Stability and uniqueness for cooperative degenerate Lotka-Volterra model
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.
cooperative degenerate Lotka-Volterra model
scientific article; zbMATH DE number 1761019
scientific article