A general formulation of the survival problem in a power-law reaction–diffusion model: Emergence of a critical parameter

Abstract

The survival of a population confined within a bounded habitat is a classical problem, traditionally analyzed in terms of the habitat size. In the linear case, persistence is ensured when the domain length exceeds a critical size lc. In nonlinear models, however survival conditions become considerably more complex and may even take less intuitive forms, such as l≤lc. In this context, Colombo and Anteneodo (2018) studied the power-law reaction–diffusion model ut=D(uν−1ux)x+auμ, with μ,ν>0, accompanied by hostile boundary conditions, determining survival thresholds in terms of habitat size for initially homogeneous populations. In this paper, we propose a general formulation of the persistence question by rewriting the power-law reaction–diffusion model in terms of suitable nondimensional variables. This approach reveals that persistence can be naturally expressed through a parameter [Formula Presented]. We show that there exists a critical value Qc depending on μ, ν and the initial distribution, such that survival occurs whenever Q≥Qc. This more intuitive condition reconciles the various survival criteria within a unified framework. To further explore this condition, we analyze two one-parameter families of initial distributions, including the homogeneous case, and apply a finite-difference scheme to estimate Qc. Conversely, for given model parameters 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 determine how concentrated the initial distribution must be to ensure population survival.