Conditions for Existence or Nonexistence of Positive Solutions to Elliptic General Model

Two species of animals are competing or cooperating in the same environment. Under whatconditions do they coexist peacefully? Or under what conditions does either one of the two speciesbecome extinct, that is, is either one of the two species excluded by the other? We investigate thisphenomenon from a mathematical point of view.In this paper we concentrate on coexistence solutions of the competition or cooperation model∆u+g(u,v) = 0∆v+h(u,v) = 0inΩ,u|∂Ω=v|∂Ω= 0.This system is the general model for the steady state of a competitive or cooperative interactingsystem depending on growth conditions forgandh. The techniques used in this paper are super-sub solutions and some detailed properties of the solution of logistic equations (See [1]