A class of invasion models in ecology with a free boundary and with cross-diffusion and self-diffusion

2021 ◽  
pp. 125318
Author(s):  
Qi-Jian Tan ◽  
Chao-Yi Pan
Keyword(s):  
Free Boundary ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Invasion Models

Evolution of cross-diffusion and self-diffusion

2009 ◽  
Vol 3 (4) ◽  
pp. 410-429 ◽  
Author(s):  
Yuan Lou ◽  
Salome Martínez
Keyword(s):  
Self Diffusion ◽  
Cross Diffusion

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.

scholarly journals Turing Patterns in a Predator-Prey System with Self-Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Hongwei Yin ◽  
Xiaoyong Xiao ◽  
Xiaoqing Wen
Keyword(s):  
Spatial Patterns ◽  
Nonlinear Diffusion ◽  
Turing Instability ◽  
Turing Patterns ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Prey System ◽  
Diffusion Parameters ◽  
Normal Diffusion

For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns.

scholarly journals Convergence of a finite volume scheme for a system of interacting species with cross-diffusion

2020 ◽  
Vol 145 (3) ◽  
pp. 473-511 ◽  
Author(s):  
José A. Carrillo ◽  
Francis Filbet ◽  
Markus Schmidtchen
Keyword(s):  
Finite Volume ◽  
Coupled System ◽  
Convergence Result ◽  
The Self ◽  
Finite Volume Scheme ◽  
Discrete Energy ◽  
Volume Scheme ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Interacting Species

Abstract In this work we present the convergence of a positivity preserving semi-discrete finite volume scheme for a coupled system of two non-local partial differential equations with cross-diffusion. The key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness. We numerically observe the convergence to reference solutions with a first order accuracy in space. Moreover we recover segregated stationary states in spite of the regularising effect of the self-diffusion. However, if the self-diffusion or the cross-diffusion is strong enough, mixing occurs while both densities remain continuous.

Turing instability in a diffusive SIS epidemiological model

2015 ◽  
Vol 08 (01) ◽  
pp. 1550006 ◽  
Author(s):  
Shaban Aly ◽  
Houari B. Khenous ◽  
Fatma Hussien
Keyword(s):  
Infectious Diseases ◽  
Modeling And Simulation ◽  
Spatial Patterns ◽  
Disease Model ◽  
Turing Instability ◽  
Nonlinear Incidence Rate ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
The Stability ◽  
Theoretical Results

Modeling and simulation of infectious diseases help to predict the likely outcome of an epidemic. In this paper, a spatial susceptible-infective-susceptible (SIS) type of epidemiological disease model with self- and cross-diffusion are investigated. We study the effect of diffusion on the stability of the endemic equilibrium with disease-induced mortality and nonlinear incidence rate. In the absence of diffusion the stationary solution stays stable but becomes unstable with respect to diffusion and that Turing instability takes place. We show that a standard (self-diffusion) system may be either stable or unstable, cross-diffusion response can stabilize an unstable standard system or decrease a Turing space (the space which the emergence of spatial patterns is holding) compared to the Turing space with self-diffusion, i.e. the cross-diffusion response is an important factor that should not be ignored when pattern emerges. Numerical simulations are provided to illustrate and extend the theoretical results.

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