Pattern Dynamics in a Spatial Predator–Prey Model with Nonmonotonic Response Function

2018 ◽  
Vol 28 (06) ◽  
pp. 1850077 ◽  
Author(s):  
Xiaoling Li ◽  
Guangping Hu ◽  
Zhaosheng Feng
Keyword(s):  
Response Function ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Weakly Nonlinear ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pattern Dynamics ◽  
Spatial Systems ◽  
Different Types ◽  
Selection Of

In this paper, we study a diffusive predator–prey system with the nonmonotonic response function. The conditions on Hopf bifurcation and Turing instability of spatial systems are obtained. Near the Turing bifurcation point we apply the weakly nonlinear analysis to derive the amplitude equations of stationary pattern, to investigate the selection of spatiotemporal pattern. It shows that different types of patterns will occur in the model under various conditions. Numerical simulations agree well with our theoretical analysis when control parameters are in the Turing space. This study may provide some deep insights into the formation and selection of patterns for diffusive predator–prey systems.

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.

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.

scholarly journals Spatiotemporal Pattern in a Self- and Cross-Diffusive Predation Model with the Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Feng Rao
Keyword(s):  
Allee Effect ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Reaction Diffusion Model ◽  
Flux Boundary Conditions ◽  
Prey Interaction

This paper proposes and analyzes a mathematical model for a predator-prey interaction with the Allee effect on prey species and with self- and cross-diffusion. The effect of diffusion which can drive the model with zero-flux boundary conditions to Turing instability is investigated. We present numerical evidence of time evolution of patterns controlled by self- and cross-diffusion in the model and find that the model dynamics exhibits a cross-diffusion controlled formation growth to spotted and striped-like coexisting and spotted pattern replication. Moreover, we discuss the effect of cross-diffusivity on the stability of the nontrivial equilibrium of the model, which depends upon the magnitudes of the self- and cross-diffusion coefficients. The obtained results show that cross-diffusion plays an important role in the pattern formation of the predator-prey model. It is also useful to apply the reaction-diffusion model to reveal the spatial predation in the real world.

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.

Stability and Bifurcation Analysis of a Delayed Discrete Predator–Prey Model

2018 ◽  
Vol 28 (09) ◽  
pp. 1850116 ◽  
Author(s):  
A. M. Yousef ◽  
S. M. Salman ◽  
A. A. Elsadany
Keyword(s):  
Stability Analysis ◽  
Local Stability ◽  
Chaotic Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Stability Analysis ◽  
Prey Model ◽  
Stability And Bifurcation Analysis ◽  
Different Types ◽  
Two Parameters

A discrete predator–prey model with delayed density dependence in the rate of growth of the prey is considered. In particular, we analyze the model presented by Kot [2005] which consists of three coupled difference equations and contains two parameters. Existence and local stability analysis of fixed points of the model are addressed. The normal form technique and perturbation method are applied to the different types of bifurcations that exist in the model being investigated. It is proved that the existence of transcritical and Neimark–Sacker bifurcations can occur in the model. In addition, the chaotic behavior of the model in the sense of Marotto is proved. To verify the results obtained analytically, we perform numerical simulations which also explore further the richer dynamics of the model.

SPATIAL ASPECT OF HUNTING COOPERATION IN PREDATORS WITH HOLLING TYPE II FUNCTIONAL RESPONSE

2018 ◽  
Vol 26 (04) ◽  
pp. 511-531 ◽  
Author(s):  
TEEKAM SINGH ◽  
SANDIP BANERJEE
Keyword(s):  
Stripe Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pattern Dynamics ◽  
Prey Model ◽  
Prey Interaction ◽  
Spatial Aspect ◽  
Mixed Pattern ◽  
Spotted Pattern ◽  
Type Ii Functional Response

In this paper, we have investigated a spatial predator–prey model with hunting cooperation in predators. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of controlling parameters. Using extensive numerical simulations, we obtain complex patterns, namely, spotted pattern, stripe pattern and mixed pattern in the Turing domain, by varying the hunting cooperation parameter in predators and carrying capacity of preys. The results focus on the effect of hunting cooperation in pattern dynamics of a diffusive predator–prey model and help us in better understanding of the dynamics of the predator–prey interaction in real environments.

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