An Exact Bifurcation Diagram for a Reaction Diffusion Equation Arising in Population Dynamics

We analyze the positive solutions to  −∆v = λv(1− v); x ∈ Ω0, ∂v ∂η + γ √ λv = 0; x ∈ ∂Ω0, where Ω0 = (0, 1) or is a bounded domain in R; n = 2, 3 with smooth boundary and |Ω0| = 1, and λ, γ are positive parameters. Such steady state equations arise in population dynamics encapsulating assumptions regarding the patch/matrix interfaces such as patch preference and movement behavior. In this paper, we will discuss the exact bifurcation diagram and stability properties for such a steady state model.