From classical to operatorial models

Mathematical models for the collective dynamics of interacting and spatially distributed populations find applications in several contexts (biology, ecology, social sciences). Their formulation depends primarily on the (continuous or discrete) description of the space. Reaction-diffusion equations have been widely used in bioecology (morphogenesis, migration of biological species, tumor growth, neuro-degenerative diseases) and in the social sciences (diffusion of opinions or decisionmaking processes), and exhibit complex behaviors (propagation of oscillatory phenomena, pattern formation caused by instability). A reaction–diffusion system exhibits diffusion-driven instability, sometimes called Turing instability, if the homogeneous steady state is stable to small perturbations in the absence of diffusion but unstable to small spatial perturbations when diffusion is present. In this thesis, we move from this classical approach, considering a so called crimo-taxis model (Epstein, 1997), and proposing two variants (Inferrera et al., 2022) enabling us to study the formation of some patterns due to instability driven by self- and cross-diffusion terms, to operatorial models built by means of some techniques typical of quantum mechanics (see Bagarello, 2012; Bagarello, 2019). The leading idea in this approach relies on the evidence, shown in the last fifteen years in several applications, that the operatorial framework provides useful tools for describing the interactions occurring within macroscopic systems. Therefore, three applications of the operatorial formalism are discussed: 1)an operatorial version of crimo-taxis model; 2)a model where two populations spatially distributed in a one–dimensional domain compete both locally and nonlocally and are able to migrate (Inferrera and Oliveri, 2022); 3) a model of a finite number of agents subjected both to cooperative and competitive interactions (Gorgone, Inferrera, and Oliveri, 2022).