A Weakly Nonlinear Theory for Pattern Formation in a Structured Model with Localized Solutions

Structured models, such as PDEs structured by age or phenotype, provide a setting to study pattern formation in heterogeneous populations. Classical tools to quantify the emergence of patterns, such as linear and weakly nonlinear analyses, pose significant mathematical challenges for these models due to sharply peaked or singular steady states. Here, we present a weakly nonlinear theory for a chemically structured (nonlocal) model which admits a base state that is spatially uniform, but exponentially localized with respect to the structured variable. Physically, the model represents a system of motile bacteria that interact through quorum sensing. Our approach utilizes WKBJ asymptotics and an analysis of the Stokes phenomenon to systematically resolve the solution structure in the limit where the steady state tends to a Dirac-delta function. Our analysis yields an amplitude equation that governs the solution dynamics near a linear instability and predicts a pitchfork bifurcation. From the amplitude equation, we deduce an effective parameter grouping whose sign determines whether the pitchfork bifurcation is subcritical or supercritical. Although we demonstrate our framework for a specific example, our techniques are applicable more generally to other structured models with sharply peaked base states.