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Coexistence theorems of steady states for predator-prey interacting systems. (English) Zbl 0655.35021

The author gives necessary and sufficient conditions for the existence of positive solutions to the following system:

Δu+uM(u,v)=0,dΔv+v(g(u)-m(v))=0,(u,v)| Ω =(0,0),Ωn,

where M satisfies the so-called prey growth rate conditions, g and m are increasing functions satisfying g(0)-m(v)<0 for v larger than some constant C. The paper includes many well-known systems and results as special cases, and some interesting examples are given.

Reviewer: C.F.Wang

MSC:
35J60Nonlinear elliptic equations
92D25Population dynamics (general)
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35A05General existence and uniqueness theorems (PDE) (MSC2000)