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Diffusion-driven instability in oscillating environments

Published online by Cambridge University Press:  26 September 2008

Jonathan A. Sherratt
Jonathan A. Sherratt
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: jas@maths.warwick.ac.uk) Centre for Mathematical Biology, Mathematical Institute, 24–29 St Giles', Oxford OXI 3LB, UK
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Abstract

Diffusion-driven instability in systems of reaction-diffusion equations is a commonly used model for pattern formation in both embryology and ecology. In ecological applications, model parameters tend to oscillate in time, because of either daily or seasonal fluctuations in the environment. I investigate the effects of such fluctuations on diffusion-driven instability by considering analytically the possibility of Turing bifurcations when the parameter values (diffusion coefficients and kinetic parameters) oscillate in time between two sets of constant values, with a period that is either very short or very long compared to the time scale of the growth and predation kinetics. I show that oscillations in the kinetics can have quite different effects from oscillations in the dispersal terms. I also discuss the comparison between the solution forms predicted by linear theory and the numerical solutions of a simple nonlinear predator-prey model.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 4 , August 1995 , pp. 355 - 372
Copyright
Copyright © Cambridge University Press 1995

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