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 $u_t=D(u^{\nu -1}{u}_x{)}_x+a{u}^{\mu }$, with $\mu ,\nu >0$, 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 $Q:=\frac{a}{D}{l}^{-\mu +\nu +2}{n}_0^{\mu -\nu }$. Indeed, there exists a critical value $Q_c$ determined by $\mu$, $\nu$ and the initial distribution parameter such that the survival condition can always be written as $Q\ge Q_c$. Notably, from this point of view, one can derive a condition for $Q$ that holds universally for our model under conditional persistence ($\mu \ge \nu$). It applies, in particular, to the case $\mu =\nu +2$, 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 $\mu$, $\nu$, $l$, $n_0$, 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.