Entropy methods and convergence to equilibrium for volume-surface reaction-diffusion systems

Abstract: We consider two volume-surface reaction-diffusion systems arising from cell biology. The first system describes the localisation of the protein Lgl in the asymmetric division of Drosophila SOP stem cells, while the second system models the JAK2/STAT5 signalling pathway. Both model systems have in common that i) different species are located in different spatial compartments, ii) the involved chemical reaction kinetics between the species satisfies a complex balance condition and iii) that the associated complex balance equilibrium is spatially inhomogeneous. By using recent advances on the entropy method for complex balanced reaction-diffusion systems, we show for both systems exponential convergence to the equilibrium with constants and rates, which can be explicitly estimated.