Global positive coexistence of a nonlinear elliptic biological interacting model (Q912801)

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.