An ecological model with a ΣΣ-shaped bifurcation curve

We consider the existence of multiple positive solutions to the steady state reaction diffusion equation with Dirichlet boundary conditions of the form: {−u″=λ[u−u2K−cu21+u2−ϵ],x∈(0,1)u(0)=0=u(1). Here 1λ is the diffusion coefficient and KK, cc and ϵϵ are positive constants. This model describes the steady states of a logistic growth model with grazing and constant yield harvesting in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. In this paper, we discuss the occurrence of a ΣΣ-shaped bifurcation diagram for positive solutions. In particular, for certain parameter values of c,K,ϵc,K,ϵ and the diffusion coefficient, we prove the existence of at least four positive solutions. We prove our results by the quadrature method.