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UBC Theses and Dissertations

Topics in the asymptotic analysis of narrow escape and quorum-sensing behavior for PDE models with biological applications Iyaniwura, Sarafa Adewale


In this thesis, we develop novel numerical and analytical techniques for calculating the MFPT for a Brownian particle to be captured by either small stationary or mobile traps inside a bounded 2-D domain. Of particular interest is identifying the optimal arrangements of small traps that minimize the average MFPT. Although the MFPT and the associated optimal trap arrangement problem have been well-studied for disk-shaped domains, there are very few analytical or numerical results available for general star-shaped domains or thin domains with large aspect ratio. We develop an embedded numerical method for both stationary and periodic mobile trap problems, based on the Closest Point Method (CPM), to perform MFPT simulations on various confining 2-D domains. Optimal trap arrangements are identified numerically through either a refined discrete sampling approach or a particle-swarm optimization procedure. To confirm some of the numerical findings, novel perturbation approaches are developed to approximate the average MFPT and identify optimal trap configurations for a class of near-disk confining domains or an arbitrary thin domain of large aspect ratio. We also analyze cell-bulk coupled ODE-PDE models for describing the communication between localized spatially segregated dynamically active signaling compartments or cells, coupled through a passive extracellular bulk diffusion field. In a 2-D bounded domain, where the cells are small disks of a common radius ε << 1, the method of matched asymptotic expansions is used in the regime of finite bulk diffusivity to construct steady-state solutions of the ODE-PDE model, and to derive a globally coupled nonlinear matrix eigenvalue problem (GCEP) that characterizes the linear stability properties of the steady-states. In the limit of large bulk diffusivity, an asymptotic analysis of the ODE-PDE model leads to a limiting ODE system for the spatial average of the bulk signaling chemical. We extend this analysis to a 3-D spherical domain, where the cells are smaller spheres localized within the domain. For Sel’kov reaction kinetics, we illustrate that a switch-like emergence of intracellular oscillations can occur through a Hopf bifurcation and that cell density plays a dual role of triggering and then quenching synchronous oscillations in the intracellular dynamics of the cells.

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