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Existence of spot and lane stationary solutions for an ant active matter PDE model

Matthias Rakotomalala, Oscar de Wit

TL;DR

The paper proves the existence of two families of nontrivial stationary densities for an ant-active-matter PDE, namely spots and lanes, by a Lyapunov–Schmidt reduction around non-Pythagorean wave numbers and a careful partial Fourier decomposition. It characterizes the bifurcation curves in terms of explicit leading-order coefficients, demonstrates their dependence on the anticipation parameter $\tau$, and establishes a stability dichotomy for the first bifurcating branch: spots are locally dynamically stable for small $\tau$ while lanes are unstable. The work provides a rigorous mathematical foundation for self-organization and lane formation in ant colonies, linking the bifurcation structure to physically interpretable patterns and symmetry constraints. Overall, the study advances non-equilibrium PDE analysis in active-matter models by delivering explicit, rigorous constructions of coexisting non-homogeneous stationary states and their stability properties.

Abstract

This paper studies the existence of multiple non-trivial stationary solutions of a partial differential equation (PDE) model introduced in [3], motivated by collective ant behavior. Previous work suggested the presence of two types of non-trivial stationary solutions for this PDE system: spot and lane solutions. In this paper, we establish the existence of these families of solutions along a bifurcation sequence as the interaction strength grows, with progressively increasing numbers of clusters and parallel lanes, respectively. Finally, we show that, for small values of the anticipation parameter, the first bifurcating spot solutions are locally dynamically stable, while the lane solutions are unstable.

Existence of spot and lane stationary solutions for an ant active matter PDE model

TL;DR

The paper proves the existence of two families of nontrivial stationary densities for an ant-active-matter PDE, namely spots and lanes, by a Lyapunov–Schmidt reduction around non-Pythagorean wave numbers and a careful partial Fourier decomposition. It characterizes the bifurcation curves in terms of explicit leading-order coefficients, demonstrates their dependence on the anticipation parameter , and establishes a stability dichotomy for the first bifurcating branch: spots are locally dynamically stable for small while lanes are unstable. The work provides a rigorous mathematical foundation for self-organization and lane formation in ant colonies, linking the bifurcation structure to physically interpretable patterns and symmetry constraints. Overall, the study advances non-equilibrium PDE analysis in active-matter models by delivering explicit, rigorous constructions of coexisting non-homogeneous stationary states and their stability properties.

Abstract

This paper studies the existence of multiple non-trivial stationary solutions of a partial differential equation (PDE) model introduced in [3], motivated by collective ant behavior. Previous work suggested the presence of two types of non-trivial stationary solutions for this PDE system: spot and lane solutions. In this paper, we establish the existence of these families of solutions along a bifurcation sequence as the interaction strength grows, with progressively increasing numbers of clusters and parallel lanes, respectively. Finally, we show that, for small values of the anticipation parameter, the first bifurcating spot solutions are locally dynamically stable, while the lane solutions are unstable.
Paper Structure (18 sections, 26 theorems, 233 equations, 3 figures, 2 tables)

This paper contains 18 sections, 26 theorems, 233 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Functional equation
  4. Symmetries of the equation
  5. Statement of the Main Result
  6. Linearized Operator
  7. Partial Fourier decomposition
  8. Spectral Analysis
  9. Analysis of Bifurcation Curves
  10. Lyapunov--Schmidt Reduction
  11. Coefficients Equations
  12. Existence of Spots and Lanes
  13. Stability of the First Supercritical Branch
  14. Analysis of Integral Quantities
  15. Integral Coefficients
  16. ...and 3 more sections

Key Result

Proposition 2.1

Given $f\in H^2_0$, we have the following commutation relation, Furthermore, $\mathrm{R}^{\pi}$ commutes with any of the terms involved in $\mathrm{F}$, and distributes on products, for $f,g \in L^2_0$.

Figures (3)

  • Figure 1: Sub-critical and super-critical bifurcation diagrams.
  • Figure 2: Representation of the Lyapunov-Schmidt functional $\mathrm{\Phi}$ as a vector field, with field lines $\Xi$ and $\Lambda$.
  • Figure 3: Numerical bifurcation diagram for $\tau=0$ on the left and $\tau = 0.5$ on the right. We used a forward-in-time version of the finite volume scheme as in bruna2025convergence. The residuals $\|\mathrm{F}(f,\chi)\|_{L^\infty}$ are of the order $10^{-6}$ or less.

Theorems & Definitions (51)

  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.2
  • Definition 3.1
  • Theorem 3.1
  • Corollary 3.1
  • Proposition 4.1
  • Proposition 4.2
  • ...and 41 more