A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat

Qiaoling  Chen; Fengquan Li; Feng  Wang

doi:10.3934/dcdsb.2016.21.13

Article Contents

2016, Volume 21, Issue 1: 13-35. Doi: 10.3934/dcdsb.2016.21.13

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A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024

Received: November 30, 2014

Revised: August 31, 2015

Published: December 2015

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Abstract

We study a diffusive logistic equation with a free boundary in time-periodic environment. To understand the effect of the diffusion rate $d$, the original habitat radius $h_0$, the spreading capability $\mu$ , and the initial density $u_0$ on the dynamics of the problem, we divide the time-periodic habitat into two cases: favorable habitat and unfavorable habitat. By choosing $d, h_0, \mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishing dichotomy and sharp criteria for the spreading and vanishing in time-periodic environment. We introduce the principal eigenvalue $\lambda_1(d, \alpha-\gamma, h_{0}, T)$ to determine the spreading and vanishing of the invasive species. We prove that if $\lambda_1(d, \alpha-\gamma, h_0, T)\leq 0$, the spreading must happen; while if $\lambda_1(d, \alpha-\gamma, h_0, T)>0$, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the diffusion rate is small or the occupying habitat is large. In an unfavorable habitat, the species vanishes if the initial density of the species is small. Moreover, when spreading occurs, the asymptotic spreading speed of the free boundary is determined.

Keywords:

Diffusive logistic problem,
free boundary,
favorable habitat and unfavorable habitat,
periodic environment,
spreading-vanishing dichotomy,
sharp criteria.

Mathematics Subject Classification: 35K57, 35K61, 35R35, 92D25.

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