Global bifurcation of solutions to diffusive logistic equations on bounded domains subject to nonlinear boundary conditions

Abstract

We consider the diffusive logistic equation

supplemented by the nonlinear boundary condition

where α is a non-negative, non-decreasing function with α([0, 1]) ⊆ [0, 1]. When regarded as an ecological model for an organism inhabiting a focal patch of its habitat, the assumptions on α are intended to capture a tendency on the part of the organism to remain in the habitat patch when it encounters the patch boundary that increases with species density. Such a mechanism has been suggested in the ecological literature as a means by which the dynamics of the organism at the scale of the patch might differ from its local dynamics within the patch. Building upon earlier examinations of the boundary-value problem by Cantrell and Cosner, we detail in this paper the global disposition of biologically relevant equilibria when both 0 and 1 (the local carrying capacity within the patch) are equilibria. Our analysis relies on global bifurcation theory and estimates for elliptic and parabolic partial differential equations.