In this article, we analyse the non-local model: ${\partial }_{t}U(t,x)=J\star U(t,x)-U(t,x)+f(x-ct,U(t,x))\text{for}t>0,\text{ and }x\in \mathbb{R},$ 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 exist critical speeds $c^{\ast ,\pm }$ and $c^{\ast \ast ,\pm }$ such that for all $-c^{\ast ,-}<c<c^{\ast ,+}$ then the population will survive and will perish when $c\ge c^{\ast \ast ,+}$ or $c\le -c^{\ast \ast ,-}$. To derive these 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 generalised principal eigenvalue ${\lambda }_p$ of the linear problem $c{\mathfrak{D}}_x\left[\phi \right]+J\star \phi -\phi +{\partial }_{s}f(x,0)\phi +{\lambda }_p\phi =0\text{in}\mathbb{R},$ is negative. ${\lambda }_p$ is a spectral quantity that we defined in the spirit of the generalised first eigenvalue of an elliptic operator. The speeds $c^{\ast ,\pm }$ and $c^{\ast \ast ,\pm }$ are then obtained through a fine analysis of the properties of ${\lambda }_p$ 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 $U(t,x)$. In the specific situation, ${\partial }_{s}f(x,0)>1$ and $J$ symmetric we also investigate the behaviour of the critical speeds $c^{\ast }$ and $c^{\ast \ast }$ with respect to the tail of the kernel $J$. We show in particular that even for very fat tailed kernel these two critical speeds exist.

在本文中，我们分析了非本地模型： ${\partial }_{吨}你(吨,X)=J\star 你(吨,X)-你(吨,X)+F(X-C吨,你(吨,X))\text{为了}吨>0,\text{ 和 }X\in \mathbb{电阻},$ 在哪里 $J$ 是一个正的连续扩散核，并且 $F(X,秒)$是描述人口增长率的异质 KPP 类型非线性。假设人口的生态位是有界的（即在一个紧凑集之外，假设环境对人口是致命的）并以恒定速度随时间移动$C$. 对于紧凑支持的分散内核$J$，假设对于 $C=0$ 人口生存，我们证明存在临界速度 $C^{＊,\pm }$ 和 $C^{＊＊,\pm }$ 以至于对于所有人 $-C^{＊,-}<C<C^{＊,+}$ 那么人口将生存并在什么时候灭亡 $C\ge C^{＊＊,+}$ 或者 $C\le -C^{＊＊,-}$. 为了得出这些结果，我们首先根据速度获得最佳持久性标准$C$对于具有漂移项的非局部问题。即，我们证明对于正速度$C$ 种群持续存在当且仅当广义主特征值 ${\lambda }_{磷}$ 线性问题 $C{\mathfrak{D}}_X\left[\phi \right]+J\star \phi -\phi +{\partial }_{秒}F(X,0)\phi +{\lambda }_{磷}\phi =0\text{在}\mathbb{电阻},$ 是否定的。 ${\lambda }_{磷}$是我们根据椭圆算子的广义第一特征值的精神定义的谱量。速度$C^{＊,\pm }$ 和 $C^{＊＊,\pm }$ 然后通过对属性的精细分析获得 ${\lambda }_{磷}$ 关于 $C$. 特别地，我们建立了它在速度方面的连续性$C$. 此外，对于任何连续有界非负初始数据，我们建立解决方案的长时间行为$你(吨,X)$. 在特定情况下，${\partial }_{秒}F(X,0)>1$ 和 $J$ 对称我们还研究了临界速度的行为 $C^{＊}$ 和 $C^{＊＊}$ 关于内核的尾部 $J$. 我们特别表明，即使对于尾非常胖的内核，这两个临界速度也存在。