Stability Of a Delayed Predator-Prey Model With Predator Migration (original) (raw)
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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:
MathLAB Journal, 2020
This study attempts to model a real life situation involving delay differential equation, in particular the predatorprey interaction. A delay of =0.01 is prescribed into the system to determine whether the system will be stable or otherwise. The results show that when an insignificant delay is introduced, its stability returns to that of an ordinary differential equation, but when the delay is significant, it results into solutions of infinite roots.
Dynamical analysis of a predator-prey interaction model with time delay and prey refuge
Nonautonomous Dynamical Systems, 2018
In this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.
Stability and bifurcation in a generalized delay prey–predator model
Nonlinear Dynamics, 2017
The present paper considers a generalized prey-predator model with time delay. It studies the stability of the nontrivial positive equilibrium and the existence of Hopf bifurcation for this system by choosing delay as a bifurcation parameter and analyzes the associated characteristic equation. The researcher investigates the direction of this bifurcation by using an explicit algorithm. Eventually, some numerical simulations are carried out to support the analytical results.
A PREY, PREDATOR AND A COMPETITOR TO THE PREDATOR MODEL WITH TIME DELAY
The aim of this paper is to study the stability analysis of three species Ecological model consists of a Prey (N 1), a predator (N 2) and a competitor (N 3) .The competitor (N 3) is competing with the Predator Species (N 2) and neutral with the prey (N 1). In this model the third species is competing with the predator for food other than the prey (N 1).In addition to that, the death rates, carrying capacities of all three species are also considered and a delay in the interaction between the prey and the predator (gestation period of the predator) is also considered. The model is characterized by a couple of integro-differential equations. All the eight equilibrium points of the model are identified and their local stability is discussed for interior equilibrium point. The global stability is studied by constructing a suitable Lyapunov's function. Suitable parameter are identified for Numerical simulation which shows that this continuous time delay model exhibits rich dynamics and time delay can further stabilize or destabilize the system.
Journal of Mathematical Analysis and Applications, 2008
In this paper, we consider a delayed reaction-diffusion equations which describes a two-species predator-prey system with diffusion terms and stage structure. By using the linearization method and the method of upper and lower solutions, we study the local and global stability of the constant equilibria, respectively. The results show that the free diffusion of the delayed reactiondiffusion equations has no effect on the populations when the diffusion is too slow; otherwise, the free diffusion has a certain influence on the populations, however, the influence can be eliminated by improving the parameters to satisfy some suitable conditions.
Dynamics of a Prey and Two Predators System with Time Delay
International journal of pure and applied mathematics, 2018
The aim of this paper is to study the prey-predator system with delay effects. Initially, the positive equilibrium point of the proposed system is derived and its local stability is discussed using Routh-Hurwitz criterion. A well suited Lyapunov function describes the global asymptotic stability of the system. To preserve the stability of the system without violating its properties the length of time delay is estimated and some important conclusions are made at the end.
Dynamical Properties of a Delay Prey-Predator Model with Disease in the Prey Species Only
Discrete Dynamics in Nature and Society, 2010
A three-dimensional ecoepidemiological model with delay is considered. We first investigate the existence and stability of the equilibria. We then study the effect of the time delay on the stability of the positive equilibrium. The existence of a Hopf bifurcation at the positive equilibrium is obtained through the study of an exponential polynomial equation with delay-dependent coefficients. Numerical simulation with a hypothetical set of data has been carried out to support the analytical findings.
Numerical Algebra, Control and Optimization, 2021
In this paper, the modelling and analysis of prey-predator model involving predation of mature prey is done using DDE. Equilibrium points are calculated and stability analysis is performed about non-zero equilibrium point. Delay parameter destabilizes the system and triggers asymptotic stability when value of delay parameter is below the critical point. Hopf bifurcation is observed when the value of delay parameter crosses the critical point. Sensitivity analysis has also been performed to look into the effect of other parameters on the state variables. The numerical results are substantiated using MATLAB.
Mathematical Biosciences, 2011
We consider a system of delay differential equations modeling the predator–prey ecoepidemic dynamics with a transmissible disease in the predator population. The time lag in the delay terms represents the predator gestation period. We analyze essential mathematical features of the proposed model such as local and global stability and in addition study the bifurcations arising in some selected situations. Threshold values for a few parameters determining the feasibility and stability conditions of some equilibria are discovered and similarly a threshold is identified for the disease to die out. The parameter thresholds under which the system admits a Hopf bifurcation are investigated both in the presence of zero and non-zero time lag. Numerical simulations support our theoretical analysis.► An eco-epidemic model. ► Effect of delay. ► Investigate possible unforeseen consequences in the human intervention. ► A threshold is identified for the disease to die out. ► Numerical simulations support our theoretical analysis.