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state-dependent diffusion

European Journal of Applied Mathematics (1)

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1 results

Pattern formation in bacterial colonies with density-dependent diffusion
JULIEN SMITH-ROBERGE, DAVID IRON, THEODORE KOLOKOLNIKOV
Journal:

European Journal of Applied Mathematics / Volume 30 / Issue 1 / February 2019

Published online by Cambridge University Press:

28 January 2018, pp. 196-218
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Recent experiments have shown that patterns can emerge in bacterial colonies genetically modified to have a drop in diffusion when population densities (detected via a quorum sensing mechanism) are sufficiently large. We examine one PDE model of this system, and construct its non-constant stationary solutions in the form of an interface for the bacterial density. We derive the equations for the interface motion and demonstrate analytically that such interface solution is stable when the diffusion rate of bacteria is large and the diffusion rate of signalling molecules, Dh, is small. We further demonstrate that increasing Dh induces a Hopf bifurcation, resulting in a loss of stability, which can lead to either complex spatio-temporal dynamics or extinction of pattern altogether. These results are confirmed by numerical simulations.