Ecological systems, nonlinear boundary conditions, and $\Sigma$-shaped bifurcation curves

Abstract

We examine a one-dimensional reaction diffusion model with a weak Allee growth rate that appears in population dynamics. We combine grazing with a certain nonlinear boundary condition that models negative density dependent dispersal on the boundary and analyze the effects on the steady states. In particular, we study the bifurcation curve of positive steady states as the grazing parameter is varied. Our results are acquired through the adaptation of a quadrature method and Mathematica computations. Specifically, we computationally ascertain the existence of $\Sigma$-shaped bifurcation curves with several positive steady states for a certain range of the grazing parameter.