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.

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.

scholarly journals Global stability of solutions in a reaction-diffusion system of predator-prey model

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4665-4672
Author(s):  
Demou Luo ◽  
Hailin Liu
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

In this article, we investigate the global asymptotic stability of a reaction-diffusion system of predator-prey model. By applying the comparison principle and iteration method, we prove the global asymptotic stability of the unique positive equilibrium solution of (1.1).

scholarly journals Effect of Diffusion and Cross-Diffusion in a Predator-Prey Model with a Transmissible Disease in the Predator Species

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Guohong Zhang ◽  
Xiaoli Wang
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Predator Population ◽  
Predator Species ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Transmissible Disease

We study a Lotka-Volterra type predator-prey model with a transmissible disease in the predator population. We concentrate on the effect of diffusion and cross-diffusion on the emergence of stationary patterns. We first show that both self-diffusion and cross-diffusion can not cause Turing instability from the disease-free equilibria. Then we find that the endemic equilibrium remains linearly stable for the reaction diffusion system without cross-diffusion, while it becomes linearly unstable when cross-diffusion also plays a role in the reaction-diffusion system; hence, the instability is driven solely from the effect of cross-diffusion. Furthermore, we derive some results for the existence and nonexistence of nonconstant stationary solutions when the diffusion rate of a certain species is small or large.

scholarly journals Pattern Formation in a Cross-Diffusive Ratio-Dependent Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Xinze Lian ◽  
Yanhong Yue ◽  
Hailing Wang
Keyword(s):  
Evolutionary Process ◽  
Spiral Wave ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Reaction Diffusion Model ◽  
Ratio Dependent

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatial distribution of the species with self- and cross-diffusion in a Holling-III ratio-dependent predator-prey model. The diffusion instability of the positive equilibrium of the model with Neumann boundary conditions is discussed. Furthermore, we present novel 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 spots, stripes, and spiral wave pattern replication, which show that reaction-diffusion model is useful to reveal the spatial predation dynamics in the real world.

CROSS-DIFFUSION INDUCED TURING PATTERNS IN A SEX-STRUCTURED PREDATOR–PREY MODEL

2012 ◽  
Vol 05 (04) ◽  
pp. 1250016 ◽  
Author(s):  
JIA LIU ◽  
HUA ZHOU ◽  
LAI ZHANG
Keyword(s):  
Steady State ◽  
Degree Theory ◽  
Reaction Diffusion ◽  
Turing Patterns ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Existence And Stability

In this paper, we consider a sex-structured predator–prey model with strongly coupled nonlinear reaction diffusion. Using the Lyapunov functional and Leray–Schauder degree theory, the existence and stability of both homogenous and heterogenous steady-states are investigated. Our results demonstrate that the unique homogenous steady-state is locally asymptotically stable for the associated ODE system and PDE system with self-diffusion. With the presence of the cross-diffusion, the homogeneous equilibrium is destabilized, and a heterogenous steady-state emerges as a consequence. In addition, the conditions guaranteeing the emergence of Turing patterns are derived.

scholarly journals Cross-Diffusion-Driven Instability in a Reaction-Diffusion Harrison Predator-Prey Model

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaoqin Wang ◽  
Yongli Cai
Keyword(s):  
Lyapunov Function ◽  
Analytical Study ◽  
Reaction Diffusion ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Global Behaviour ◽  
Prey Model ◽  
Model Dynamics ◽  
Spatially Distributed

We present a theoretical analysis of processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with cross-diffusion in a Harrison-type predator-prey model. We analyze the global behaviour of the model by establishing a Lyapunov function. We carry out the analytical study in detail and find out the certain conditions for Turing’s instability induced by cross-diffusion. And the numerical results reveal that, on increasing the value of the half capturing saturation constant, the sequences “spots → spot-stripe mixtures → stripes → hole-stripe mixtures → holes” are observed. The results show that the model dynamics exhibits complex pattern replication controlled by the cross-diffusion.

scholarly journals Global Behavior of Solutions in a Predator-Prey Cross-Diffusion Model with Cannibalism

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Meijun Chen ◽  
Shengmao Fu ◽  
Xiaoli Yang
Keyword(s):  
Diffusion Model ◽  
Uniform Boundedness ◽  
Reaction Diffusion ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Global Existence Of Solutions ◽  
Reaction Diffusion Model

The global asymptotic behavior of solutions in a cross-diffusive predator-prey model with cannibalism is studied in this paper. Firstly, the local stability of nonnegative equilibria for the weakly coupled reaction-diffusion model and strongly coupled cross-diffusion model is discussed. It is shown that the equilibria have the same stability properties for the corresponding ODE model and semilinear reaction-diffusion model, but under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability occurs. Secondly, the uniform boundedness and the global existence of solutions for the model with SKT-type cross-diffusion are investigated when the space dimension is one. Finally, the global stability of the positive equilibrium is established by constructing a Lyapunov function. The result indicates that, under certain conditions on reaction coefficients, the model has no nonconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough.

scholarly journals Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Weiming Wang ◽  
Yongli Cai ◽  
Yanuo Zhu ◽  
Zhengguang Guo
Keyword(s):  
Pattern Formation ◽  
Allee Effect ◽  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
Model Dynamics ◽  
And Diffusion

We investigate the spatiotemporal dynamics induced by Allee effect in a reaction-diffusion predator-prey model. In the case without Allee effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee effect is not simple, but rich and complex.

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