American Institute of Mathematical Sciences

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August  2001, 1(3): 339-362. doi: 10.3934/dcdsb.2001.1.339

Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation

R.A. Satnoianu 1, , Philip K. Maini 1, , F.S. Garduno 2, and  J.P. Armitage 3,

1. 

Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OX1 3LB Oxford, United Kingdom, United Kingdom
2. 

Departamento de Matematicas, Facultad de Ciencias, UNAM, Ciudad Universitaria, Circuito Exterior, México, 04510 D.F., México, Mexico
3. 

Department of Biochemistry, University of Oxford, Oxford OX1 3QU, United Kingdom

Received  November 2000 Revised  April 2001 Published  May 2001

We study a reaction diffusion model recently proposed in [5] to describe the spatiotemporal evolution of the bacterium Bacillus subtilis on agar plates containing nutrient. An interesting mathematical feature of the model, which is a coupled pair of partial differential equations, is that the bacterial density satisfies a degenerate nonlinear diffusion equation. It was shown numerically that this model can exhibit quasi-one-dimensional constant speed travelling wave solutions. We present an analytic study of the existence and uniqueness problem for constant speed travelling wave solutions. We find that such solutions exist only for speeds greater than some threshold speed giving minimum speed waves which have a sharp profile. For speeds greater than this minimum speed the waves are smooth. We also characterise the dependence of the wave profile on the decay of the front of the initial perturbation in bacterial density. An investigation of the partial differential equation problem establishes, via a global existence and uniqueness argument, that these waves are the only long time solutions supported by the problem. Numerical solutions of the partial differential equation problem are presented and they confirm the results of the analysis.
Keywords: phase plane analysis, bacterial chemotaxis, travelling waves, Degenerate diffusion, nonlinear coupled parabolic equations..
Mathematics Subject Classification: 37C4.
Citation: R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
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