A revisit to the diffusive logistic model with free boundary condition
A revisit to the diffusive logistic model with free boundary condition
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DOI：
10.3934/dcdsb.2016.21.837
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发表时间：
2016
影响因子：
1.2
通讯作者：
Wenzhen Gan;Peng Zhou
中科院分区：
文献类型：
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作者：
Wenzhen Gan;Peng Zhou
这篇简短的论文重新审视了一个自由边界问题，它被用来描述一个新的或入侵物种的传播。我们的主要目标是了解潜在的长时间动力学行为如何响应初始数据。为此，我们将初始函数参数化为$u_0=\sigma\phi^*$，其中$\sigma$被视为可变参数，$\phi^*$是给定函数。我们的主要结果表明，当扩散率较小时，对于任何$\sigma>0$，物种都可以长期存在（称为扩散）;而当扩散率较大时，对于小$\sigma>0$，物种最终将走向灭绝（称为消失）。也许更有趣的是，对于一些中间扩散速率，在（0，\infty）$中出现了一个尖锐的阈值$\sigma^*\，使得如果$0\sigma ^*$，则会发生消失。这个结果可以看作是对[8]中定理1.2的改进。
This short paper revisits a free boundary problem which is used to describe the spreading of a new or invasive species. Our main goal is to understand how the underlying long-time dynamical behaviors response to the initial data. To this end, we parameterize the initial function as $u_0=\sigma\phi^*$, where $\sigma$ is regarded as a variable parameter and $\phi^*$ is a given function. Our main result suggests that when the diffusion rate is small, the species can persist in the long run (called spreading) for any $\sigma>0$; while if the diffusion rate is large, the species will go to extinction finally (called vanishing) for small $\sigma>0$. Maybe of more interest is that for some intermediate diffusion rates, there appears a sharp threshold value $\sigma^*\in(0, \infty)$ such that vanishing happens provided $0\sigma^*$. This result can be seen as an improvement of Theorem 1.2 in [8].