An ecological model with a capital sigma -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: [inline image] Here [inline image] is the diffusion coefficient and K, c and epsilon 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 capital sigma -shaped bifurcation diagram for positive solutions. In particular, for certain parameter values of c,K, epsilon and the diffusion coefficient, we prove the existence of at least four positive solutions. We prove our results by the quadrature method.