A Climate Shift Model with Free Boundary: Enhanced Invasion

Abstract

We examine how climate change enhances the spreading of an invading species through a nonlinear diffusion equation of the form $u_{t}=du_{xx}+A\left( x-ct\right) u-bu^{2}$ with a free boundary, where climate change causes more favourable environment shifting into the habitat of the species with a constant speed $c>0$. The free boundary represents the invading front of the expanding population range. We show that the long-time dynamics of this model obeys a spreading-vanishing dichotomy, which is best illustrated by using a suitably parameterised family of initial functions $u_0^\sigma$ increasing continuously in $\sigma$: there exists a critical value $\sigma _*\in (0,\infty )$ so that the species vanishes ultimately when $\sigma \in (0, \sigma _*]$, and it spreads successfully when $\sigma >\sigma ^*$. However, when spreading is successful, there exist two threshold speeds $c_0<c_1$ that divide the spreading profile into strikingly different patterns. For example, when $c<c_0$ the profile of the population density function u(t, x) approaches a propagating pair composed of a traveling wave with speed c and a semi-wave with speed $c_0$; when $c_0<c<c_1$, it approaches a semi-wave with speed c, and when $c>c_1$, it approaches a semi-wave with speed $c_1$.