  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.
