Article contents
Effect of the location of a protection zone in a reaction–diffusion model
Published online by Cambridge University Press: 14 June 2023
Abstract
In this paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1< L_2$
, and the general Robin condition is imposed on $x=0$
(i.e. $u(t,\,0)=bu_x(t,\,0)$
with $b\geq 0$
). 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 $L^*$
. Based on the properties of $L_*$
, we obtain the precise strategies for an optimal protection zone: if $b$
is large (i.e. $b\geq 1/\sqrt {-g'(0)}$
), the protection zone should start from somewhere near $0$
; while if $b$
is small (i.e. $b< 1/\sqrt {-g'(0)}$
), then the protection zone should start from somewhere away from $0$
, and as far away from $0$
as possible.
Keywords
- Type
- Research Article
- Information
- Copyright
- Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230613084331222-0506:S0308210523000525:S0308210523000525_inline556.png?pub-status=live)
- 1
- Cited by