Controlling Chaos and Neimark-Sacker Bifurcation in a Discrete-Time Predator-Prey System

A discrete-time predator–prey system incorporating fear effect of the prey with the predator has other food resource is proposed in this paper. The trivial equilibrium and the predator free equilibrium are both unstable. A set of sufficient conditions for the global attractivity of prey free equilibrium and interior equilibrium are established by using iteration scheme and the comparison principle of difference equations. Our study shows that due to the fear of predation, the prey species will be driven to extinction while the predator species tends to be stable since it has other food resource, i.e., the prey free equilibrium may be globally stable under some suitable conditions. Numeric simulations are provided to illustrate the feasibility of the main results.

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Bifurcations in a discrete time Lotka-Volterra predator-prey system

Discrete & Continuous Dynamical Systems - B ◽

10.3934/dcdsb.2006.6.559 ◽

2006 ◽

Vol 6 (3) ◽

pp. 559-572 ◽

Cited By ~ 11

Author(s):

Xiaoli Liu ◽

◽

Dongmei Xiao

Keyword(s):

Discrete Time ◽

Predator Prey ◽

Prey System

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Four positive periodic solutions of a discrete time delayed predator–prey system with nonmonotonic functional response and harvesting

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2008.09.009 ◽

2008 ◽

Vol 56 (12) ◽

pp. 3015-3022 ◽

Cited By ~ 14

Author(s):

Daowei Hu ◽

Zhengqiu Zhang

Keyword(s):

Periodic Solutions ◽

Functional Response ◽

Discrete Time ◽

Positive Periodic Solutions ◽

Predator Prey ◽

Prey System ◽

Nonmonotonic Functional Response

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Discrete-Time Predator-Prey Interaction with Selective Harvesting and Predator Self-Limitation

Journal of Mathematics ◽

10.1155/2020/6737098 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Zhihua Chen ◽

Qamar Din ◽

Muhammad Rafaqat ◽

Umer Saeed ◽

Muhammad Bilal Ajaz

Keyword(s):

Discrete Time ◽

Chaotic Behavior ◽

Period Doubling ◽

State Feedback Control ◽

Time Model ◽

Predator Prey ◽

Selective Harvesting ◽

Predator Prey Model ◽

Prey System ◽

Prey Interaction

Selective harvesting plays an important role on the dynamics of predator-prey interaction. On the other hand, the effect of predator self-limitation contributes remarkably to the stabilization of exploitative interactions. Keeping in view the selective harvesting and predator self-limitation, a discrete-time predator-prey model is discussed. Existence of fixed points and their local dynamics is explored for the proposed discrete-time model. Explicit principles of Neimark–Sacker bifurcation and period-doubling bifurcation are used for discussion related to bifurcation analysis in the discrete-time predator-prey system. The control of chaotic behavior is discussed with the help of methods related to state feedback control and parameter perturbation. At the end, some numerical examples are presented for verification and illustration of theoretical findings.

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Cooperative hunting in a discrete predator–prey system

International Journal of Biomathematics ◽

10.1142/s1793524520500631 ◽

2020 ◽

Vol 13 (07) ◽

pp. 2050063

Author(s):

Yunshyong Chow ◽

Sophia R.-J. Jang ◽

Hua-Ming Wang

Keyword(s):

Growth Rate ◽

Discrete Time ◽

Reproductive Number ◽

Model Parameters ◽

Predator Population ◽

Sufficient Condition ◽

Predator Prey ◽

Cooperative Hunting ◽

Prey System ◽

Dependent Growth

We propose and investigate a discrete-time predator–prey system with cooperative hunting in the predator population. The model is constructed from the classical Nicholson–Bailey host-parasitoid system with density dependent growth rate. A sufficient condition based on the model parameters for which both populations can coexist is derived, namely that the predator’s maximal reproductive number exceeds one. We study existence of interior steady states and their stability in certain parameter regimes. It is shown that the system behaves asymptotically similar to the model with no cooperative hunting if the degree of cooperation is small. Large cooperative hunting, however, may promote persistence of the predator for which the predator would otherwise go extinct if there were no cooperation.

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