The diffusive logistic equation with a free boundary and sign-changing coefficient

TL;DR

This paper analyzes a diffusive logistic model with a free boundary in a heterogeneous environment, establishing criteria for species spread or extinction and estimating the spreading speed when spread occurs.

Contribution

It introduces sharp criteria for spreading and vanishing in a heterogeneous environment with a free boundary, advancing understanding of invasive species dynamics.

Findings

Established a spreading-vanishing dichotomy.

Derived sharp criteria for spreading and vanishing.

Estimated the asymptotic spreading speed.

Abstract

This short paper concerns a diffusive logistic equation with the heterogeneous environment and a free boundary, which is formulated to study the spread of an invasive species, where the free boundary represents the expanding front. A spreading-vanishing dichotomy is derived, namely the species either successfully spreads to the right-half-space as time $t\to\infty$ and survives (persists) in the new environment, or it fails to establish and will extinct in the long run. The sharp criteria for spreading and vanishing is also obtained. When spreading happens, we estimate the asymptotic spreading speed of the free boundary.