3 years ago

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

Jerome Goddard II, Quinn A. Morris, Stephen B. Robinson, Ratnasingham Shivaji

Abstract

We analyze the positive solutions to $ \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 \(\Omega_{0}=(0,1)\) or is a bounded domain in \(\mathbb{R}^{n}\) , \(n =2,3\) , with smooth boundary and \(|\Omega_{0}|=1\) , 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.

Publisher URL: https://link.springer.com/article/10.1186/s13661-018-1090-z

DOI: 10.1186/s13661-018-1090-z

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