Global positive coexistence of a nonlinear elliptic biological interacting model
The author studies an elliptic predator-prey system of the Dirichlet problem of the type \[ (1) \Delta u+u M(u,v)=0,\;d\Delta v+v(g(u)- m(v))=0,\;(u,v)|_{\partial \Omega}=(0,0). \] Under some assumptions on \(M\), \(g\) and \(m\), and providing that the domain \(\Omega\) is large, he gives a necessary and sufficient condition for the coexistence of positive solutions to (1). This necessary and sufficient condition is equivalent to the condition that the corresponding o.d.e. system \[ du/dt=u M(u,v),\;dv/dt=v(g(u)-m(v)) \] has positive equilibrium \({\tilde u}>0\), \({\tilde v}>0\). So, when \(\Omega\) is large, this condition does not depend on the shape of \(\Omega\) and is algebraically computable. Also, some stability properties of the positive solutions to (1) are studied.
necessary and sufficient condition for the coexistence of positive solutions
scientific article; zbMATH DE number 4145731
scientific article