scholarly journals Turing-Hopf Bifurcation in the Predator-prey Model with Cross-diffusion Considering Two Different Prey Behaviours Transition

2021 ◽  
Author(s):  
Yehu Lv
Keyword(s):  
Hopf Bifurcation ◽  
Boundedness Of Solutions ◽  
Hopf Bifurcation Point ◽  
Spatiotemporal Model ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
Stability And Instability

Abstract In this paper, we study the Turing-Hopf bifurcation in the predator-prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analyzing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by computing and analyzing the normal form on the center manifold associated with the Turing-Hopf bifurcation, we investigate the dynamical classification near the Turing-Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.

Spatiotemporal patterns induced by cross-diffusion in predator–prey model with prey herd shape effect

2020 ◽  
Vol 13 (04) ◽  
pp. 2050030 ◽  
Author(s):  
Salih Djilali
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Spatiotemporal Patterns ◽  
Shape Effect ◽  
Hopf Bifurcation Point ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Impact

In this paper, we investigate a predator–prey model with herd behavior and cross-diffusion subject to the zero flux boundary conditions. First, the temporal behavior of the model has been investigated, where Hopf bifurcation has been obtained. Then, by analyzing the characteristic equation it has been proved that the cross-diffusion generate a complex dynamics such as Hopf bifurcation, Turing instability, even Turing–Hopf bifurcation. Further, the impact of the prey herd shape on the spatiotemporal patterns has been discussed. Furthermore, by computing and analyzing the normal form associated with the Turing–Hopf bifurcation point, the spatiotemporal dynamics near the Turing–Hopf bifurcation point has been discussed and also justified by some numerical simulations.

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

Symmetry ◽  
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.

Spatial Pattern of Ratio-Dependent Predator–Prey Model with Prey Harvesting and Cross-Diffusion

2019 ◽  
Vol 29 (03) ◽  
pp. 1950036 ◽  
Author(s):  
R. Sivasamy ◽  
M. Sivakumar ◽  
K. Balachandran ◽  
K. Sathiyanathan
Keyword(s):  
Temporal Dynamics ◽  
Intake Rate ◽  
Diffusion Term ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonlinear Harvesting ◽  
Ratio Dependent ◽  
The Cross

This study focuses on the spatial-temporal dynamics of predator–prey model with cross-diffusion where the intake rate of prey is per capita predator according to ratio-dependent functional response and the prey is harvested through nonlinear harvesting strategy. The permanence analysis and local stability analysis of the proposed model without cross-diffusion are analyzed. We derive the conditions for the appearance of diffusion-driven instability and global stability of the considered model. Also the parameter space for Turing region is specified by keeping the cross-diffusion coefficient as one of the crucial parameters. Numerical simulations are given to justify the proposed theoretical results and to show that the cross-diffusion term plays a significant role in the pattern formation.

scholarly journals Instability Induced by Cross-Diffusion in a Predator-Prey Model with Sex Structure

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Shengmao Fu ◽  
Lina Zhang
Keyword(s):  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Sex Structure ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Ode System ◽  
Prey Model ◽  
Weakly Coupled ◽  
Coupled Reaction

In this paper, we consider a cross-diffusion predator-prey model with sex structure. We prove that cross-diffusion can destabilize a uniform positive equilibrium which is stable for the ODE system and for the weakly coupled reaction-diffusion system. As a result, we find that stationary patterns arise solely from the effect of cross-diffusion.

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