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

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.

Spatiotemporal Dynamics of a Diffusive Leslie–Gower Predator–Prey Model with Ratio-Dependent Functional Response

2015 ◽  
Vol 25 (05) ◽  
pp. 1530014 ◽  
Author(s):  
Hong-Bo Shi ◽  
Shigui Ruan ◽  
Ying Su ◽  
Jia-Fang Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Spatiotemporal Dynamics ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Ratio Dependent ◽  
Theoretical Results

This paper is devoted to the study of spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey system with ratio-dependent Holling type III functional response under homogeneous Neumann boundary conditions. It is shown that the model exhibits spatial patterns via Turing (diffusion-driven) instability and temporal patterns via Hopf bifurcation. Moreover, the existence of spatiotemporal patterns is established via Turing–Hopf bifurcation at the degenerate points where the Turing instability curve and the Hopf bifurcation curve intersect. Various numerical simulations are also presented to illustrate the theoretical results.

scholarly journals Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 17
Author(s):  
Ruizhi Yang ◽  
Qiannan Song ◽  
Yong An
Keyword(s):  
Phase Diagram ◽  
Hopf Bifurcation ◽  
Functional Response ◽  
Dynamic Properties ◽  
Equation System ◽  
Turing Instability ◽  
Bifurcation Curve ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Characteristic Roots

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.

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

2015 ◽  
Vol 08 (01) ◽  
pp. 1550013 ◽  
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.

Turing Instability and Bifurcation in a Diffusion Predator–Prey Model with Beddington–DeAngelis Functional Response

2018 ◽  
Vol 28 (09) ◽  
pp. 1830029 ◽  
Author(s):  
Wei Tan ◽  
Wenwu Yu ◽  
Tasawar Hayat ◽  
Fuad Alsaadi ◽  
Habib M. Fardoun
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Steady State Solution ◽  
Turing Instability ◽  
Rigorous Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Global Positive Solution ◽  
The Stability

In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response with or without diffusion. For this system, we give a complete and rigorous analysis of the dynamics including the existence of a global positive solution, the stability/Turing instability and the Hopf bifurcation. In the meanwhile, we show, via numerical simulations, that there appears Hopf bifurcation, steady state solution and Turing–Hopf bifurcation with the changes of some parameters of the system.

Bifurcation Analysis of a Diffusive Predator–Prey Model with Monod–Haldane Functional Response

2019 ◽  
Vol 29 (11) ◽  
pp. 1950152
Author(s):  
Qiannan Song ◽  
Ruizhi Yang ◽  
Chunrui Zhang ◽  
Leiyu Tang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Response ◽  
Turing Instability ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Spatially Inhomogeneous ◽  
Prey Model ◽  
Steady State Solutions ◽  
Theoretical Results

In this paper, we consider a diffusive predator–prey model with Monod–Haldane functional response. We study the Turing instability and Hopf bifurcation of the coexisting equilibriums. We investigate the Turing–Hopf bifurcation through some key bifurcation parameters. In addition, we obtain a normal form for the Turing–Hopf bifurcation. Finally, we show numerical simulations to illustrate the theoretical results. For parameters around the critical value of the Turing–Hopf bifurcation, we demonstrate that the predator–prey model exhibits complex spatiotemporal dynamics, including spatially homogeneous periodic solutions, spatially inhomogeneous periodic solutions, and spatially inhomogeneous steady-state solutions.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

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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