Oscillatory Dynamics for PDE Models Coupling Bulk Diffusion and Dynamically Active Compartments: Theory, Numerics and Applications

We formulate and investigate a relatively new modeling paradigm by which spatially segregated dynamically active units communicate with each other through a signaling molecule that diffuses in the bulk medium between active units. The modeling studies start with a simplified setting in a onedimensional space, where two dynamically active compartments are located at boundaries of the domain and coupled through the feedback term to the local dynamics together with flux boundary conditions at the two ends. For the symmetric steady state solution, in-phase and anti-phase synchronizations are found and Hopf bifurcation boundaries are studied using a winding number approach as well as parameter continuation methods of bifurcation theory in the case of linear coupling. Numerical studies show the existence of double Hopf points in the parameter space where center manifold and normal form theory are used to reduce the dynamics into a system of amplitude equations, which predicts the configurations of the Hopf bifurcation and stability of the two modes near the double Hopf point. The system with a periodic chain of cells is studied using Floquet theory. For the case of a single active membrane bound component, rigorous spectral results for the onset of oscillatory dynamics are obtained and in the finite domain case, a weakly nonlinear theory is developed to predict the local branching behavior near the Hopf bifurcation point. A previously developed model by Gomez et al.[23] is analyzed in detail, where the phase diagrams and the Hopf frequencies at onset are provided analytically with slow-fast type of local kinetics. A coupled cell-bulk system, with small signaling compartments, is also studied in the case of a two-dimensional bounded domain using the method of asymptotic expansions. In the very large diffusion limit we reduce the PDE cell-bulk system to a finite dimensional dynamical system, which is studied both analytically and numerically. When the diffusion rate is not very large, we show the effect of spatial distribution of cells and find the dependence of the quorum sensing threshold on influx rate.