We introduce a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic PDE system describing the evolution of the population density and the chemical signal. The key novelty lies in the transport term for the population, which depends on the second-order derivatives of the chemical field. This term is derived as a limiting anticipation-reaction mechanism for an infinitesimally small ant. We establish global existence and uniqueness of solutions, as well as the propagation of regularity of the initial data. We then analyze the long-time behavior of the system: we prove the existence of a compact global attractor and show that the homogeneous steady state becomes nonlinearly unstable under an inviscid instability criterion. Additionally, we provide a lower bound on the dimension of the attractor. Conversely, we prove that for sufficiently small interaction strength, the homogeneous steady state is globally asymptotically stable. Finally, we present several numerical simulations illustrating the model's dynamics.

References: Curvature in chemotaxis: A model for ant trail pattern formation, Charles Bertucci, Matthias Rakotomalala, Milica Tomasevic Existence and dimensional lower bound for the global attractor of a PDE model for ant trail formation, Matthias Rakotomalala, Oscar de Wit