Abstract
A reaction-diffusion model is presented in which spatial structure is maintained by means of a diffusive mechanism more general than classical Fickian diffusion. This generalized diffusion takes into account the diffusive gradient (or gradient energy) necessary to maintain a pattern even in a single diffusing species. The approach is based on a Landau-Ginzburg free energy model. A problem involving simple logistic kinetics is fully analyzed, and a nonlinear stability analysis based on a multi-scale perturbation method shows bifurcation to non-uniform states.
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Part of this work was done while at the Mathematical Institute, Oxford University as a Senior Visiting Fellow supported by the Science Research Council of Great Britain under grant GR/B31378
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Cohen, D.S., Murray, J.D. A generalized diffusion model for growth and dispersal in a population. J. Math. Biology 12, 237–249 (1981). https://doi.org/10.1007/BF00276132
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DOI: https://doi.org/10.1007/BF00276132