Positive steady-states for two interacting species models with linear self-cross diffusions
type: Item
instance of: scholarly article
title: Positive steady-states for two interacting species models with linear self-cross diffusions
zbMATH Open document ID: 1065.35119
author: Kimun Ryu
author: Inkyung Ahn
published in: Discrete and Continuous Dynamical Systems
publication date: 2003-10-12T00:00:00Z
review text: The authors consider the following coupled system of elliptic partial differential equations: \[ -\Delta[(d_1+ \beta_{11} u+\beta_{12}\upsilon)u]=(a_1-b_{11}u-b_{12}\upsilon)u \quad\text{in }\Omega, \] \[ -\Delta[(d_2+\beta_{21}u+ \beta_{22}\upsilon)\upsilon]=(a_2\pm b_{21}u-b_{22}\upsilon)\upsilon \quad\text{in }\Omega, \] \[ \kappa_1\frac{\partial u}{\partial n}+ \tau_1u=0\quad\text{on }\partial \Omega, \] \[ \kappa_2\frac{\partial\upsilon}{\partial n} + \tau_2\upsilon=0\quad\text{on }\partial \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial \Omega\) and \(d_i, \beta_{ij}, a_i, b_{ij}, \kappa_i, \tau_i\) for \(i,j=1,2\) are nonnegative constants. They use the methods of fixed point index on cones in Banach spaces and that of decomposing operators to obtain sufficient conditions for the existence of positive solutions. Furthermore, a priori bounds for the positive solutions are obtained.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 47N60
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 47H11
zbMATH DE Number: 1941564
zbMATH Keywords: existence
zbMATH Keywords: nonlinear system of elliptic partial differential equations
zbMATH Keywords: predator-prey model
zbMATH Keywords: competition model
zbMATH Keywords: positive coexistence
zbMATH Keywords: semi-trivial solution
zbMATH Keywords: theory of fixed point index
zbMATH Keywords: decomposing operator
reviewed by: Mohamed O. El-Doma
MaRDI profile type: Publication
full work available at URL: dcds.2003.9.1049
OpenAlex ID: W2020482432
DOI: 10.3934/DCDS.2003.9.1049
description: scientific article; zbMATH DE number 1941564