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 u0σ 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 u(t,x) approaches a propagating pair consisting of a traveling wave with speed c and a semi-wave with speed c0.

中文翻译：

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通过自由边界模型在变化的气候下传播：恶化环境的入侵

在本文中，我们考虑了一个空间维度上的自由边界模型，该模型描述了受气候变化影响的物种的扩散，其中有利的环境以恒定的速度远离C > 0取而代之的是一个恶化但仍然有利的环境。我们获得两个阈值速度C1 < C0以及模型长期动态的完整分类，揭示案例之间的显着差异0 < C < C1,C = C1,C1 < C < C0和C ≥ C0. 例如，当C1 < C < C0, 对于适当参数化的初始函数族你0σ在不断增加σ, 我们证明存在0 < σ* < σ* < ∞这样物种最终会消失，当σ ∈ (0,σ*], 它以渐近速度传播C1什么时候σ ∈ (σ*,σ*), 它以强制速度传播C什么时候σ = σ*，并且传播速度很快C0什么时候σ > σ*. 此外，在最后一种情况下，扩展前沿以渐近速度传播C0, 人口密度函数的轮廓你(吨,X)接近由具有速度的行波组成的传播对C和一个有速度的半波C0.