Stability and uniqueness for cooperative degenerate Lotka-Volterra model
type: Item
instance of: scholarly article
title: Stability and uniqueness for cooperative degenerate Lotka-Volterra model
zbMATH Open document ID: 1043.35078
DOI: 10.1016/S0362-546X(01)00138-9
author: Manuel Delgado
author: Antonio Suárez
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-06-27T00:00:00Z
review text: The cooperative degenerate Lotka-Volterra model considered here is as follows: \[ \begin{cases} u_t-\Delta u=u^{1/m}(\lambda-u^{1/m}+ bv^{1/m})\\ v_t-\Delta v=v^{1/m}(\mu-v^{1/m}+ cu^{1/m}) \end{cases} \quad\text{in }\mathbb{R}^+ \times\Omega, \] with zero Dirichlet boundary conditions and nonnegative initial data; \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\lambda,\mu\in\mathbb{R}\), \(b,c>0\) and \(m>1\). Since the system is cooperative \((b,c>0)\), it has some order preserving properties which allows one to use the method of sub- and supersolutions. Indeed, cooperative systems share many properties of single equations. Accordingly, a large part of the paper is concerned with setting up the spectral framework for a cooperative weakly coupled elliptic system \({\mathcal L}U=M(x)U+ \lambda U\) in \(\Omega\) and \(U=0\) on \(\partial\Omega\), with \(U:\Omega\to\mathbb{R}^n\), similar as for the linearization of a single equation. Special for the present case is the singularity of the coefficients at \(\partial\Omega\). The main result states conditions on the parameters such that either an (asymptotically stable) positive steady state exists or blow-up in finite time occurs.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 1761019
zbMATH Keywords: cooperative degenerate Lotka-Volterra model
zbMATH Keywords: order preserving properties
zbMATH Keywords: sub- and supersolutions
zbMATH Keywords: positive steady state
zbMATH Keywords: blow-up in finite time
reviewed by: Guido Sweers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1761019