From Turing Patterns to Spike Self-Replication and Nucleation in a Growing-Domain Tulip Breaking Virus Model
A recently proposed reaction-diffusion model for the tulip breaking virus (TBV) has been shown to generate striped pigmentation patterns on growing tulip petals. In this paper, we investigate the formation of localized spike patterns in this model in the asymptotic limit of small viral diffusivity. It is known that two distinct bifurcation mechanisms can occur which generate new spikes as the domain half-length $L$ slowly increases: spike self-replication and spike nucleation. Self-replication occurs through a saddle-node bifurcation of the inner spike core problem, whereas nucleation is associated with a fold of the nonlinear outer problem. Using matched asymptotic expansions, we derive critical conditions for these mechanisms and construct phase diagrams that predict which mechanism occurs first as $L$ increases, or whether nucleation is absent. We further combine these results with a Turing instability analysis of the homogeneous steady state to obtain a phase diagram of pattern-forming mechanisms. An interesting finding is that, for the TBV model, the Turing-unstable region lies entirely within the parameter regime where spike nucleation is possible. These results provide further insight into the mechanisms underlying the formation and evolution of pigmentation patterns in virus-infected tulips.