The role of the initial distribution in population survival within a bounded habitat

Abstract

In this paper, we analyze the role of initial conditions in population persistence. Specifically, we consider the reaction-diffusion equation $u_t\,=\,D\,(u^{ν-1}\,u_x)_x\,+\,a\,u^μ$, with $μ,ν>0$, accompanied by hostile boundary conditions and examine two families of one-parametric initial distributions, including homogeneous distributions. The model was previously studied by Colombo and Anteneodo (2018). They determined appropriate habitat sizes $l$ for the survival of a population, whose individuals are initially placed homogeneously within the full habitat domain with a total initial population $n_0$. We show that the survival condition can be naturally formulated in terms of the parameter $Q:=\frac{a}{D}l^{-μ+ν+2}n_0^{μ-ν}$. Indeed, there exists a critical value $Q_c$ determined by $μ$, $ν$ and the initial distribution parameter such that the survival condition can always be written as $Q\geq Q_c$. Notably, from this point of view, one can derive a condition for $Q$ that holds universally for our model under conditional persistence ($μ\geqν$). It applies, in particular, to the case $μ=ν+2$, which was not addressed in the previously mentioned work. Nevertheless, in this case $Q=\frac{a}{D}n_0^2$, therefore survival depends solely on the total population, not on the habitat size. We apply a finite-difference scheme to estimate $Q_c$. Conversely, given a population whose evolution is determined by $μ$, $ν$, $l$, $n_0$, and the growth and diffusion coefficients $a$ and $D$ (and consequently the value of $Q$) we use the numerical algorithm to estimate the initial distribution to ensure population survival.