SPIKE DYNAMICS IN 2-COMPONENT AND 3-COMPONENT REACTION DIFFUSION SYSTEMS

Abstract

In this thesis, we study spike dynamics and stability in different reaction-diffusion systems. Since localized patterns are "far-from-equilibrium" structures, the classical Turing-type stability analysis is not applicable. Instead, we apply the method of matched asymptotic expansions and nonlocal eigenvalue problems to analyze these singular perturbed PDEs. In the first part of the thesis, we investigate an SIRS model with spatial diffusion and nonlinear incidence rates. We show that for small diffusion rate of the infected class, the infected population tends to be highly localized. We then study three distinct destabilization mechanisms, as well as a transition from localized spikes to plateau solutions. In all cases, the stability thresholds are computed asymptotically and are verified by numerical experiments. In the second part we study the effect of noise on spike dynamics for the Gierer-Meinhardt model. When spatial-temporal noise is introduced in the activator equation, we derive a stochastic ODE that describes the motion of a single spike. For small noise level, the spike can deviate from the domain center but remains "trapped" within a subinterval. For larger noise levels, the spike undergoes large excursions that eventually collide with the domain boundary. We then derive the expected time for the spike to collide with the boundary. In third part we propose an extension of the Klausmeier model to two plant species that consume water at different rates. We are interested in how the competition for water affects stability of plant patches. We find a finite range of precipitation rate for which two species can co-exist. Outside of that range, the frugal species outcompetes the thirsty species. There is sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. In the end, an analysis is undertaken of the formation and stability of localized patterns in the Schnakenberg model with source terms in both the activator and inhibitor fields. Single-spike patterns are constructed and we then derive the nonlocal eigenvalue problem and study a Hopf bifurcation in the amplitudes of the spike.