scholarly journals Hopf bifurcation and global stability of a diffusive Gause-type predator–prey models

2016 ◽  
Vol 72 (10) ◽  
pp. 2620-2635 ◽  
Author(s):  
Yunfei Lv ◽  
Yongzhen Pei ◽  
Rong Yuan
Keyword(s):  
Hopf Bifurcation ◽  
Global Stability ◽  
Predator Prey ◽  
Predator Prey Models

scholarly journals Boundedness and Global Stability for a Predator-Prey System with the Beddington-DeAngelis Functional Response

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
Wahiba Khellaf ◽  
Nasreddine Hamri
Keyword(s):  
Global Stability ◽  
Functional Response ◽  
Qualitative Behavior ◽  
Uniform Persistence ◽  
Boundedness Of Solutions ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Prey System ◽  
Predator Prey Models ◽  
Type Ii Functional Response

We study the qualitative behavior of a class of predator-prey models with Beddington-DeAngelis-type functional response, primarily from the viewpoint of permanence (uniform persistence). The Beddington-DeAngelis functional response is similar to the Holling type-II functional response but contains a term describing mutual interference by predators. We establish criteria under which we have boundedness of solutions, existence of an attracting set, and global stability of the coexisting interior equilibrium via Lyapunov function.

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.

scholarly journals Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Normal Form Method ◽  
Predator Prey Models ◽  
The Stability

A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained 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 Global Stability of a Predator-Prey Model with Ivlev-Type Functional Response

2020 ◽  
Vol 99 (99) ◽  
pp. 1-12
Author(s):  
Yinshu Wu ◽  
Wenzhang Huang
Keyword(s):  
Global Stability ◽  
Functional Response ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models ◽  
Global And Local

A predator-prey model with Ivlev-Type functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recently developed approach shows that for certain classes of models, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing a detailed analytic analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium, which enable us to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional responses.

THE EFFECTS OF HARVESTING AND TIME DELAY ON PREDATOR-PREY SYSTEMS WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE

2012 ◽  
Vol 05 (01) ◽  
pp. 1250008 ◽  
Author(s):  
XINYOU MENG ◽  
HAIFENG HUO ◽  
XIAOBING ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Global Stability ◽  
Functional Response ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Combined Effects ◽  
Predator Prey ◽  
Region Of Stability ◽  
Two Parameter

The combined effects of harvesting and time delay on predator-prey systems with Beddington–DeAngelis functional response are studied. The region of stability in model with harvesting of the predator, local stability of equilibria and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation due to the two-parameter geometric criteria developed by Ma, Feng and Lu [Discrete Contin. Dyn. Syst. Ser B9 (2008) 397–413]. The global stability of the positive equilibrium is investigated by the comparison theorem. Furthermore, local stability of steady states and the existence of Hopf bifurcation for prey harvesting are also considered. Numerical simulations are given to illustrate our theoretical findings.

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