The role of protection zone on species spreading governed by a reaction-diffusion model with strong Allee effect

TL;DR

This paper investigates how the size and configuration of protection zones influence species survival and spreading in a reaction-diffusion model with strong Allee effect, revealing critical thresholds and optimal protection strategies.

Contribution

It introduces a model combining strong Allee effect with a protection zone governed by Fisher-KPP dynamics, identifying critical zone lengths and comparing protection zone configurations.

Findings

Existence of two critical zone lengths $L_*$ and $L^*$ for species dynamics.

Protection zones larger than $L_*$ promote species spreading.

Connected protection zones are more effective than separate ones.

Abstract

It is known that a species dies out in the long run for small initial data if its evolution obeys a reaction of bistable nonlinearity. Such a phenomenon, which is termed as the strong Allee effect, is well supported by numerous evidence from ecosystems, mainly due to the environmental pollution as well as unregulated harvesting and hunting. To save an endangered species, in this paper we introduce a protection zone that is governed by a Fisher-KPP nonlinearity, and examine the dynamics of a reaction-diffusion model with strong Allee effect and protection zone. We show the existence of two critical values $0<L_*\leq L^*$, and prove that a vanishing-transition-spreading trichotomy result holds when the length of protection zone is smaller than $L_*$; a transition-spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$; only spreading happens when the length of protection zone is larger than $L^*$. This suggests that the protection zone works when its length is larger than the critical value $L_*$. Furthermore, we compare two types of protection zone with the same length: a connected one and a separate one, and our results reveal that the former is better for species spreading than the latter.