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.

DOI: 10.1186/s13661-018-1090-z

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