Hopf bifurcation and global stability for a delayed predator–prey system with stage structure for predator

2008 ◽  
Vol 202 (2) ◽  
pp. 721-729 ◽  
Author(s):  
Shujing Gao ◽  
Lansun Chen ◽  
Zhidong Teng
Keyword(s):  
Hopf Bifurcation ◽  
Global Stability ◽  
Stage Structure ◽  
Predator Prey ◽  
Prey System

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

2019 ◽  
Vol 17 (1) ◽  
pp. 141-159 ◽  
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.

scholarly journals Dynamical Analysis in a Delayed Predator-Prey System with Stage-Structure for Both the Predator and the Prey

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

A predator-prey system with two delays and stage-structure for both the predator and the prey is considered. Sufficient conditions for the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. Specially, the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from the Hopf bifurcation are determined by applying the normal form theory and center manifold argument. Some numerical simulations for justifying the theoretical analysis are also provided.

scholarly journals Global Stability and Hopf Bifurcation of a Predator-Prey Model with Time Delay and Stage Structure

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Lingshu Wang ◽  
Guanghui Feng
Keyword(s):  
Hopf Bifurcation ◽  
Global Stability ◽  
Stage Structure ◽  
Predator Prey ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Predator Prey Model ◽  
Persistence Theory ◽  
Lasalle Invariant Principle ◽  
Type Ii Functional Response

A delayed predator-prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of persistence theory on infinite dimensional systems, it is proved that the system is permanent. By using Lyapunov functions and the LaSalle invariant principle, the global stability of each of the feasible equilibria of the model is discussed. Numerical simulations are carried out to illustrate the main theoretical results.

A diffusive predator–prey system with additional food and intra-specific competition among predators

2018 ◽  
Vol 11 (04) ◽  
pp. 1850060 ◽  
Author(s):  
Ruizhi Yang ◽  
Ming Liu ◽  
Chunrui Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Global Stability ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Delay System ◽  
Additional Food ◽  
Predator Prey ◽  
Prey System ◽  
Boundary Equilibrium ◽  
The Stability

In this paper, a diffusive predator–prey system with additional food and intra-specific competition among predators subject to Neumann boundary condition is investigated. For non-delay system, global stability, Turing instability and Hopf bifurcation are studied. For delay system, instability and Hopf bifurcation induced by time delay and global stability of boundary equilibrium are discussed. By the theory of normal form and center manifold method, the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived.

scholarly journals Hopf Bifurcation of a Predator-Prey System with Delays and Stage Structure for the Prey

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
The Stability

This paper is concerned with a Holling type III predator-prey system with stage structure for the prey population and two time delays. The main result is given in terms of local stability and bifurcation. By choosing the time delay as a bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. In particular, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form method and center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are also included.

scholarly journals Hopf bifurcation analysis in a stage-structure predator–prey model with two time delay

2022 ◽  
Vol 355 ◽  
pp. 03048
Author(s):  
Bochen Han ◽  
Shengming Yang ◽  
Guangping Zeng
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Time Delays ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Predator Model

In this paper, we consider a predator-prey system with two time delays, which describes a prey–predator model with parental care for predators. The local stability of the positive equilibrium is analysed. By choosing the two time delays as the bifurcation parameter, the existence of Hopf bifurcation is studied. Numerical simulations show the positive equilibrium loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold.

scholarly journals A Stochastic Predator-Prey System with Stage Structure for Predator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shufen Zhao ◽  
Minghui Song
Keyword(s):  
Global Existence ◽  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Stability In Probability ◽  
Prey System ◽  
Theoretical Results

The authors introduce stochasticity into a predator-prey system with Beddington-DeAngelis functional response and stage structure for predator. We present the global existence and positivity of the solution and give sufficient conditions for the global stability in probability of the system. Numerical simulations are introduced to support the main theoretical results.

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