Hopf Bifurcation and Singularity Induced Bifurcation in a Leslie-Gower Predator-Prey System with Nonlinear Harvesting

A singular delayed biological economic predator-prey system with and without stochastic fluctuation is proposed. The conditions of singularity induced bifurcation are gained, and a state feedback controller is designed to eliminate such bifurcation. Furthermore, saddle-node bifurcation is also showed. Next, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of roots of the corresponding characteristic equation, and the hybrid control strategy is used to control the occurrence of Hopf bifurcation. In addition, some explicit formulas determining the spectral densities of the populations and harvest effort are given when the system is considered with stochastic fluctuation. Finally, numerical simulations are illustrated to verify the theoretical results.

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Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

Advances in Difference Equations ◽

10.1186/s13662-020-02879-4 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Heping Jiang ◽

Huiping Fang ◽

Yongfeng Wu

Keyword(s):

Hopf Bifurcation ◽

Neumann Boundary ◽

Herd Behavior ◽

Predator Prey ◽

Predator Prey Model ◽

Dynamical Behaviors ◽

Prey System ◽

Prey Model ◽

Homogeneous Neumann Boundary Condition ◽

The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

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Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

Complexity ◽

10.1155/2020/4285294 ◽

2020 ◽

Vol 2020 ◽

pp. 1-19

Author(s):

Kankan Sarkar ◽

Subhas Khajanchi ◽

Prakash Chandra Mali ◽

Juan J. Nieto

Keyword(s):

Growth Rate ◽

Hopf Bifurcation ◽

Dynamical Behavior ◽

Prey Population ◽

Uniform Persistence ◽

Functional Responses ◽

Normal Form Theory ◽

Predator Prey ◽

Center Manifold Theorem ◽

Prey System

In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

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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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Stability and Hopf bifurcation in a reaction–diffusion predator–prey system with interval biological parameters and stage structure

Nonlinear Dynamics ◽

10.1007/s11071-014-1775-5 ◽

2014 ◽

Vol 79 (3) ◽

pp. 1797-1816 ◽

Cited By ~ 4

Author(s):

Hongyong Zhao ◽

Ling Wang

Keyword(s):

Hopf Bifurcation ◽

Reaction Diffusion ◽

Stage Structure ◽

Biological Parameters ◽

Predator Prey ◽

Prey System

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Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting

Abstract and Applied Analysis ◽

10.1155/2014/624162 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Fengying Wei ◽

Lanqi Wu ◽

Yuzhi Fang

Keyword(s):

Hopf Bifurcation ◽

Center Manifold ◽

Hopf Bifurcations ◽

Normal Form Theory ◽

Predator Prey ◽

Center Manifold Theorem ◽

Prey System ◽

Theoretical Predictions ◽

Form Theory ◽

The Stability

A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delayτpasses through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.

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