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arXiv · 2609.14679

A free boundary model for invasive and native species under shifting climate in the weak competition case

Phuoc Vinh Dinh·Phuong Le·Tien Dung Nguyen
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Abstract

We study a free boundary problem for a diffusive Lotka--Volterra competition system describing the invasion of a new species into the habitat of a native competitor, in a habitat that is shifted from unfavourable to favourable at a constant speed $c>0$ by climate change. Only the invader feels the shifting environment and only its range is governed by a Stefan-type free boundary, while the native species occupies the whole half line. We work throughout in the weak competition regime, in which the two species may coexist. We prove a spreading--vanishing dichotomy: either the invader spreads and the pair converges to the coexistence steady state $(u^*,v^*)$, or the invader vanishes and the native species recovers its carrying capacity. In the vanishing case we obtain the explicit bound $\lim_{t\to\infty}h(t)\le\fracπ{2}\sqrt{d_1c_2/(a_1c_2-a_2c_1)}$, and we give criteria guaranteeing each alternative. When spreading occurs, we determine the exact asymptotic spreading speed: $\lim_{t\to\infty}h(t)/t=\min\{c,c_0\}$, where $c_0$ is the spreading speed of the corresponding homogeneous weak competition system. In particular the invasion is slowed down both by the competitor and by the climate shift, and the slower of the two mechanisms is the one that determines the speed. Numerical simulations illustrate the results.

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Phuoc Vinh Dinh, Phuong Le, Tien Dung Nguyen. 2026-09-13. A free boundary model for invasive and native species under shifting climate in the weak competition case. https://arxiv.org/abs/2609.14679

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