On multiplicity and stability of positive solutions of a diffusive prey-predator model
The paper deals with the existence, stability and number of positive solutions of the following prey-predator system with diffusion: \[ \left\{{\begin{aligned} &-\Delta u=u\left( a-u-\dfrac{v}{1+mu}\right)\quad \text{ in }\Omega,\\ &-\Delta v=v\left( b-\dfrac{mv}{m+u}\right)\quad \text{ in }\Omega,\\ &u=0\quad \text{ on }\partial\Omega,\\ &v=0\quad \text{ on }\partial\Omega.\\ \end{aligned}}\right.\tag{1} \] Here \(a,b,m>0\) are constants and \(m\) is large. Denote by \(\lambda_{1},\lambda_{2}\) the first and the second eigenvalue of \(-\Delta\) in \(\Omega\), subject to the homogeneous Dirichlet boundary condition. The main result of the paper is: Theorem A. Let \(b>\lambda_{1}\) and \(\Omega\) be fixed. Then there exists a large \(M\), depending only on \(b\) and \(\Omega\), such that for each \(m\geq M\), there is a unique constant \(\widetilde{a}\in(\lambda_{1},b)\) with \(\widetilde{a}\rightarrow\lambda_{1}\) as \(m\rightarrow \infty\) and (i) (1) has a positive solution if and only if \(a\geq\widetilde{a}\); (ii) (1) has a unique positive solution if \(a=\widetilde{a}\) or \(a\in(b,\infty)\). Moreover, the unique positive solution is asymptotically stable if \(a\in(b,\infty)\); (iii) (1) has at least two positive solutions if \(a\in(\widetilde{a},b)\). Furthermore, when \(b<\lambda_{2}\), (1) has exactly two positive solutions, one of which is asymptotically stable and the other is unstable.
scientific article; zbMATH DE number 5016179
scientific article