scholarly journals Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xinze Lian ◽  
Shuling Yan ◽  
Hailing Wang
Keyword(s):  
Time Delay ◽  
Pattern Formation ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
The Stability ◽  
Self Replication

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

scholarly journals Pattern Formation in a Predator-Prey Model with Both Cross Diffusion and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Boli Xie ◽  
Zhijun Wang ◽  
Yakui Xue
Keyword(s):  
Time Delay ◽  
Spatial Dynamics ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Real Model ◽  
Diffusion Controlled ◽  
And Diffusion

A predator-prey model with both cross diffusion and time delay is considered. We give the conditions for emerging Turing instability in detail. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits a delay and diffusion controlled formation growth not only of spots and stripe-like patterns, but also of the two coexist. The obtained results show that this system has rich dynamics; these patterns show that it is useful for the diffusive predation model with a delay effect to reveal the spatial dynamics in the real model.

scholarly journals Pattern Formation in a Cross-Diffusive Holling-Tanner Model

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Weiming Wang ◽  
Zhengguang Guo ◽  
R. K. Upadhyay ◽  
Yezhi Lin
Keyword(s):  
Pattern Formation ◽  
Sufficient Conditions ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Diffusion Controlled ◽  
Model Dynamics ◽  
Flux Boundary Conditions ◽  
Spatially Distributed

We present a theoretical analysis of the processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self- as well as cross-diffusion in a Holling-Tanner predator-prey model; the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained; Hopf and Turing bifurcation in a spatial domain is presented, too. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self- as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to spots, but also to strips, holes, and stripes-spots replication. And the methods and results in the present paper may be useful for the research of the pattern formation in the cross-diffusive model.

Turing Instability and Hopf Bifurcation in a Modified Leslie–Gower Predator–Prey Model with Cross-Diffusion

2018 ◽  
Vol 28 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Walid Abid ◽  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
Ahmed Aghriche
Keyword(s):  
Spatial Dynamics ◽  
Real Life ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

This paper is concerned with some mathematical analysis and numerical aspects of a reaction–diffusion system with cross-diffusion. This system models a modified version of Leslie–Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator–prey model to detect the spatial dynamics in the real life.

Hopf Bifurcation of a Delayed Predator–Prey Model with Nonconstant Death Rate and Constant-Rate Prey Harvesting

2018 ◽  
Vol 28 (14) ◽  
pp. 1850179 ◽  
Author(s):  
Fengrong Zhang ◽  
Xinhong Zhang ◽  
Yan Li ◽  
Changpin Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Differential Equations ◽  
Positive Constant ◽  
Death Rate ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Constant Rate ◽  
The Stability

This paper is concerned with a delayed predator–prey model with nonconstant death rate and constant-rate prey harvesting. We mainly study the impact of the time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively. By choosing time delay [Formula: see text] as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay passes some critical values. In addition, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out to depict our theoretical results.

Local Bifurcations and Optimal Theory in a Delayed Predator–Prey Model with Threshold Prey Harvesting

2015 ◽  
Vol 25 (07) ◽  
pp. 1540015 ◽  
Author(s):  
Israel Tankam ◽  
Plaire Tchinda Mouofo ◽  
Abdoulaye Mendy ◽  
Mountaga Lam ◽  
Jean Jules Tewa ◽  
...  
Keyword(s):  
Time Delay ◽  
Piecewise Linear ◽  
Optimal Harvesting ◽  
Threshold Policy ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Bifurcations ◽  
Prey Model ◽  
Linear Threshold ◽  
The Stability

We investigate the effects of time delay and piecewise-linear threshold policy harvesting for a delayed predator–prey model. It is the first time that Holling response function of type III and the present threshold policy harvesting are associated with time delay. The trajectories of our delayed system are bounded; the stability of each equilibrium is analyzed with and without delay; there are local bifurcations as saddle-node bifurcation and Hopf bifurcation; optimal harvesting is also investigated. Numerical simulations are provided in order to illustrate each result.

Pattern Dynamics in a Predator–Prey Model with Schooling Behavior and Cross-Diffusion

2019 ◽  
Vol 29 (11) ◽  
pp. 1950146
Author(s):  
Wen Wang ◽  
Shutang Liu ◽  
Zhibin Liu ◽  
Da Wang
Keyword(s):  
Spatial Dynamics ◽  
Multiple Scale ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Schooling Behavior ◽  
Pattern Dynamics ◽  
Prey Model ◽  
Pattern Formations

In this paper, a diffusive predator–prey model is considered in which the predator and prey populations both exhibit schooling behavior. The system’s spatial dynamics are captured via a suitable threshold parameter, and a sequence of spatiotemporal patterns such as hexagons, stripes and a mixture of the two are observed. Specifically, the linear stability analysis is applied to obtain the conditions for Hopf bifurcation and Turing instability. Then, employing the multiple-scale analysis, the amplitude equations near the critical point of Turing bifurcation are derived, through which the selection and stability of pattern formations are investigated. The theoretical results are verified by numerical simulations.

Hopf Bifurcation in a Delayed Diffusive Leslie–Gower Predator–Prey Model with Herd Behavior

2019 ◽  
Vol 29 (04) ◽  
pp. 1950055
Author(s):  
Fengrong Zhang ◽  
Yan Li ◽  
Changpin Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we consider a delayed diffusive predator–prey model with Leslie–Gower term and herd behavior subject to Neumann boundary conditions. We are mainly concerned with the impact of time delay on the stability of this model. First, for delayed differential equations and delayed-diffusive differential equations, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated respectively. It is observed that when time delay continues to increase and crosses through some critical values, a family of homogeneous and inhomogeneous periodic solutions emerge. Then, the explicit formula for determining the stability and direction of bifurcating periodic solutions are also derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are shown to support the analytical results.

scholarly journals Drivers of pattern formation in a predator–prey model with defense in fearful prey

2021 ◽  
Author(s):  
Purnedu Mishra ◽  
Barkha Tiwari
Keyword(s):  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Direct Method ◽  
Turing Instability ◽  
Steady States ◽  
Predator Prey ◽  
Random Movement ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fear And Anxiety

AbstractExistence of predator is routinely used to induce fear and anxiety in prey which is well known for shaping entire ecosystem. Fear of predation restricts the development of prey and promotes inducible defense in prey communities for the survival. Motivated by this fact, we investigate the dynamics of a Leslie–Gower predator prey model with group defense in a fearful prey. We obtain conditions under which system possess unique global-in-time solutions and determine all the biological feasible states of the system. Local stability is analyzed by linearization technique and Lyapunov direct method has been applied for global stability analysis of steady states. We show the occurrence of Hopf bifurcation and its direction at the vicinity of coexisting equilibrium point for temporal model. We consider random movement in species and establish conditions for the stability of the system in the presence of diffusion. We derive conditions for existence of non-constant steady states and Turing instability at coexisting population state of diffusive system. Incorporating indirect prey taxis with the assumption that the predator moves toward the smell of prey rather than random movement gives rise to taxis-driven inhomogeneous Hopf bifurcation in predator–prey model. Numerical simulations are intended to demonstrate the role of biological as well as physical drivers on pattern formation that go beyond analytical conclusions.

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