A symbiotic self-cross diffusion model (Q397189)

The paper is concerned with the coexistence states for the Lotka-Volterra symbiotic model with self-diffusion and cross-diffusion in one species NEWLINE\[NEWLINE\begin{cases}-\Delta u=u(\lambda-u+bv)\,\,\,\text{in}\,\,\,\Omega,\\ -\Delta[(1+\alpha v+\beta u)v]=v(\mu-v+cu)\,\,\,\text{in}\,\,\,\Omega,\\ u=v=0\,\,\,\text{on}\,\,\,\partial\Omega,\end{cases}\leqno(1)NEWLINE\]NEWLINE where \(\Omega\subset \mathbb{R}^N\), \(N\geq 1\), is a bounded and smooth domain, \(\lambda,\,\mu\in\mathbb{R}\) and \(b,\,c,\,\alpha,\,\beta>0\). The authors prove the existence and nonexistence of solutions for problem \((1)\) and then they investigate the behavior of the set of positive solutions when the cross-diffusion parameter \(\beta\) or the self-diffusion parameter \(\alpha\) is large.