In this paper, we analyze the role of initial conditions in population persistence. Specifically, we consider the reaction-diffusion equation uₜ\, =\, D\, (u^ν-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: =aDl^-μ+ν+2n₀^μ-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 (μν). It applies, in particular, to the case μ=ν+2, which was not addressed in the previously mentioned work. Nevertheless, in this case Q=aDn₀², 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.
Rosa et al. (Sun,) studied this question.