非均匀移动栖息地中的速率诱导临界点及临界速率确定
Rate-Induced Tipping in a Non-Uniformly Moving Habitat and Determination of the Critical Rate
AI总结：
研究非均匀移动栖息地中的速率诱导临界点，通过分析反应-扩散方程确定临界速率及灭绝机制。
摘要：
一个由于环境变化而移动的栖息地，如果移动速率过大，可能导致灭绝。我们使用一个带有非自治反应项的标量反应-扩散方程，代表一个从一个渐近位置移动到另一个位置的局部栖息地，作为研究这一现象的背景。移动由位移d和速率参数r来表征。系统在两个渐近栖息地位置都允许三个稳态：一个稳定的灭绝状态u0*=0，一个不稳定的脉冲（所谓的边缘状态）u1*(x)>0，导致阿勒效应，以及一个稳定的脉冲（被占基础状态）u2*(x)>u1*(x)，对应于处于承载力的繁荣种群。针对特定模型的数值模拟确定了临界位移d*，对于d>d*，展示了存在临界速率rc(d)，即速率诱导临界点发生的速率：对于r>rc，初始繁荣的种群由于栖息地移动过快而灭绝。我们为两种极限情况提供了分析结果。对于r<<1，解跟踪移动基础状态，误差为O(r)。对于r>>1，解收敛到灭绝状态，前提是d足够大。对于d太小，无论r如何都不会发生临界点。数值模拟补充并扩展了这些分析结果。在临界速率rc(d)时，我们识别了基础状态在过去渐近位置与边缘状态在未来渐近位置之间的脉冲-脉冲异宿临界连接。我们还建立了临界速率的唯一性和作为r变化时异宿临界连接的非退化性。
英文摘要：
A habitat that is moving due to environmental change may result in tipping to extinction if the rate at which it moves is too great. We use a scalar reaction-diffusion equation with a non-autonomous reaction term, representing a spatially localized habitat moving from one asymptotic location to another, as a context for studying this phenomenon. The movement is characterized by displacement $d$ and rate parameter $r$. The system admits three steady states in both asymptotic habitat locations: a stable extinction state $u_0^*=0$, an unstable pulse (so-called edge state) $u_1^*(x)>0$, which gives rise to the Allee effect, and a stable pulse (populated base state) $u_2^*(x)>u_1^*(x)$, which corresponds to a thriving population at its carrying capacity. Numerical simulations for a specific model identify a critical displacement $d^*$ and, for $d > d^*$, demonstrate the existence of a \textit{critical rate} $r_c(d)$ at which rate-induced tipping occurs: for $r> r_c$ an initially thriving population becomes extinct due to habitat movement being too rapid. We provide analytical results for two limiting cases. For $r\ll 1$, solutions track the moving base state with error $O(r)$. For $r\gg 1$, solutions converge to the extinction state provided $d$ is sufficiently large. For $d$ too small, no tipping occurs regardless of $r$. Numerical simulations complement and extend these analytical results. At the critical rate $r=r_c(d)$, we identify a pulse-to-pulse heteroclinic connection between the base state at the past asymptotic location and the edge state at the future asymptotic location of the habitat. We also establish the uniqueness of this critical rate and non-degeneracy of the heteroclinic connection as $r$ varies.
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