Extinction of bistable populations is affected by the shape of their initial spatial distribution

Abstract

The question of whether biological populations survive or are eventually driven to extinction has long been examined using mathematical models. In this work we study population survival or extinction using a stochastic, discrete lattice-based random walk model where individuals undergo movement, birth and death events. The discrete model is defined on a two-dimensional hexagonal lattice with periodic boundary conditions. A key feature of the discrete model is that crowding effects are introduced by specifying two different crowding functions that govern how local agent density influences movement events and birth/death events. The continuum limit description of the discrete model is a nonlinear reaction-diffusion equation, and we focus on crowding functions that lead to linear diffusion and a bistable source term that is often associated with the strong Allee effect. Using both the discrete and continuum modelling tools we explore the complicated relationship between the long-term survival or extinction of the population and the initial spatial arrangement of the population. In particular, we study different spatial arrangements of initial distributions: (i) a well-mixed initial distribution where the initial density is independent of position in the domain; (ii) a vertical strip initial distribution where the initial density is independent of vertical position in the domain; and, (iii) several forms of two-dimensional initial distributions where the initial population is distributed in regions with different shapes. Our results indicate that the shape of the initial spatial distribution of the population affects extinction of bistable populations. All software required to solve the discrete and continuum models used in this work are available on GitHub.