Pattern Formation in a Bacterial Colony Model

Xinze Lian,Hailing Wang,Guichen Lu

doi:10.1155/2014/149801

Abstract

We investigate the spatiotemporal dynamics of a bacterial colony model. Based on the stability analysis, we derive the conditions for Hopf and Turing bifurcations. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by parameters in the model and find that the model dynamics exhibit a diffusion controlled formation growth to spots, holes and stripes pattern replication, which show that the bacterial colony model is useful in revealing the spatial predation dynamics in the real world.

Highlights

Spatial patterns which are formed by some kinds of bacterial colonies present an interesting structure during their growth conditions
Pattern formation of a spatial model for the growth of bacterial colonies with the two-dimensional space is investigated. Based on both mathematical analysis and numerical simulations, we have found that its spatial pattern includes periodic solutions from Hopf bifurcation and the spotted and striped patterns from Turing bifurcation
It seems interesting to know that the dynamics vary when the parameter moves across the diagram

Summary

Introduction

Spatial patterns which are formed by some kinds of bacterial colonies present an interesting structure during their growth conditions. Kawasaki et al [7] have developed a reaction-diffusion model and have shown the patterns by using the computer simulations. Braverman [4] have introduced a model of prey-predator type with Holling-II functional response under the situation of a renewable nutrient. We consider the model with the consumption term of nutrient in a Holling III functional response. ΓV, where r is the intrinsic nutrient growth rate, M is the carrying capacity of the environment for the nutrient (prey), γ is the bacteria (predator) mortality rate, κ, θ, and γ0 are parameters of the Holling Type III functional response, and DV is the nutrient diffusion coefficient.

Modeling Analysis and Mathematical Setup

Bifurcation Analysis

Pattern Formation

Concluding and Remarks