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We consider the mean first passage time (MFPT) of a two-dimensional diffusing particle to a small trap with a distribution of absorbing and reflecting sections. High-order asymptotic formulas for the MFPT and the fundamental eigenvalue of the Laplacian are derived which extend previously obtained results and show how the orientation of the trap affects the(More)
Spatiotemporal Turing-Hopf pinning solutions near the codimension-two Turing-Hopf point of the one-dimensional Brusselator model are studied. Both the Turing and Hopf bifurcations are supercritical and stable. The pinning solutions exhibit coexistence of stationary stripes of near critical wavelength and time-periodic oscillations near the characteristic(More)
A hybrid asymptotic-numerical method is formulated and implemented to accurately calculate the mean first passage time (MFPT) for the expected time needed for a predator to locate small patches of prey in a 2-D landscape. In our analysis, the movement of the predator can have both a random and a directed component, where the diffusivity of the predator is(More)
We present three examples of delayed bifurcations for spike solutions of reaction-diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analyzed in the It was found that the instability would not be fully realized until the system had entered well into the unstable regime. The(More)
On a finite three-dimensional domain Ω, a hybrid asymptotic-numerical method is employed to analyze the existence, linear stability, and slow dynamics of localized quasi-equilibrium multi-spot patterns of the Schnakenberg activator-inhibitor model with bulk feed rate A in the singularly perturbed limit of small diffusivity ε 2 of the activator component. By(More)
For certain singularly perturbed two-component reaction-diffusion (RD) systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behavior in terms of some parameter β in the system. For some such systems, such as the Gray-Scott model, a spike self-replication behavior is observed as a parameter varies across the(More)
For a dissipative variant of the two-dimensional Gross-Pitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the Thomas-Fermi radius. The instability occurs as a result of a(More)
Spot patterns, whereby the activator field becomes spatially localized near certain dynamically-evolving discrete spatial locations in a bounded multi-dimensional domain, is a common occurrence for two-component reaction-diffusion (RD) systems in the singular limit of a large diffusivity ratio. In previous studies of 2-D localized spot patterns for various(More)