I will present asymptotic analyses of two different types of problems involving diffusive processes. The first type involves computing mean first passage times (MFPT) of a random walker inside a finite two-dimensional domain with small targets. Such problems may act as simplified models of search processes, e.g., a protein diffusing in the cytoplasm of a cell before reaching the nucleus. While studies in the past have focused on stationary targets, I will discuss recent results involving mobile targets. In particular, I present the derivation of the corresponding mathematical problem for the MFPT, and for a simple case, give full results for optimal parameters of target motion that minimize the MFPT. I will also introduce a hybrid-asymptotic numerical method for computing the full distribution of first passage times.

In the second part, I will discuss examples of delayed bifurcations in reaction-diffusion systems in both the weakly and fully nonlinear regimes. The delay effect results as the system passes slowly from a stable to unstable regime, and was previously analyzed in ODEs by [Mandel, Erneux, 1987]. For spike solutions in the fully nonlinear regime, I will demonstrate that delay can be analytically quantified for a special class of problems in which the linear stability problem is explicitly solvable. In the weakly nonlinear regime, in the context of a simplified Klausmeier model for vegetation patterns, I will analyze how addition of random noise can affect the magnitude of delay. In both regimes, I show that delay can play a critical role in determining the eventual fate of the system.