宏观异质环境中局部受控种群模型的扩散速度
Spreading speed of locally regulated population models in macroscopically heterogeneous environments
AI总结：
研究了宏观异质环境中的格点分支随机游走模型，发现在特定渐近极限下前沿位置收敛于显式 ODE 解，并证明极限不可交换。
摘要：
我们考虑一种格点分支随机游走，它具有现场竞争，并处于在空间和时间上具有宏观尺度 $1/\varepsilon$ 异质性的环境中。这可以作为一个在宏观尺度异质的栖息地（如山脉、温度或降水梯度等）中生物种群空间动态的模型。该模型包含另一个参数 $K$，用于衡量局部种群密度。我们研究在先 $\varepsilon\to 0$ 然后 $K\to\infty$ 的极限下的模型。在这种渐近状态下，我们证明了重新缩放的前沿位置作为时间的函数收敛于一个显式 ODE 的解。我们进一步讨论了与另一种流行的种群动态模型 Fisher-KPP 方程的关系，该方程在 $K\to\infty$ 的极限下出现。结合关于 Fisher-KPP 方程的已知结果，我们的结果特别表明，极限 $\varepsilon\to0$ 和 $K\to\infty$ 通常不可交换。我们推测，当 $\log K$ 和 $1/\varepsilon$ 同阶时，会出现一个插值状态。
英文摘要：
We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\varepsilon$ in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient\ldots). The model incorporates another parameter, $K$, which is a measure of the local population density. We study the model in the limit when first $\varepsilon\to 0$ and then $K\to\infty$. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit $K\to\infty$. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits $\varepsilon\to0$ and $K\to\infty$ do not commute in general. We conjecture that an interpolating regime appears when $\log K$ and $1/\varepsilon$ are of the same order.
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