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

• Main Authors: Jerome Goddard II, Quinn A. Morris, Stephen B. Robinson, Ratnasingham Shivaji
• Format: Article
• Language: English
• Published: SpringerOpen 2018-11-01
• Series: Boundary Value Problems
• Subjects: Mathematical biology; Reaction–diffusion model; Bifurcation; Stability
• Online Access: http://link.springer.com/article/10.1186/s13661-018-1090-z

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.