Hopf bifurcation for a predator–prey model with age structure

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.

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Hopf bifurcation for a predator–prey model with age structure

Applied Mathematical Modelling ◽

10.1016/j.apm.2015.09.015 ◽

2016 ◽

Vol 40 (2) ◽

pp. 726-737 ◽

Cited By ~ 14

Author(s):

Hui Tang ◽

Zhihua Liu

Keyword(s):

Hopf Bifurcation ◽

Age Structure ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model

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Stability and Bifurcation Analysis in a Predator–Prey Model with Age Structure and Two Delays

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421500243 ◽

2021 ◽

Vol 31 (02) ◽

pp. 2150024

Author(s):

Yujia Wang ◽

Dejun Fan ◽

Junjie Wei

Keyword(s):

Hopf Bifurcation ◽

Age Structure ◽

Time Delays ◽

Semigroup Theory ◽

Abstract Cauchy Problem ◽

Transcendental Equation ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Two Delays

In this paper, a predator–prey model with age structure, Beddington–DeAngelis functional response and time delays is considered. Using a geometric method for studying transcendental equation with two delays, we conduct detailed analysis on the distribution of the roots for the characteristic equation of the model. Then, applying the integrated semigroup theory and the Hopf bifurcation theorem for an abstract Cauchy problem within a nondense domain, we proved the existence of Hopf bifurcation for the model. Stability switches can also occur, as the two time delays pass through a continuous curve in the parameter plane. To illustrate the theoretical results, numerical simulations are presented.

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Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽

10.3390/sym13050785 ◽

2021 ◽

Vol 13 (5) ◽

pp. 785

Author(s):

Hasan S. Panigoro ◽

Agus Suryanto ◽

Wuryansari Muharini Kusumawinahyu ◽

Isnani Darti

Keyword(s):

Hopf Bifurcation ◽

Fractional Derivative ◽

Equilibrium Point ◽

Power Law ◽

Equilibrium Points ◽

Essential Difference ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

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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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