Existence and dimensional lower bound for the global attractor of a PDE model for ant trail formation
Matthias Rakotomalala, Oscar de Wit
TL;DR
The paper investigates a nonlinear PDE model for ant trail formation on a torus, coupling a phase-space density $f$ to a chemical field $c$ through a curvature-driven interaction $B_\tau[c]$. It combines semigroup theory, spectral analysis of linearized operators, and nonlinear instability tools to show the existence of a compact global attractor and to characterize stability: nonlinear instability and a lower bound on attractor dimension under a linear-instability condition, and global asymptotic stability when the interaction strength is sufficiently small. The main contributions are a rigorous attractor theory for both parabolic-parabolic and parabolic-elliptic couplings, a detailed linear instability analysis around the homogeneous steady state, and a dimensional lower bound $4k$ on the unstable manifold when the instability inequality $\chi(2\pi k\tau+1) > \lambda(\gamma+4\pi\sigma_c k^2)$ holds with small diffusion parameters. Together, these results provide a mathematical foundation for the emergence of nontrivial ant-trail patterns from the proposed PDE model and quantify how parameter regimes control pattern formation versus stability.
Abstract
We study the asymptotic behavior of a nonlinear PDE model for ant trail formation, which was introduced in [3]. We establish the existence of a compact global attractor and prove the nonlinear instability of the homogeneous steady state under an inviscid instability condition. We also provide a dimensional lower bound on the attractor. Alternatively, we prove that if the interaction parameter is sufficiently small, the homogeneous steady state is globally asymptotically stable.