An exact bifurcation diagram for a reaction–diffusion equation arising in population dynamics

Abstract

Abstract We analyze the positive solutions to {−Δv=λv(1−v);Ω0,∂v∂η+γλv=0;∂Ω0, $$ \textstyle\begin{cases} - \Delta v = \lambda v(1-v); & \Omega_{0}, \\ \frac{\partial v}{\partial\eta} + \gamma\sqrt{\lambda} v =0 ; & \partial\Omega_{0}, \end{cases} $$ where Ω0=(0,1) $\Omega_{0}=(0,1)$ or is a bounded domain in Rn $\mathbb{R}^{n}$, n=2,3 $n =2,3$, with smooth boundary and |Ω0|=1 $|\Omega_{0}|=1$, 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.

Keywords

• Mathematical biology
• Reaction–diffusion model
• Bifurcation
• Stability