A Climate Shift Model with Free Boundary: Enhanced Invasion

We examine how climate change enhances the spreading of an invading species through a nonlinear diffusion equation of the form <i>u_t</i> = <i>du_xx</i>+A (<i>x</i> − <i>ct</i>) <i>u</i>−<i>bu</i>² with a free boundary, where climate change causes more favourable environment shifting into the habitat of the species with a constant speed <i>c</i> > 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 <i>u</i>₀^σ increasing continuously in <i>σ</i>: there exists a critical value <i>σ</i>_∗ ∈ (0,∞) so that the species vanishes ultimately when <i>σ</i> ∈ (0, <i>σ</i>_∗], and it spreads successfully when <i>σ</i> > <i>σ</i>^∗ . However, when spreading is successful, there exist two threshold speeds <i>c</i>₀ < <i>c</i>₁ that divide the spreading profile into strikingly different patterns. For example, when <i>c</i> < <i>c</i>₀ the profile of the population density function <i>u</i>(t, x) approaches a propagating pair composed of a traveling wave with speed <i>c</i> and a semi-wave with speed <i>c</i>₀; when <i>c</i>₀ < <i>c</i> < <i>c</i>₁, it approaches a semi-wave with speed <i>c</i>, and when <i>c</i> > <i>c</i>₁, it approaches a semi-wave with speed <i>c</i>₁.