Systems describing the long-range interaction between individuals have
attracted a lot of attention in the last years, in particular in relation with
living systems. These systems are quadratic, written under the form of
transport equations with a nonlocal self-generated drift. We establish the
localisation limit, that is the convergence of nonlocal to local systems, when
the range of interaction tends to 0. These theoretical results are sustained by
numerical simulations. The major new feature in our analysis is that we do not
need diffusion to gain compactness, at odd with the existing literature. The
central compactness result is provided by a full rank assumption on the
interaction kernels. In turn, we prove existence of weak solutions for the
resulting system, a cross-diffusion system of quadratic type