Oscillatory Instabilities of a One-Spot Pattern in the Schnakenberg Reaction-Diffusion System in $3$-D Domains

For an activator-inhibitor reaction-diffusion system in a bounded three-dimensional domain $\Omega$ of $O(1)$ volume and small activator diffusivity of $O(\varepsilon^2)$, we employ a hybrid asymptotic-numerical method to investigate two instabilities of a localized one-spot equilibrium that result from Hopf bifurcations: an amplitude instability leading to growing oscillations in spot amplitude, and a translational instability leading to growing oscillations of the location of the spot's center $\mathbf{x}_0 \in \Omega$. Here, a one-spot equilibrium is one in which the activator concentration is exponentially small everywhere in $\Omega$ except in a localized region of $O(\varepsilon)$ about $\mathbf{x}_0 \in \Omega$ where its concentration is $O(1)$. We find that the translation instability is governed by a $3\times 3$ nonlinear matrix eigenvalue problem. The entries of this matrix involve terms calculated from certain Green's functions, which encode information about the domain's geometry. In this nonlinear matrix eigenvalue system, the most unstable eigenvalue determines the oscillation frequency at onset, while the corresponding eigenvector determines the direction of oscillation. We demonstrate the impact of domain geometry and defects on this instability, providing analytic insights into how they select the preferred direction of oscillation. For the amplitude instability, we illustrate the intricate way in which the Hopf bifurcation threshold $\tau_H$ varies with a feed-rate parameter $A$. In particular, we show that the $\tau_H$ versus $A$ relationship possesses two saddle-nodes, with different branches scaling differently with the small parameter $\varepsilon$. All asymptotic results are confirmed by finite elements solutions of the full reaction-diffusion system.