Stability and Dynamics of Invasion Fronts in Spatially Extended Systems

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This project concerns the development of mathematical techniques to predict and explain features of front propagation into unstable states with applications in physics, biology and chemistry. Unstable states arise frequently in applied problems as system parameters are changed or new species are introduced into the problem. The subsequent dynamics are often dominated by the formation of moving interfaces called invasion fronts that propagate into the unstable state at fixed speed and select some secondary state in their wake. This proposal is built around expanding knowledge of these processes with a specific goal of elucidating mechanisms leading to the emergence of invasion fronts and then leveraging this knowledge to make predictions in actual systems of interest. One portion of this work will develop new mathematical approaches to study the effect of localized forcing on the spreading speed of interfaces in partial differential equation (PDE) models. The second portion will develop predictions and general principles for the dynamics of instabilities spreading over complex networks. Two complementary avenues of research are proposed. One area of proposed work concerns stability of invasion fronts with a focus on understanding the role of external forcing in the selection of the inter-facial velocity and subsequent dynamics. Since invasion fronts propagate into unstable states, stability analysis requires perturbations of these fronts to be sufficiently localized. This proposal will study problems where the forcing function is also localized, but not sufficiently localized to fit into the stability framework of the homogeneous equation. Success in carrying out the projects described will lead to new mathematical approaches to the study of stability of traveling waves and will clarify and explain some novel phenomena that will be of interest to researchers working in applied problems. A second area of proposed work concerns invasion fronts where space is taken to be a complex network with local dynamics occurring at each node with coupling between nodes via a graph Laplacian. The goal here is to leverage knowledge of front propagation in the PDE setting to make concrete predictions related to the spread of instabilities over networks. Applications include the spread of global epidemics and that of invasive species in discrete environments. Successful completion of this work will provide researchers working in this area with new tools to estimate arrival times, provide a rigorous theoretical framework by which to understand these spreading phenomena and provide insight into the way features of complex networks influence arrival times of invasions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.