Effects of source and loss terms on the wave-pinning description of cell polarisation

TL;DR

This paper analyzes how source and loss terms influence wave-pinning in cell polarization models, revealing complex bifurcation structures and localized solutions through analytical and numerical methods.

Contribution

It extends the minimal wave-pinning model by incorporating source and loss terms, exploring their effects on stability and pattern formation.

Findings

Introduction of source and loss terms breaks mass conservation.

Identification of subcritical bifurcations leading to localized solutions.

Asymptotic analysis links pulse solutions to front solutions in the conservative limit.

Abstract

A system of two Schnakenberg-like reaction-diffusion equations is investigated analytically and numerically. The system has previously been used as a minimal model for concentrations of GTPases involved in the process of cell polarisation. Source and loss terms are added, breaking the mass conservation, which was shown previously to be responsible for the generation of stable fronts via a so-called wave-pinning mechanism. The extended model gives rise to a unique homogeneous equilibrium in the parameter region of interest, which loses stability via a pattern formation, or Turing bifurcation. The bistable character of the reaction terms ensures that this bifurcation is subcrtical for sufficiently small values of the driving parameter multiplying the nonlinear kinetics. This subcriticality leads to the onset of a multitude of localised solutions, through the homoclinic snaking mechanism. As the driving parameter is further decreased, the multitude of solutions transforms into a single pulse through a Belyakov-Devaney transition in which there is the loss of a precursive pattern. An asymptotic analysis is used to probe the conservative limit in which the source and loss terms vanish. Matched asymptotic analysis shows that on an infinite domain the pulse solution transitions into a pair of fronts, with an additional weak quadratic core and exponential tails. On a finite domain, the core and tails disappear, leading to the mere wave-pinning front and its mirror image.