A Mathematical Characterization for Patterns of a Keller‐Segel Model with a Cubic Source Term

Shengmao Fu and Ji Liu

Advances in Mathematical Physics, 2013, vol. 2013, issue 1

Abstract: This paper deals with a Neumann boundary value problem for a Keller‐Segel model with a cubic source term in a d‐dimensional box (d = 1, 2, 3), which describes the movement of cells in response to the presence of a chemical signal substance. It is proved that, given any general perturbation of magnitude δ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order ln(1/δ). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model.

Date: 2013
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https://doi.org/10.1155/2013/934745

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlamp:v:2013:y:2013:i:1:n:934745

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