Abstract: We present a well-posedness theory for weak measure solutions of the Cauchy problem for a system of two nonlocal interaction equation, that can be interpreted as a continuum model for interacting species. We provide globally in time existence, uniqueness and stability when the system presents a certain symmetry in the interaction between the two species, using the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. We show the finite-time total collapse of the solution for compactly supported initial measures. In addition we prove existence in the general case using variational steepest descent approximation schemes, showing the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Wasserstein distance.