The effect of slow spatial processes on emerging spatiotemporal patterns

Abstract

We consider a two-component system of evolutionary partial differential equations posed on a bounded domain. Our system is pattern forming, with a small stationary pattern bifurcating from the background state. It is also equipped with a multiscale structure, manifesting itself through the presence of spectrum close to the origin. Spatial processes are associated with long time scales and affect the nonlinear pattern dynamics strongly. To track these dynamics past the bifurcation, we develop an asymptotics-based method complementing and extending rigorous center manifold reduction. Using it, we obtain a complete analytic description of the pattern stability problem in terms of the linear stability of the background state. Through this procedure, we portray with precision how slow spatial processes can destabilize small patterns close to onset. We further illustrate our results on a model describing phytoplankton whose growth is co-limited by nutrient and light. Localized colonies forming at intermediate depths are found to be subject to oscillatory destabilization shortly after emergence, whereas boundary-layer type colonies at the bottom persist. These analytic results are in agreement with numerical simulations for the full model, which we also present.