2 research outputs found
Acute myeloid leukaemia with aberrant expression of pancytokeratin: A diagnostic pitfall
Localised spatial structures in the Thomas model
The Thomas model is a system of two reactionâdiffusion equations which was originally proposed in the context of enzyme kinetics. It was subsequently realised that it offers a plausible chemical mechanism for the generation of coat markings on mammals. To that end previous investigations have focused on establishing the conditions for the Turing instability and on following the associated patterns as the bifurcation parameter increases through the instability. In this paper we use modern ideas from the theory of dynamical systems to systematically investigate the formation of localised structures in this model. The Turing instability is found to exhibit a degeneracy at which it changes from being subcritical to supercritical (or vice versa). Associated with such degenerate points is a heteroclinic connection which is a crucial requirement for the generation of localised patterns. Localised solutions containing a single âspikeâ are associated with the so-called BelyakovâDevaney transition. Localised solutions containing either multiple âpeaksâ or multiple âholesâ are associated with homoclinic snaking bifurcations. These structures are investigated using continuation software. The snaking bifurcations are found to collapse when the system crosses the BelyakovâDevaney transition. The isolated spike solutions collapse at a codimension two heteroclinic connection to create the so-called collapsed snaking. We show that the temporal stability of the localised solutions depends upon the presence of Hopf bifurcations which in turn are controlled by the diffusion ratio. For small values of the ratio only solutions in the spike region are destabilised by a Hopf bifurcation. For larger values of the diffusion ratio Hopf bifurcations destabilise most of the localised patterns