scholarly journals Stable Periodic Orbits for a Predator–Prey Model with Delay

2000 ◽  
Vol 249 (2) ◽  
pp. 324-339 ◽  
Author(s):  
Mario Cavani ◽  
Marcos Lizana ◽  
Hal L. Smith
Keyword(s):  
Periodic Orbits ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Stable Periodic

BIFURCATION CONTROL OF A PREDATOR–PREY MODEL BASED ON NUTRITION KINETICS

2013 ◽  
Vol 06 (04) ◽  
pp. 1350019
Author(s):  
LICHUN ZHAO ◽  
JINGNA LIU ◽  
WEI GAO
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Trajectory ◽  
Loss Of Stability ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Model Based ◽  
Prey Model ◽  
Catastrophic Loss ◽  
Stable Periodic ◽  
Existence Conditions

The existence conditions of Hopf bifurcation for a predator–prey model based on nutrition kinetics are given. The two results may appear as follows: one is that the model has a stable periodic trajectory from Hopf bifurcation, which shows the system is in an ecological balance; the other is that periodic trajectory from Hopf bifurcation is unstable, which indicates the system is in a sharp or catastrophic loss of stability. For the latter, a bifurcation controller is designed to make the periodic trajectory stable. Finally, some simulations are carried out to prove the results.

Bifurcation Analysis of a Predator–Prey Model with Age Structure

2020 ◽  
Vol 30 (08) ◽  
pp. 2050114
Author(s):  
Yuting Cai ◽  
Chuncheng Wang ◽  
Dejun Fan
Keyword(s):  
Hopf Bifurcation ◽  
Age Structure ◽  
Cauchy Problems ◽  
Maturation Period ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Stable Periodic ◽  
The Stability

In this paper, a predator–prey model with age structure in predator is studied. Using maturation period as the varying parameter, we prove the existence of Hopf bifurcation for the model and calculate the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The method we employed includes Hopf bifurcation theorem, center manifolds and normal form theory for the abstract Cauchy problems with nondense domain. Under a certain set of parameter values, it turns out that subcritical Hopf bifurcation may occur, indicating that the increment of maturation period could stabilize the steady state, which is initially unstable and enclosed by a stable periodic solution. In addition, stability switches will also take place. Numerical simulations are finally carried out to show the theoretical results.

scholarly journals A Predator-Prey Model in the Chemostat with Holling Type II Response Function

2020 ◽  
Vol 9999 (9999) ◽  
pp. 1-22
Author(s):  
Tedra Bolger ◽  
Brydon Eastman ◽  
Madeleine Hill ◽  
Gail Wolkowicz
Keyword(s):  
Type Ii ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Asymptotic Stability ◽  
Prey Model ◽  
Prey Interaction ◽  
Stable Periodic ◽  
Unstable Solutions

A model of predator-prey interaction in a chemostat with Holling Type II functional and numerical response functions ofthe Monod or Michaelis-Menten form is considered. It is proved that local asymptotic stability of the coexistence equilibriumimplies that it is globally asymptotically stable. It is also shown that when the coexistence equilibrium exists but is unstable,solutions converge to a unique, orbitally asymptotically stable periodic orbit. Thus the range of the dynamics of the chemostatpredator-prey model is the same as for the analogous classical Rosenzweig-MacArthur predator-prey model with Holling TypeII functional response. An extension that applies to other functional rsponses is also given.

Nonlinear Dynamical Behaviour in a Predator-Prey Model with Harvesting

2017 ◽  
Vol 7 (2) ◽  
pp. 376-395
Author(s):  
Wei Liu ◽  
Yaolin Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Sufficient Conditions ◽  
Dynamical Behaviour ◽  
Nonlinear Dynamical ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Damped Oscillations ◽  
Prey Model ◽  
The Stability

AbstractWe investigate the stability and periodic orbits of a predator-prey model with harvesting. The model has a biologically-meaningful interior, an attractor undergoing damped oscillations, and can become destabilised to produce periodic orbits via a Hopf bifurcation. Some sufficient conditions for the existence of the Hopf bifurcation are established, and a stability analysis for the periodic solutions using a Lyapunov function is presented. Finally, some computer simulations illustrate our theoretical results.

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