# American Institute of Mathematical Sciences

May  2016, 21(3): 837-847. doi: 10.3934/dcdsb.2016.21.837

## A revisit to the diffusive logistic model with free boundary condition

 1 School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7

Received  February 2015 Revised  July 2015 Published  January 2016

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 $0<\sigma\leq\sigma^*$ and spreading happens provided $\sigma>\sigma^*$. This result can be seen as an improvement of Theorem 1.2 in [8].
Citation: Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837
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