scholarly journals Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Willian Cintra ◽  
Carlos Alberto dos Santos ◽  
Jiazheng Zhou
Keyword(s):  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Type Ii ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Coexistence States ◽  
Global Bifurcation Theory ◽  
Type Ii Functional Response

<p style='text-indent:20px;'>In this paper, we present results about existence and non-existence of coexistence states for a reaction-diffusion predator-prey model with the two species living in a bounded region with inhospitable boundary and Holling type II functional response. The predator is a specialist and presents self-diffusion and cross-diffusion behavior. We show the existence of coexistence states by combining global bifurcation theory with the method of sub- and supersolutions.</p>

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.

Coexistence states in a cross-diffusion system of a predator–prey model with predator satiation term

2018 ◽  
Vol 28 (11) ◽  
pp. 2131-2159 ◽  
Author(s):  
Willian Cintra ◽  
Cristian Morales-Rodrigo ◽  
Antonio Suárez
Keyword(s):  
A Priori ◽  
Diffusion System ◽  
Predator Satiation ◽  
Linear Diffusion ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Coexistence States ◽  
Predator Model

In this paper, we study the existence and non-existence of coexistence states for a cross-diffusion system arising from a prey–predator model with a predator satiation term. We use mainly bifurcation methods and a priori bounds to obtain our results. This leads us to study the coexistence region and compare our results with the classical linear diffusion predator–prey model. Our results suggest that when there is no abundance of prey, the predator needs to be a good hunter to survive.

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.

Solitary waves in excitable systems with cross-diffusion

2005 ◽  
Vol 461 (2064) ◽  
pp. 3711-3730 ◽  
Author(s):  
Vadim N Biktashev ◽  
Mikhail A Tsyganov
Keyword(s):  
Asymptotic Behaviour ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
System Of Equations ◽  
Soliton Interaction ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Excitable Systems

We consider a FitzHugh–Nagumo system of equations where the traditional diffusion terms are replaced with linear cross-diffusion of components. This system describes solitary waves that have unusual form and are capable of quasi-soliton interaction. This is different from the classical FitzHugh–Nagumo system with self-diffusion, but similar to a predator–prey model with taxis of populations on each other's gradient which we considered earlier. We study these waves by numerical simulations and also present an analytical theory, based on the asymptotic behaviour which arises when the local dynamics of the inhibitor field are much slower than those of the activator field.

Bifurcation of steady-state solutions in predator-prey and competition systems

1984 ◽  
Vol 97 ◽  
pp. 21-34 ◽  
Author(s):  
J. Blat ◽  
K. J. Brown
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Existence Of Solutions ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Predator Prey ◽  
Interacting Species ◽  
Global Bifurcation Theory ◽  
Steady State Solutions

SynopsisWe discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.

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