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We consider a spatial multi-type branching model in which individuals migratein geographic space according to random walks and reproduce according to astate-dependent branching mechanism which can be sub-, super- or criticaldepending on the local intensity of individuals of the different types. Themodel is a Lotka-Volterra type model with a spatial component and is related totwo models studied in \cite{BlathEtheridgeMeredith2007} as well as to earlierwork in \cite{Etheridge2004} and in \cite{NeuhauserPacala1999}. Our main focusis on the diffusion limit of small mass, locally many individuals and rapidreproduction. This system differs from spatial critical branching systems sinceit is not density preserving and the densities for large times do not depend onthe initial distribution but mainly on the carrying capacities.  We prove existence of the infinite particle model and the system ofinteracting diffusions as solutions of martingale problems or systems ofstochastic equations. In the exchangeable case in which the parameters are nottype dependent we show uniqueness of the solutions. For that purpose weestablish a new exponential duality.

Multi-type spatial branching models for local self-regulation I: Construction and an exponential duality