A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment

TL;DR

This paper extends the analysis of a diffusive logistic equation with a free boundary in a time-periodic environment, demonstrating that previous spreading-vanishing results hold even with sign-changing growth rates.

Contribution

It generalizes existing models by including sign-changing intrinsic growth rates and confirms that key spreading-vanishing criteria remain valid.

Findings

Spreading-vanishing dichotomy is preserved under sign-changing growth rates.

Sharp criteria for spreading and vanishing are established.

Asymptotic spreading speed estimates are confirmed.

Abstract

This paper concerns a diffusive logistic equation with a free boundary and sign-changing intrinsic growth rate in heterogeneous time-periodic environment, in which the variable intrinsic growth rate may be "very negative" in a "suitable large region". Such a model can be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. In the case of higher space dimensions with radial symmetry and the intrinsic growth rate has a positive lower bound, this problem has been studied by Du, Guo and Peng . They established a spreading-vanishing dichotomy, the sharp criteria for spreading and vanishing and estimate of the asymptotic spreading speed. In the present paper, we show that the above results are retained for our problem. This paper has been submitted to Journal of Functional Analysis in August 5, 2014 (JFA-14-548).