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

TL;DR

The paper develops a weakly nonlinear framework to analyze pattern formation in structured PDEs with base states that are spatially uniform yet exponentially localized in the internal state, by combining WKBJ asymptotics with a Stokes-phenomenon analysis to resolve Dirac-delta limits as $\varepsilon\to0^+$. Focusing on a quorum-sensing–driven, nonlocal bacterial model, it derives a pitchfork-type amplitude equation near the instability, with an explicit coefficient $\mu$ determining whether the bifurcation is subcritical or supercritical. The authors compute inner- and outer-region corrections, derive solvability conditions, and connect these to nonlocal integral terms via Laplace methods, yielding quantitative predictions for nonuniform steady states and phase separation that agree with numerical continuations. The approach is demonstrated to be broadly applicable to other structured models with localized internal states and nonlocal couplings, offering a systematic route to capture weakly nonlinear patterning beyond traditional analyses. Overall, the work provides a rigorous, general toolkit for predicting the onset and nature of patterns in complex, structured populations where base states concentrate in internal variables.

Abstract

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 framework that extends classical tools to structured PDE models in settings where the base state is spatially uniform, but exponentially localized in the structured variable. 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. To demonstrate our method, we consider a chemically structured (nonlocal) model of motile bacteria that interact through quorum sensing. For this example, 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 broadly applicable.

Figures (3)

• Figure 1: a): sample bifurcation diagrams when the pitchfork bifurcation is supercritical and subcritical. Local nontrivial branches (black curves) from weakly nonlinear theory are shown near the pitchfork bifurcation. Red and blue curves with $\Delta\rho>0$ are computed from numerical solutions; the rest is from theory. Steady states are computed via continuation, while stability is inferred from time-dependent calculations. b) and c): representative steady state profile for $\mathcal{D}_*'\approx -1.5$, and $\varepsilon=0.002$ (green circle, top left panel), computed numerically. Systematic comparison of theory with numerical results for $\rho^*=0.65$ and $\varepsilon\to0^+$. Relative errors in the leading order term of $\mathcal{D}_{0*}'$ in \ref{['eqn:bif_condition']} d) and the coefficient $b$ in \ref{['eqn:b_def']} e). We plot a straight line passing through the origin to demonstrate that the error is $\mathcal{O}(\varepsilon)$. The numerical value of $b$ is determined by fitting a square root to the numerically computed points on the nontrivial branch near the pitchfork bifurcation.
• Figure 2: Phase diagram showing the order parameter $\Delta\rho/\rho^*$ in the phase-separated state ($\Delta\rho=0$ for the uniform solution). The yellow curve (Eq. \ref{['eqn:bif_condition1']}) traces out the bifurcation point; the spatially uniform state is linearly unstable above this curve and stable below. The bifurcation becomes subcritical as $\rho^*$ is decreased through $\rho^*\approx0.265$, defined by $\mu=0$ in \ref{['eqn:wna_sub_super_cond']} (yellow star). The area wherein $\Delta\rho>0$ below the yellow curve denotes the region where phase separation is dynamically accessible (via finite amplitude perturbation), but the uniform state is linearly stable.
• Figure 3: Numerically computed, non-uniform, steady state profile $\rho(x)$ and slices of $n(x,u)$ at fixed $x$, $(\rho^*,\mathcal{D}_*')=(0.65,-2)$. As $\varepsilon\to0^+$, the distribution of internal states remains regular where spatial gradients are order one, but may become singular where spatial gradients are small.