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On the weakly nonlinear analysis of coupled bulk-surface reaction-diffusion systems : theory, numerics, applications Paquin-Lefebvre, Frédéric


Motivated by the spatial segregation of intracellular proteins between the cytoplasm and the cellular membrane, we investigate the spatio-temporal dynamics of coupled bulk-surface reaction-diffusion models. For such models, a passive diffusion process occurring inside a bounded bulk domain is coupled to a nonlinear reaction-diffusion process restricted to the domain boundary. We first consider a model with 2-D circular bulk geometry, for which we perform a systematic weakly nonlinear analysis of spatio-temporal patterns. Amplitude equations near Hopf, pitchfork and codimension-two pitchfork-Hopf bifurcations are derived from a multi-scale expansion. For both Brusselator and Schnakenberg surface kinetics, we detect a variety of sub- and supercritical bifurcations leading to the formation of stationary Turing patterns and radially symmetric oscillations. Unstable mixed-mode oscillations near codimension-two pitchfork-Hopf bifurcations are also identified. We then consider the synchronized oscillatory dynamics of two identical pointlike boundaries, modeled by nonlinear ODEs, that are spatially segregated and coupled via a 1-D PDE bulk diffusion field. For such coupled PDE--ODE systems, we perform a weakly nonlinear analysis near Hopf bifurcations associated with even and odd perturbations. This allows us to detect stable in-phase and anti-phase oscillations when assuming Sel’kov kinetics in each compartment. We also consider the PDE diffusive coupling of two chaotic Lorenz oscillators. Through a computation of Lyapunov exponents, we establish that complete synchronization of the two chaotic trajectories is possible whenever the coupling strength and the bulk diffusivity are large enough. Lastly, we study a mass-conserved membrane-bulk model for a specific intracellular protein pattern-forming system, the Cdc42-GEF system. For both 1-D and 2-D bulk geometries, we investigate the formation of spatio-temporal patterns as diffusion levels and membrane-bulk coupling rates are varied. As a result of mass conservation, only anti-phase oscillations are observed on a 1-D interval, with no instabilities leading to in-phase oscillations detected. For the 2-D circular bulk case, no instabilities leading to radially symmetric oscillations are found. Instead, a variety of stationary Turing patterns and traveling waves, that interact near Bogdanov-Takens bifurcations, are observed.

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