Non-hermitian localization and population biology

TLDR: This work proposes a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states, and describes singular scaling behavior described by a $(d\ensuremath{-}1)$-dimensional generalization of the noisy Burgers' equation.

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 Schr\"odinger 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 $(d\ensuremath{-}1)$-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