Population Dynamics and Non-Hermitian Localization

TL;DR

This paper reviews how non-Hermitian localization phenomena, influenced by convection, affect population dynamics models with spatially varying growth, providing exact spectral results and analyzing boundary condition effects.

Contribution

It introduces a detailed analysis of non-Hermitian localization in population models, including exact spectral solutions and the impact of boundary conditions.

Findings

Convection introduces a constant imaginary vector potential in growth models.

Exact spectral and winding number results are derived for one-dimensional random growth.

Periodic boundary conditions yield results representative of broad growth problem classes.

Abstract

We review localization with non-Hermitian time evolution as applied to simple models of population biology with spatially varying growth profiles and convection. Convection leads to a constant imaginary vector potential in the Schroedinger-like operator which appears in linearized growth models. We illustrate the basic ideas by reviewing how convection affects the evolution of a population influenced by a simple square well growth profile. Results from discrete lattice growth models in both one and two dimensions are presented. A set of similarity transformations which lead to exact results for the spectrum and winding numbers of eigenfunctions for random growth rates in one dimension is described in detail. We discuss the influence of boundary conditions, and argue that periodic boundary conditions lead to results which are in fact typical of a broad class of growth problems with convection.