Species survival and scaling laws in hostile and disordered environments

TL;DR

This study investigates species survival in hostile, disordered environments using the Fisher equation, analyzing phase diagrams, critical patch sizes, and the impact of topology and diffusion on survival probabilities.

Contribution

It introduces a measure for critical patch size based on participation ratio and compares survival dynamics across 1D and fractal topologies with finite-size scaling analysis.

Findings

Fractal topology requires higher fluctuations for survival.

The PDF of critical patch sizes is universal in 1D but not in fractal basins.

Diffusion significantly affects survival probabilities in fractal environments.

Abstract

In this work we study the likelihood of survival of single-species in the context of hostile and disordered environments. Population dynamics in this environment, as modeled by the Fisher equation, is characterized by negative average growth rate, except in some random spatially distributed patches that may support life. In particular, we are interested in the phase diagram of the survival probability and in the critical size problem, i.e., the minimum patch size required for surviving in the long time dynamics. We propose a measure for the critical patch size as being proportional to the participation ratio (PR) of the eigenvector corresponding to the largest eigenvalue of the linearized Fisher dynamics. We obtain the (extinction-survival) phase diagram and the probability distribution function (PDF) of the critical patch sizes for two topologies, namely, the one-dimensional system and the fractal Peano basin. We show that both topologies share the same qualitative features, but the fractal topology requires higher spatial fluctuations to guarantee species survival. We perform a finite-size scaling and we obtain the associated scaling exponents. In addition, we show that the PDF of the critical patch sizes has an universal shape for the 1D case in terms of the model parameters (diffusion, growth rate, etc.). In contrast, the diffusion coefficient has a drastic effect on the PDF of the critical patch sizes of the fractal Peano basin, and it does not obey the same scaling law of the 1D case.