Diffusion driven instability in an inhomogeneous domain |
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Authors: | Debbie L Benson Jonathan A Sherratt Philip K Maini |
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Institution: | (1) Centre for Mathematical Biology, Mathematical Institute, 24-29 St Giles', OX1 3LB Oxford, UK |
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Abstract: | Diffusion driven instability in reaction-diffusion systems has been proposed as a mechanism for pattern formation in numerous
embryological and ecological contexts. However, the possible effects of environmental inhomogeneities has received relatively
little attention. We consider a general two species reaction-diffusion model in one space dimension, with one diffusion coefficient
a step function of the spatial coordinate. We derive the dispersion relation and the solution of the linearized system. We
apply our results to Turing-type models for both embryogenesis and predator-prey interactions. In the former case we derive
conditions for pattern to be isolated in one part of the domain, and in the latter we introduce the concept of “environmental
instability”. Our results suggest that environmental inhomogeneity could be an important regulator of biological pattern formation. |
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