Turing bifurcations with a temporally varying diffusion coefficient

This paper is concerned with the possibility of Turing bifurcations in a reaction-diffusion system in which the diffusion coefficient of one species varies periodically in time. This problem was introduced and investigated numerically by Timm and Okubo (J. Math. Biol. 30, 307, 1992) in the context of predator-prey interactions in plankton populations. Here, I consider the simple case in which the temporal variation in diffusivity has a square-tooth form, alternating between two constant values, with a period that is long compared with the time scale of the kinetics. The analysis is valid for any set of reaction kinetics. I derive explicit expressions for the Floquet multipliers that determine the stability of the steady state, and thereby obtain the conditions for diffusion driven instability to occur. These conditions imply that, depending on the kinetics, the homogeneous equilibrium may be either more or less stable than when the diffusion coefficient is a constant equal to the mean of the variable diffusivity. I go on to consider the form of the solution when diffusion driven instability does occur, and I use perturbation theory to determine the effect of a small temporal variation in the diffusion coefficient on the spatial wavelength of the pattern that results from diffusion driven instability.

[1]  A. Turing The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[2]  J. L. Jackson,et al.  Dissipative structure: an explanation and an ecological example. , 1972, Journal of theoretical biology.

[3]  H. Meinhardt,et al.  Applications of a theory of biological pattern formation based on lateral inhibition. , 1974, Journal of cell science.

[4]  Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equations and the steady state solutions , 1975 .

[5]  L. Segel,et al.  Hypothesis for origin of planktonic patchiness , 1976, Nature.

[6]  Colin Norman US Budget: Escaping the ‘New Realism’ , 1976, Nature.

[7]  G. Iooss,et al.  Elementary stability and bifurcation theory , 1980 .

[8]  J. Murray A Pre-pattern formation mechanism for animal coat markings , 1981 .

[9]  Jonathan Roughgarden,et al.  Spatial heterogeneity and interspecific competition , 1982 .

[10]  N. Shigesada Spatial Distribution of Rapidly Dispersing Animals in Heterogeneous Environments , 1984 .

[11]  Chris Cosner,et al.  The effects of spatial heterogeneity in population dynamics , 1991 .

[12]  P. Maini,et al.  Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients , 1992 .

[13]  A. Ōkubo,et al.  Diffusion-driven instability in a predator-prey system with time-varying diffusivities , 1992 .

[14]  Philip K. Maini,et al.  Diffusion driven instability in an inhomogeneous domain , 1993 .

[15]  P. Maini,et al.  Analysis of pattern formation in reaction diffusion models with spatially inhomogenous diffusion coefficients , 1993 .