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Bisexual branching diffusions

We study the limiting behaviour of large systems of two types of Brownian particles undergoing bisexual branching. Particles of each type generate individuals of both types, and the respective branching law is asymptotically critical for the two-dimensional system, while being subcritical for each individual population. The main result of the paper is that the limiting behaviour of suitably scaled sums and differences of the two populations is given by a pair of measure and distribution valued processes which, together, determine the limit behaviours of the individual populations. Our proofs are based on the martingale problem approach to general state space processes. The fact that our limit involves both measure and distribution valued processes requires the development of some new methodologies of independent interest.