Pattern formation during growth development: models, numerics and applications
Mathematical modelling, numerical analysis and simulations of spatial patterning during growth development in developmental biology and biomedicine is an emerging young research area with significant potential of elucidating mechanisms for pattern formation on real biological evolving skin surfaces. Since the seminal work of Turing in 1952 which showed that a system of reacting and diffusing chemical morphogens could evolve from an initially uniform spatial distribution to concentration profiles that vary spatially - a spatial pattern - many models have been proposed on stationary domains exploiting the generalised patterning principle of short-range activation, long-range inhibition elucidated by Meinhardt of which the Turing model is an example. Turing's hypothesis was that one or more of the morphogens played the role of a signaling chemical, such that cell fate is determined by levels of morphogen concentration. However, our recent results show that in the presence o f domain growth, short-range inhibition, long-range activation as well as activator-activator mechanisms have the potential of giving rise to the formation of patterns only during growth development of the organism. These results offer us a unique opportunity to study non-standard mechanisms, either experimentally or hypothetically, for pattern formation on evolving surfaces, a largely unchartered research area. In this talk I will discuss modelling, numerical analysis, computations and applications of the models for pattern formation during growth.