Effect of maturation and gestation delays in a stage structure predator prey model (original) (raw)
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Chaos, Solitons & Fractals, 2013
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.
Stability and Hopf Bifurcation Analysis of a Stage-Structured Predator–Prey Model with Delay
Axioms
In this work, a Lotka–Volterra type predator–prey system with time delay and stage structure for the predators is proposed and analyzed. By using the permanence theory for infinite dimensional system, we get that the system is permanent if some conditions are satisfied. The local and global stability of the positive equilibrium is presented. The existence of Hopf bifurcation around the positive equilibrium is observed. Further, by using the normal form theory and center manifold approach, we derive the explicit formulas determining the stability of bifurcating periodic solutions and the direction of Hopf bifurcation. Numerical simulations are carried out by Matlab software to explain the theoretical results. We find that combined time delay and stage structure can affect the dynamical behavior of the system.
Nonlinear Dynamics, 2013
This paper describes a delay induced preypredator system with stage structure for prey. The dynamical characteristics of the system are rigorously studied using mathematical tools. The coexistence equilibria of the system is determined and the dynamic behavior of the system is investigated around coexistence equilibria. Sufficient conditions are derived for the global stability of the system. The optimal harvesting problem is formulated and solved in order to achieve the sustainability of the system, keeping the ecological balance, and maximize the monetary social benefit. Maturation time delay of prey is incorporated and the existence of Hopf bifurcation phenomenon is examined at the coexistence equilibria. It is shown that the time delay can cause a stable
International Journal of Applied and Computational Mathematics, 2018
In this paper, a new stage structured prey-predator model with Monod-Haldane functional response is proposed and the stages for predator have been considered. The proposed mathematical model consists of three nonlinear delay differential equations to describe the interaction among prey, immature predator and mature predator populations. Two time delays viz. feedback delay and gestation delay have been used as the bifurcation parameter. A rigorous mathematical analysis has been carried out by considering all possible cases for both the delays. Conditions for local stability and Hopf bifurcation have been investigated in all cases. Furthermore, by using normal form method, Riesz representing theory and central manifold theorem, formulae are derived for the direction of Hopf bifurcation and stability of bifurcating periodic solutions. Global stability analysis is carried out and the numerical simulation to validate the theory is also executed at the end.
A stage-structured predator–prey model with distributed maturation delay and harvesting
A stage-structured predator–prey system with distributed matura-tion delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium.
International Journal of Dynamics and Control, 2014
In this paper we have considered a prey-predator type fishery model with stage-structure for predator and harvesting of both the prey and predator species. Our study shows that, using the harvesting effort as control, it is possible to break the cycle behaviour of the system and drive it to required steady state. It is also shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. We also investigate the stability of the bifurcating limit cycles and direction of Hopf bifurcation by applying the normal form method and the center manifold theorem. Lastly some numerical simulations are carried out to validate our analytical results.
Iranian Journal of Mathematical Sciences and Informatics, 2020
A mathematical model describing the dynamics of a delayed stage structure prey-predator system with prey refuge is considered. The existence, uniqueness and boundedness of the solution are discussed. All the feasible equilibrium points are determined. The stability analysis of them are investigated. By employing the time delay as the bifurcation parameter, we observed the existence of Hopf bifurcation at the positive equilibrium. The stability and direction of the Hopf bifurcation are determined by utilizing the normal form method and the center manifold reduction. Numerical simulations are given to support the analytic results.
Stability and bifurcation of a prey–predator model with time delay
Comptes Rendus Biologies, 2009
In this article a system of retarded differential equations is proposed as a predator-prey model. We investigate the model, representing a resource (prey) and a two predator system with delay due to gestation. The response function is assumed here to be concave in nature. Since global stability of positive equilibrium is of great interest, we provide sufficient conditions in terms of parameters of the system to guarantee it. By the simulation process the bifurcation occurring are discussed in terms of two bifurcation parameters. We have also shown that the time delay can cause a stable equilibrium to become unstable and even switching of stabilities. Numerical simulations are given to illustrate the results. To cite this article:
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...