Hopf Bifurcations in a Predator–Prey Model with an Omnivore

2019 ◽  
Vol 18 (3) ◽  
pp. 1201-1224 ◽  
Author(s):  
Yongjun Li ◽  
Valery G. Romanovski
Keyword(s):  
Hopf Bifurcations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

Stability and Bifurcation Analysis in a Delayed Predator-Prey Model with Stage-Structure and Harvesting

2010 ◽  
Vol 143-144 ◽  
pp. 1358-1363
Author(s):  
Zhi Chao Jiang ◽  
Ming Wei Nie
Keyword(s):  
Bifurcation Analysis ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Stability And Bifurcation Analysis ◽  
Characteristic Values ◽  
The Stability

In this paper, we investigate a delayed stage-structured predator-prey model with continuous harvesting on prey. Positivity and boundness of solutions and sufficient conditions of the stability of equilibria are obtained. Using and as bifurcation parameters, the existence of Hopf bifurcations at equilibria is established by analyzing the distribution of the characteristic values.

Spatial Periodic Solutions in a Delayed Diffusive Predator–Prey Model with Herd Behavior

2015 ◽  
Vol 25 (11) ◽  
pp. 1550155 ◽  
Author(s):  
Chaoqun Xu ◽  
Sanling Yuan
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Critical Conditions ◽  
Hopf Bifurcations ◽  
Herd Behavior ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

A delayed diffusive predator–prey model with herd behavior subject to Neumann boundary conditions is studied both theoretically and numerically. Applying Hopf bifurcation analysis, we obtain the critical conditions under which the model generates spatially nonhomogeneous bifurcating periodic solutions. It is shown that the spatially homogeneous Hopf bifurcations always exist and that the spatially nonhomogeneous Hopf bifurcations will arise when the diffusion coefficients are suitably small. The explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorems for partial functional differential equations.

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