4pm - 5pm
Wednesday 28 November 2018
Heteroclinic to homoclinic and homoclinic to homoclinic transition in reaction-diffusion systems
Inspired by a minimal reaction-diffusion model for cell polarisation exhibiting stable fronts as polarisation profiles, we study the consequences of breaking the conservation of mass by introducing generic source and sink terms.
University of Surrey
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- Nicolas Verschueren Van Rees, Bristol
In this non-conservative regime, the fronts disappear and several time-independent solutions are observed instead: patterns, localised patterns and single peak solutions. A characterisation of these solutions, in a two-parameter space, is provided. Two transitions are studied in this context.
The first transition corresponds to the connection between these new solutions and the fronts observed in the conservative regime. This transition is studied using numerical (continuation and solving of the full PDE) and analytical (matched asymptotics) tools.
The second transition is between localised pattern solutions (organised through the snaking scenario) and the single peak solutions. This transition is reported numerically in two additional systems and analysed via the Shilinkov-type analysis of a generic two parameter reversible four-dimensional system of ODE. This analysis provides analytical predictions which are in good agreement with numerical observations in two of the systems.