Non-reciprocal interactions enhance heterogeneity

We study a process of pattern formation for a generic model of species anchored to the nodes of a network where local reactions take place, and that experience non-reciprocal non-local long-range interactions, encoded by the network directed links. By assuming the system to exhibit a stable homogeneous equilibrium whenever only local interactions are considered, we prove that such equilibrium can turn unstable once suitable non-reciprocal non-local long-range interactions are allowed for. Stated differently, we propose sufficient conditions allowing for patterns to emerge by using a non-symmetric coupling, while initial perturbations about the homogeneous equilibrium always fade away by assuming reciprocal coupling, namely the latter is stable. The instability, precursor of the emerging spatio-temporal patterns, can be traced back, via a linear stability analysis, to the complex spectrum of an interaction non-symmetric Laplace operator. The proposed theory is then applied to several paradigmatic dynamical models largely used in the literature to study the emergence of patterns or synchronisation. Taken together, our results pave the way for the understanding of the many and heterogeneous patterns of complexity found in ecological, chemical or physical systems composed by interacting parts, once no diffusion takes place.