一个种群能在一个变化的环境中利用非本地扩散生存吗
Can a population survive in a shifting environment using non-local dispersion
摘要：
本文分析了非局部模型：$\partial$ t U（t，x）= J $\star$ U（t，x）-- U（t，x）+ f（x -- ct，U（t，x）），其中t > 0，x $\in$ R，J是正的连续扩散核，f（x，s）是描述种群增长率的异质KPP型非线性.假设种群的生态位是有界的（即在一个紧集之外，假设环境对种群是致命的），并以恒定的速度c随着时间推移而移动。对于紧支撑的扩散核J，假设对于c = 0，种群生存，我们证明了存在临界速度c *，$\pm$和c * *，$\pm$，使得对于所有-c *，- < c < c *，+，那么种群将生存，当c $\ge$ c * *，+或c $\le$ -c * *，--时将灭亡.为了得到这个结果，我们首先得到了一个最优的持久性准则依赖于速度c的非局部问题的漂移项。也就是说，我们证明了，对于一个积极的速度c的人口持续当且仅当广义主特征值$λ$ p的线性问题cDx [PDMATHAMT] + J $\star$ PDMATHAMT-PDMATHAMT + $\partial$ s f（x，0）PDMATHAMT + $λ$ p $Φ$ = 0在R，是负的。$λ$ p是我们根据椭圆算子的广义第一本征值定义的谱量。然后，通过精细分析$λ$ p相对于c的性质，获得速度c *，$\pm$和c * *，pm。特别地，我们建立了它关于速度c的连续性。此外，对于任何连续有界非负初值，我们建立了解U（t，x）的长时间行为。
英文摘要：
In this article, we analyse the non-local model : $\partial$ t U (t, x) = J $\star$ U (t, x) -- U (t, x) + f (x -- ct, U (t, x)) for t > 0, and x $\in$ R, where J is a positive continuous dispersal kernel and f (x, s) is a heterogeneous KPP type non-linearity describing the growth rate of the population. The ecological niche of the population is assumed to be bounded (i.e. outside a compact set, the environment is assumed to be lethal for the population) and shifted through time at a constant speed c. For compactly supported dispersal kernels J, assuming that for c = 0 the population survive, we prove that there exists a critical speeds c * ,$\pm$ and c * * ,$\pm$ such that for all --c * ,-- < c < c * ,+ then the population will survive and will perish when c $\ge$ c * * ,+ or c $\le$ --c * * ,--. To derive this results we first obtain an optimal persistence criteria depending of the speed c for non local problem with a drift term. Namely, we prove that for a positive speed c the population persists if and only if the generalized principal eigenvalue $λ$ p of the linear problem cD x [$Φ$] + J $\star$ $Φ$ -- $Φ$ + $\partial$ s f (x, 0)$Φ$ + $λ$ p $Φ$ = 0 in R, is negative. $λ$ p is a spectral quantity that we defined in the spirit of the generalized first eigenvalue of an elliptic operator. The speeds c * ,$\pm$ and c * * ,pm are then obtained through a fine analysis of the properties of $λ$ p with respect to c. In particular, we establish its continuity with respect to the speed c. In addition, for any continuous bounded non-negative initial data, we establish the long time behaviour of the solution U (t, x).
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