On the activation-inhibition mechanism of pattern formation
Rouman Anguelov (Department of Mathematics and Applied Mathematics, University of Pretoria)
(April 16, 2019 10:20 AM - 11:05 AM)
The most widely known mechanism of pattern formation is the one based on the Turing instability theorem. The model involves two species (chemical, biological, artificial) one of them called activator and the other inhibitor, with typically unique stable equilibrium in a space independent conditions. The Turing theorem states that when the difference of the diffusion coefficients of the activator and the inhibitor is sufficiently large, the spatially uniform equilibrium is unstable. Since the solutions are bounded, typically any solution eventually approaches a stable spatially non-uniform state, referred to as a pattern. The Turing mechanism from 1952 was rediscovered for biological species in 1972 in Gierer and Meinhardt's Theory of Biological Pattern formation. Gierer and Meinhardt arrived independently to the same model as Turing, and probably contrary to popular belief, the Turing mechanism is not the only way to represent self-activation and lateral inhibition on which the Gierer and Meinhardt pattern formation theory is based. Particularly in population dynamics, a principal inhibitor could be the lack of resource. The self-activation concept of populations can be identified as local (of family or population group) conspecific support. We show that, even if the competing species/groups have the same demographics and interaction, their co-existence can be destabilized by sufficient level of conspecific support. When considering large number of species, the conspecific support destabilizes the co-existence equilibrium, thus producing a pattern of extinction and varied levels of existence. Upscaling the model to continuous space variable leads to a model of pattern formation via local self-activation and lateral inhibition, where these two factors are represented via integral operators. This approach was first pioneered to model patterns in tiger bush. As illustration we consider the modelling the formation of heterocyst cells in the filaments of the algae Anabaena.