# Non-Hermitian localization and population biology

Original entry: Naveen Sinha, APPHY 226, Spring 2009

## Contents

### Non-Hermetian localization and population biology

D.R. Nelson and N. Shnerb

Phys. Rev. E 58: 1383 (1998).

## Soft matter keywords

Schoedinger, Burgers, population

## Summary

The quantitative analysis of population (e.g. frequency of a gene, number of a individuals in a particular species) has received much attention in recent years. These phenomoneon are governed by the well-known equations for diffusion and convection, with an addition growth term:

<math>\frac{\partial{c}}{\partial{t}} + (v \cdot \nabla) c = f(c) + D \nabla^2{c}</math>.

In this case c is a vector representing the reactants, which can be bacteria, nutrients, or other biologically relevant variables. D is a matrix of the diffusivities and f represents the non-linear reaction kinetics. The second term, involving v, is the convective flux with velocity v and is analogous to equations form hydrodynamics.

Despite all the existing literature, little has been done to understand inhomogeneities in the underlying medium (e.g. nutrients in the environment). The goal of the paper is to study the effect of such fluctuations of the dynamics of a single species. The modified reaction-diffusion equation becomes:

<math>\frac{\partial{c}}{\partial{t}} + (v \cdot \nabla) c = f(c) + D \nabla^2{c} + [a + U(x)]c(x,t) - b c^2 (x,t)</math>.

In this case, the function f(c) from above is written more explicitly with a representing the difference between birth and death rates, b representing some self-limiting process (e.g. over-population), and U incorporating the spatial inhomogeneity.

## Soft matter aspects

This system illustrates a competition between the disorder and the diffusion term, which prevent any steady state solution. This spatial variation can be decreased with the convective term, which corresponds to an averaging of the environment as the individual (e.g. bacteria) explores its surroundings. An abrupt phase change is observed between the disordered and relatively smooth state due to non-linear effects. The authors draw parallels to the fluctuations observed in quantum mechanical systems.

*Written by: Naveen N. Sinha*