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

In this paper, we analyze the role of initial conditions in population persistence. Specifically, we consider the reaction-diffusion equation uₜ\,=\,D\,(uν-1\,uₓ)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 n₀. We show that the survival condition can be naturally formulated in terms of the parameter Q:=a/Dl-μ+ν+2n₀μ-ν. Indeed, there exists a critical value Qc determined by $μ$, $ν$ and the initial distribution parameter such that the survival condition can always be written as Q≥ Qc. Notably, from this point of view, one can derive a condition for Q that holds universally for our model under conditional persistence (μ≥ν). It applies, in particular, to the case $μ=ν+2$, which was not addressed in the previously mentioned work. Nevertheless, in this case Q=a/Dn₀², therefore survival depends solely on the total population, not on the habitat size. We apply a finite-difference scheme to estimate Qc. Conversely, given a population whose evolution is determined by $μ$, $ν$, l, n₀, 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.