Spreading under shifting climate by a free boundary model: Invasion of deteriorated environment

<p>In this paper, we consider a free boundary model in one space dimension which describes the spreading of a species subject to climate change, where favorable environment is shifting away with a constant speed c > 0 and replaced by a deteriorated yet still favorable environment. We obtain two threshold speeds c₁ < c₀ and a complete classification of the long-time dynamics of the model, which reveals significant differences between the cases 0 < c < c₁, c = c₁, c₁ < c < c₀ and c ≥ c₀. For example, when c₁ < c < c₀, for a suitably parameterized family of initial functions <i>u</i>σ0 increasing continuously in σ, we show that there exists 0 < σ_⁎ < σ^⁎ < ∞ such that the species vanishes eventually when σ ∈ (0, σ_⁎], it spreads with asymptotic speed c₁ when σ ∈ (σ_⁎, σ^⁎), it spreads with forced speed c when σ = σ^⁎, and it spreads with speed c₀ when σ > σ^⁎. Moreover, in the last case, while the spreading front propagates with asymptotic speed c₀, the profile of the population density function <i>u</i>(<i>t, x</i>) approaches a propagating pair consisting of a traveling wave with speed c and a semi-wave with speed c₀.</p>