# A free boundary problem for spreading under shifting climate

## TL;DR

This paper studies a free boundary model for invasive species spreading under shifting climate conditions, revealing a critical speed that determines whether the species spreads or vanishes, with detailed long-term behavior classification.

## Contribution

It introduces a refined free boundary model for species spread under climate shift and classifies the long-term dynamics based on the shifting speed.

## Key findings

- Existence of a critical shifting speed c_0 for spreading or vanishing.
- Spreading profile determined by a semi-wave with forced speed c when c < c_0.
- Spreading profile determined by a semi-wave with speed c_0 when c ≥ c_0.

## Abstract

In this paper we consider a free boundary problem which models the spreading of an invasive species whose spreading is enhanced by the changing climate. We assume that the climate is shifting with speed c and obtain a complete classification of the long-time dynamical behaviour of the species. The model is similar to that in [9] with a slight refinement in the free boundary condition. While [9], like many works in the literature, investigates the case that unfavourable environment is shifting into the favourable habitat of the concerned species, here we examine the situation that the unfavourable habitat of an invasive species is replaced by a favourable environment with a shifting speed c. We show that a spreading-vanishing dichotomy holds, and there exists a critical speed$c_0$ such that when spreading happens in the case $c < c_0$, the spreading profile is determined by a semi-wave with forced speed c, but when $c \geq c_0$, the spreading profile is determined by the usual semi-wave with speed $c_0$.