Non-Hermitian localization and population biology

David R. Nelson, Nadav M. Shnerb

Research output: Contribution to journalArticlepeer-review

235 Scopus citations

Abstract

The time evolution of spatial fluctuations in inhomogeneous [Formula Presented]-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 Schrödinger 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 [Formula Presented] dimensions exhibits singular scaling behavior described by a [Formula Presented]-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.

Original languageEnglish
Pages (from-to)1383-1403
Number of pages21
JournalPhysical Review E
Volume58
Issue number2
DOIs
StatePublished - 1998
Externally publishedYes

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