang, Permanence of a predator–prey system with stage structure and time delay, Appl. Math. Comput. 201 (2008) pp. 65–71]. By applying the standard comparison theorem, some novel results concern with the extinction of the system and partial survival of the predator (prey) species, respectively, are obtained. One of the interesting finding is that for this system, the extinction of prey may not necessarily lead to extinction of predator.