We propose a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic local PDE system, for the population density and the chemical field. The main novelty lies in the transport term of the population density, which depends on the second-order derivatives of the chemical field. This term is derived as an anticipation-reaction steering mechanism of an infinitesimally small ant as its size approaches zero. We establish global-in-time existence and uniqueness for the model, as well as the propagation of regularity from the initial data. Then, we build a numerical scheme and present various examples that provide hints of trail formation.
After presenting a brief introduction to chemotaxis and its similarities with mean-field games systems, we will derive the PDE model, provide a sketch of the proof for the main results, and introduce a numerical scheme for the McKean-Vlasov equation.
