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.

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.

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.

scholarly journals A new study for the global asymptotic stability of a general predator-prey model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Mathematical Analysis ◽  
Global Asymptotic Stability ◽  
Lyapunov Stability Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Theoretical Results

In a previous paper [L. M. Ladino, E. I. Sabogal, Jose C. Valverde, General functional response and recruitment in a predator-prey system with capture on both species, Math. Methods Appl. Sci. 38(2015) 2876-2887], a mathematical model for a predator-prey model with general functional response and recruitment including capture on both species was formulated and analyzed. However, the global asymptotic stability (GAS) of this model was only partially resolved. In the present paper, we provide a rigorously mathematical analysis for the complete GAS of the predator-prey model. By using the Lyapunov stability theory in combination with some nonstandard techniques of mathematical analysis for dynamical systems, the GAS of equilibria of the model is determined fully. The obtained results not only provide an important improvement for the population dynamics of the predator-prey model but also can be extended to study its modified versions in the context of fractional-order derivatives. The theoretical results are supported and illustrated by a set of numerical examples.

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.

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

2018 ◽  
Vol 11 (04) ◽  
pp. 1850060 ◽  
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.

scholarly journals Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response

2004 ◽  
Vol 2004 (2) ◽  
pp. 325-343 ◽  
Author(s):  
Lin-Lin Wang ◽  
Wan-Tong Li
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Globally Asymptotical Stability ◽  
Ratio Dependent

The existence of positive periodic solutions for a delayed discrete predator-prey model with Holling-type-III functional responseN1(k+1)=N1(k)exp{b1(k)−a1(k)N1(k−[τ1])−α1(k)N1(k)N2(k)/(N12(k)+m2N22(k))},N2(k+1)=N2(k)exp{−b2(k)+α2(k)N12(k−[τ2])/(N12(k−[τ2])+m2N22(k−[τ2]))}is established by using the coincidence degree theory. We also present sufficient conditions for the globally asymptotical stability of this system when all the delays are zero. Our investigation gives an affirmative exemplum for the claim that the ratio-dependent predator-prey theory is more reasonable than the traditional prey-dependent predator-prey theory.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
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.

A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH DELAY AND HARVESTING

2010 ◽  
Vol 18 (02) ◽  
pp. 437-453 ◽  
Author(s):  
A. K. MISRA ◽  
B. DUBEY
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Functional Response ◽  
Bifurcation Parameter ◽  
Discrete Delay ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper a predator-prey model with discrete delay and harvesting of predator is proposed and analyzed by considering ratio-dependent functional response. Conditions of existence of various equilibria and their stability have been discussed. By taking delay as a bifurcation parameter, the system is found to undergo a Hopf bifurcation. Numerical simulations are also performed to illustrate the results.

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