A prey-predator model with migrations and delays

Al-Darabsah, Isam; Tang, Xianhua; Yuan, Yuan

doi:10.3934/dcdsb.2016.21.737

Cited by 19 publications

(3 citation statements)

References 32 publications

(56 reference statements)

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“…In fact, there are many literatures on predator-prey model with delays, we refer to Refs. [12,[23][24][25][26] and the references therein. Among them, effects of one delay on systems have been discussed widely.…”

Section: Introductionmentioning

confidence: 99%

Two delays induce Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system

Niu

Wei

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, the dynamics of a modified Leslie-Gower predator-prey system with two delays and diffusion is considered. By calculating stability switching curves, the stability of positive equilibrium and the existence of Hopf bifurcation and double Hopf bifurcation are investigated on the parametric plane of two delays. Taking two time delays as bifurcation parameters, the normal form on the center manifold near the double Hopf bifurcation point is derived, and the unfoldings near the critical points are given. Finally, we obtain the complex dynamics near the double Hopf bifurcation point, including the existence of quasi-periodic solutions on a 2-torus, quasi-periodic solutions on a 3-torus, and strange attractors.Diffusive predator-prey models with delays have been investigated widely, and the delay induced Hopf bifurcation analysis has been well studied. However, the study about bifurcation analysis of predator-prey models with two simultaneously varying delays has not been well established. Neither the Hopf bifurcation theorem with two parameters nor the derivation process of normal form for two delays induced double Hopf bifurcation has been proposed in literatures. In this paper, we investigate a diffusive Leslie-Gower model with two delays, and carry out Hopf and double Hopf bifurcation analysis of the model. Applying the method of studying characteristic equation with two delays, we get the stability switching curves and the crossing direction, after which we give the Hopf bifurcation theorem in two-parameter plane for the first time. Under some condition, the intersections of two stability switching curves are double Hopf bifurcation points. To figure out the dynamics near the double Hopf bifurcation point, we calculate the normal form on the center manifold. The derivation process of normal form we use in this paper can be extended to other models with two delays, one delay, or without delay.

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Section: Introductionmentioning

confidence: 99%

Two delays induce Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system

Niu

Wei

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…e.g. [3,9,17]). The existence of a saturated equilibrium depends on the properties of the controlled community matrix M .…”

Section: )mentioning

confidence: 99%

Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls

Muroya¹,

Faria²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we apply a Lyapunov functional approach to Lotka-Volterra systems with infinite delays and feedback controls and establish that the feedback controls have no influence on the attractivity properties of a saturated equilibrium. This improves previous results by the authors and others, where, while feedback controls were used mostly to change the position of a unique saturated equilibrium, additional conditions involving the controls had to be assumed in order to preserve its global attractivity. The situation of partial extinction is further analysed, for which the original system is reduced to a lower dimensional one which maintains its global dynamics features.

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“…Qiu and Mitsui [13] considered a predator-prey system with diffusion and time delay in an npatch environment, where prey disperses between n patches of a heterogeneous environment with barriers between patches where a predator cannot cross. Al-Darabsah et al [14] proposed a prey-predator model in multiple patches through the stage-structured maturation time delay with migrations among patches. However, they only discussed the existence of equilibrium points and the uniform persistence for the special case of two patches.…”

Section: Introductionmentioning

confidence: 99%

A Stage-Structured Predator-Prey Model in a Patchy Environment

2020

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In this paper, we propose a stage-structured predator-prey model with migrations among patches in an n-patch environment. The net reproduction number for each patch in isolation is obtained along with the net reproduction number of the system of patches, ℛ0. Inequalities describing the relationship among these numbers are also given. Furthermore, threshold dynamics determined by ℛ0 is established: the predator dies out if ℛ0<1 while the predator persists if ℛ0>1. Focusing on the case with two patches, we obtain that the dispersal decreases the net reproduction number ℛ0. By numerical simulations, we find that the dispersal may be a good thing or a bad thing because the dispersal could make the predator population thrive or extinct, and hence we might seek steady state in the ecological environment by controlling parameters related to the prey and the predator.

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A prey-predator model with migrations and delays

Cited by 19 publications

References 32 publications

Two delays induce Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system

Two delays induce Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system

Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls

A Stage-Structured Predator-Prey Model in a Patchy Environment

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