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UBC Theses and Dissertations

An asymptotic and numerical analysis of localized solutions to some linear and nonlinear pattern formation problems with heterogeneities Wong, Ka Wah


Reaction-diffusion (RD) systems are indispensable models to understand pattern formation. To reproduce sophisticated pattern-forming behaviors that resemble observed biological patterns, we can increase the complexity of a RD model by introducing spatial heterogeneity. In the first part of this thesis, we consider two-component RD systems in the singular limit of a large diffusivity ratio for which localized spot patterns occur. We will use a hybrid asymptotic-numerical approach to analyze the existence, linear stability, and dynamics of localized spot patterns in the presence of certain spatial heterogeneities. More specifically, for the Schnakenberg model, we will investigate the effect on spot patterns that occur for distinct types of localized spatial heterogeneity for which spots are either repelled from or attracted to, respectively. For the Klausmeier model, we study the effect of water advection on the dynamics and steady-state behavior of localized patches or ``spots'' of vegetation. Moreover, we study the effect of a slowly varying rainfall rate on spot dynamics, and the induced bifurcation with delays. In many previous studies of spot instabilities, it has been observed from numerical PDE simulations that a linear shape-deforming instability of a localized spot is the trigger of a nonlinear spot-replication event in the absence of heterogeneity. To provide a theoretical basis for these observations, we derive an amplitude equation from a weakly nonlinear analysis which confirms that a peanut-shaped instability of a spot is subcritical. In the second part of this thesis, we study mean first passage time (MFPT) for a Brownian particle to be captured by small circular traps in a 2-D confining domain. Our focus is to understand how the deviations from a radially symmetric domain, which represents a domain heterogeneity in a general sense, alters the optimal spatial configuration of a collection of small circular traps that minimizes the average MFPT. In this direction, we develop a numerical method and perform asymptotic analysis to approximate the MFPT for general 2-D domains. In particular, by deriving a new explicit analytical formula for the Neumann Green's function, we demonstrate the full power of these tools for an elliptical domains of arbitrary aspect ratio.

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