|
We examine how climate change enhances the spreading of an invading species through a nonlinear diffusion equation of the form ut = duxx+A (x − ct) u−bu2 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 u0σ increasing continuously in σ: there exists a critical value σ∗ ∈ (0,∞) so that the species vanishes ultimately when σ ∈ (0, σ∗], and it spreads successfully when σ > σ∗ . However, when spreading is successful, there exist two threshold speeds c0 < c1 that divide the spreading profile into strikingly different patterns. For example, when c < c0 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 c0; when c0 < c < c1, it approaches a semi-wave with speed c, and when c > c1, it approaches a semi-wave with speed c1. |
|