Non-Hermitian Localization and Population Biology

TL;DR

This paper explores how environmental heterogeneities and convection influence localization phenomena in biological populations, revealing a transition between localized and extended states and uncovering universal scaling behaviors.

Contribution

It introduces a novel analogy between population dynamics with disorder and non-Hermitian quantum systems, proposing a delocalization transition and analyzing singular scaling in high convection regimes.

Findings

Localized states dominate in certain conditions

A delocalization transition exists at a critical convection threshold

Universal singularities appear in the density of states near the band edge

Abstract

The time evolution of spatial fluctuations in inhomogeneous d-dimensional biological systems is analyzed. A single species continuous growth model, in which the population disperses via diffusion and convection is considered. Time-independent environmental heterogeneities, such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate. Linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size. Using an analogy with a Schrodinger equation subject to a constant imaginary vector potential, we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states. In the limit of high convection velocity, the linearized growth problem in $d$ dimensions exhibits singular scaling behavior described by a (d-1)-dimensional generalization of the noisy Burgers' equation, with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane. The Burgers mapping leads to unusual transverse spreading of convecting delocalized populations.