| Abstract: |
| We introduce a new chemotaxis model 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 second-order derivatives of the chemical field. This term is derived as the limit of an anticipation-reaction mechanism for an infinitesimally small ant. We establish global existence and uniqueness of solutions, as well as the propagation of the regularity of the initial data. We then analyze the long-time behavior of the system: we prove the existence of the 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. |
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