Mathematical Sciences: Studies in Pattern Formation and CellMovement

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This project is for theoretical and computational studies of pattern formation and cell movement with the aims of (i) elucidating the types of mechanisms that can produce the observed sequence of condensations leading to the humerus, the radius and ulna, and the digits in developing vertebrate limbs, (ii) developing microscopic models for non-interacting cells whose dynamics depend on external concentration fields and which exhibit various behavioral responses to these fields, (iii) developing continuum descriptions for the population-level behavior of such cells, and (iv) developing models for the motion of cells in dense aggregates. The principal investigator plans to analyze a model for pattern formation that incorporates cell movement by random motion and chemotaxis, the production and diffusion of the chemotatic species, and two types of cells that are interconvertible. A major objective is to understand how these processes interact to produce spatial patterns. Analysis and numerical computations will serve to delimit the parameter values for which pattern can develop in an initially-homogeneous tissue, and numerical continuation techniques will be used to determine how the stable patterns depend on parameters. The effects of other processes and the role of specialized regions of the limb will be investigated in this context.