Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator–prey models

In this paper, a diffusive predator–prey system with a functional response that increases in both predator and prey densities is considered. By analyzing the characteristic roots of the partial differential equation system, the Turing instability and Hopf bifurcation are studied. In order to consider the dynamics of the model where the Turing bifurcation curve and the Hopf bifurcation curve intersect, we chose the diffusion coefficients d1 and β as bifurcating parameters. In particular, the normal form of Turing–Hopf bifurcation was calculated so that we could obtain the phase diagram. For parameters in each region of the phase diagram, there are different types of solutions, and their dynamic properties are extremely rich. In this study, we have used some numerical simulations in order to confirm these ideas.

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SPATIOTEMPORAL DYNAMICS OF A DIFFUSIVE LESLIE-TYPE PREDATOR–PREY MODEL WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE

Journal of Biological System ◽

10.1142/s0218339020500175 ◽

2020 ◽

Vol 28 (03) ◽

pp. 785-809

Author(s):

YAN LI ◽

LINYAN ZHANG ◽

DAGEN LI ◽

HONG-BO SHI

Keyword(s):

Hopf Bifurcation ◽

Functional Response ◽

Homogeneous Model ◽

Neumann Boundary ◽

Turing Instability ◽

Spatiotemporal Dynamics ◽

Predator Prey ◽

Predator Prey Model ◽

Local Asymptotic Stability ◽

Diffusive Model

In this paper, we study the spatiotemporal dynamics of a diffusive Leslie-type predator–prey system with Beddington–DeAngelis functional response under homogeneous Neumann boundary conditions. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the diffusive model, firstly, it is shown that Turing (diffusion-driven) instability occurs which induces spatial inhomogeneous patterns. Next, it is proved that the diffusive model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the diffusive model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Numerical simulations are also presented to verify the theoretical results.

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

Abstract and Applied Analysis ◽

10.1155/2012/363051 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 4

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.

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Stability and Hopf bifurcation analysis of a diffusive predator–prey model with Smith growth

International Journal of Biomathematics ◽

10.1142/s1793524515500138 ◽

2015 ◽

Vol 08 (01) ◽

pp. 1550013 ◽

Cited By ~ 5

Author(s):

M. Sivakumar ◽

M. Sambath ◽

K. Balachandran

Keyword(s):

Boundary Condition ◽

Hopf Bifurcation ◽

Bifurcation Analysis ◽

Neumann Boundary Condition ◽

Local Stability ◽

Neumann Boundary ◽

Turing Instability ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model

In this paper, we consider a diffusive Holling–Tanner predator–prey model with Smith growth subject to Neumann boundary condition. We analyze the local stability, existence of a Hopf bifurcation at the co-existence of the equilibrium and stability of bifurcating periodic solutions of the system in the absence of diffusion. Furthermore the Turing instability and Hopf bifurcation analysis of the system with diffusion are studied. Finally numerical simulations are given to demonstrate the effectiveness of the theoretical analysis.

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

International Journal of Biomathematics ◽

10.1142/s1793524518500602 ◽

2018 ◽

Vol 11 (04) ◽

pp. 1850060 ◽

Cited By ~ 1

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.

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Pattern Formation in a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

Discrete Dynamics in Nature and Society ◽

10.1155/2013/454209 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Bo Yang

Keyword(s):

Boundary Condition ◽

Hopf Bifurcation ◽

Turing Instability ◽

Turing Patterns ◽

Positive Equilibrium ◽

Flux Boundary Condition ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Ratio Dependent

The spatiotemporal dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition are investigated analytically and numerically. The asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation around the positive equilibrium are shown; the conditions of Turing instability are obtained. And with the help of numerical simulations, it is found that the model exhibits complex pattern replication: stripes, spots-stripes mixtures, and spots Turing patterns.

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