Global dynamics of a predator-prey model with stage structure for the predator (original) (raw)

Permanence and Stability of a Stage-Structured Predator–Prey Model

Journal of Mathematical Analysis and Applications, 2001

A predator᎐prey model with a stage structure for the predator which improves the assumption that each individual predator has the same ability to capture prey is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey. We obtain conditions that determine the permanence of the populations and the extinction of the populations. Furthermore, we establish necessary and sufficient conditions for the local stability of the positive equilibrium of the model. By exploiting the monotonicity of one equation of the model, we obtain conditions for the global attractivity of the positive equilibrium, which allow for long delay as long as the predator birth rate is large or the death rate of immature predators is small. By constructing Liapunov functionals, we also obtain conditions under which the positive equilibrium is globally stable when the delay is small. ᮊ

Global stability for a stage-structured predator-prey model

Mathematical Sciences Research Journal, 2006

The asymptotic behavior of a stagestructured predator-prey system is studied using the theory of finite dimensional competitive systems. Using natural conditions on the persistency constant for prey and on the reproduction rate of the mature predators, it is found that the system under consideration has a unique positive equilibrium, which is globally asymptotically stable. Some considerations on the uniform pcrsistency of the system are also included. Key-words: Stage structure, predator-prey model, global stability, competitive ...

Dynamics of a predator-prey model with stage-structure on both species and anti-predator behavior

Informatics in Medicine Unlocked, 2018

In this paper, we have formulated and studied a stage-structure predator-prey model. Here, we consider stage-structure on both prey as well as predator population which means that the prey population is divided into two sub populations such as juvenile prey and adult prey, on the other hand, the predator population is also divided into two sub populations such as juvenile predator and adult predator. It is assumed that only adult predator have the ability to predation and they consume both juvenile prey as well as adult prey. Here, it is also considered that the growth rate of juvenile prey depends upon the adult prey population i.e., the juvenile prey has no reproduction capability. Also, Holling II and Holling IV response function have been used for the consumption of juvenile prey and adult prey by adult predator respectively. It is also considered two types of factors such as anti-predator behavior and group defense to formulate our proposed model. Mathematically, we have analyzed the positivity and boundedness of solutions, existence of equilibria, stability of the proposed system around these equilibrium points and Hopf bifurcation of interior equilibrium point. Finally, some numerical simulations have been presented to validate our theoretical results.

Dynamics of a Stage-Structured Predator-Prey Model

International Journal of Applied Physics and Mathematics, 2017

This paper deals with the dynamics of a stage-structured predator-prey system. The immature and mature prey are predated by the predator for which modified Holling type II functional response is considered in the model. The solution of the system is positive and bounded. Stability analysis has been discussed about all possible feasible equilibrium points. The origin and boundary equilibrium points are shown to be globally asymptotically stable. The parameters are identified for which system also admits trans-critical bifurcation about these points. The occurrence of Hopf bifurcation has been shown through numerical simulation about positive interior point. Persistence condition is obtained.

Dynamics and Numerical Simulation of Stage Structure Prey-Predator Models

Proceedings of the International Conference on Science and Technology (ICST 2018), 2018

In the present paper, we studied a prey-predator model with stage-structure for prey. Prey populations which are divided into subpopulations, one is immature prey and the other is mature prey. We investigate the stability of prey-predator models as well as local analysis of four equilibrium obtained and analyzed how the dynamical behavior if the solution is stable or unstable. Numerical simulations are performed from the system around the equilibrium point to show dynamic behavior around that point with several different conditions andinitial values.

Stability Analysis of a Stage Structure Prey-Predator Model

Dirasat, 2011

In this paper, a stage structure prey-predator model with a Beddington-DeAngelis type of functional response is proposed and analyzed. The local stability analysis of the system is carried out. The occurrence of a simple Hopf bifurcation is investigated. By the use of a suitable Lyapunov function, the global dynamics of the axial equilibrium point are discussed. The basin of attraction of the positive equilibrium point is determined. The global dynamics of the system are investigated by using numerical simulations. It is observed that the system has only two types of dynamics, approaches to a global stable point or to a global stable limit cycle.

A Stage Structured Predator-Prey model and its Dependence on Through-Stage Delay and Death Rate

2003

The work of Aiello and Freedman on a single species growth with stage structure has received much attention in the literature in recent years. Their model predicts a positive steady state as the global attractor and thus suggests that stage structure does not generate the sustained oscillations frequently observed in nature. This work inevitably stirred some controversy. Subsequent works by other authors suggest that the time delay to adulthood should be state dependent and careful formulation of such state dependent time delay can lead to models that produce periodic solutions. We review this work from a fresh biological angle: growth is a combined result of birth and death processes, both of which are closely linked to the resource supply which is dynamic in nature. From this basic standpoint, we formulate a general and robust predator-prey model with stage structure with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and computatio...

Global Dynamics of a Predator-Prey Model with Stage Structure and Delayed Predator Response

Discrete Dynamics in Nature and Society, 2013

A Holling type II predator-prey model with time delay and stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. The existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and the sufficient conditions are obtained for the global stability of the coexistence equilibrium.

A stage structured predator-prey model and its dependence on maturation delay and death rate

Journal of Mathematical Biology, 2004

Many of the existing models on stage structured populations are single species models or models which assume a constant resource supply. In reality, growth is a combined result of birth and death processes, both of which are closely linked to the resource supply which is dynamic in nature. From this basic standpoint, we formulate a general and robust predator-prey model with stage structure with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and computational study. Our work indicates that if the juvenile death rate (through-stage death rate) is nonzero, then for small and large values of maturation time delays, the population dynamics takes the simple form of a globally attractive steady state. Our linear stability work shows that if the resource is dynamic, as in nature, there is a window in maturation time delay parameter that generates sustainable oscillatory dynamics.