A diffusive predator-prey system with prey refuge and predator cannibalism

A diffusive predator-prey system with prey refuge is studied analytically and numerically. The Turing bifurcation is analyzed in detail, which in turn provides a theoretical basis for the numerical simulation. The influence of prey refuge and group defense on the equilibrium density and patterns of species under the condition of Turing instability is explored by numerical simulations, and this shows that the prey refuge and group defense have an important effect on the equilibrium density and patterns of species. Moreover, it can be obtained that the distributions of species are more sensitive to group defense than prey refuge. These results are expected to be of significance in exploration for the spatiotemporal dynamics of ecosystems.

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Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge

Advances in Difference Equations ◽

10.1186/s13662-021-03222-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Yumin Wu ◽

Fengde Chen ◽

Caifeng Du

Keyword(s):

Lyapunov Function ◽

Differential Equations ◽

Comparison Theorem ◽

Sufficient Conditions ◽

Oscillation Theory ◽

Type Ii ◽

Prey Refuge ◽

Predator Prey ◽

Predator Prey Model ◽

Prey System

AbstractIn this paper, we consider a nonautonomous predator–prey model with Holling type II schemes and a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are obtained. By applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained.

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Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

Open Mathematics ◽

10.1515/math-2019-0014 ◽

2019 ◽

Vol 17 (1) ◽

pp. 141-159 ◽

Cited By ~ 9

Author(s):

Zaowang Xiao ◽

Zhong Li ◽

Zhenliang Zhu ◽

Fengde Chen

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Sufficient Conditions ◽

Stage Structure ◽

Prey Refuge ◽

Predator Species ◽

Predator Prey ◽

Predator Prey Model ◽

Prey System ◽

Bifurcation And Stability

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

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The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

Abstract and Applied Analysis ◽

10.1155/2013/168340 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Shaoli Wang ◽

Zhihao Ge

Keyword(s):

Hopf Bifurcation ◽

Logistic Growth ◽

Positive Equilibrium ◽

Prey Refuge ◽

Normal Form Theory ◽

Center Manifold Reduction ◽

Predator Prey ◽

Prey System ◽

Form Theory ◽

The Stability

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

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