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Global positive coexistence of a nonlinear elliptic biological interacting model. (English) Zbl 0698.92022
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.
Reviewer: A.Canada

92D25 Population dynamics (general)
35J65 Nonlinear boundary value problems for linear elliptic equations
35B99 Qualitative properties of solutions to partial differential equations
Full Text: DOI
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