Scaling Limits of Bisexual Galton-Watson Processes

Bisexual Galton-Watson processes are discrete Markov chains wherereproduction events are due to mating of males and females. Owing to thisinteraction, the standard branching property of Galton-Watson processes islost. We prove tightness for conveniently rescaled bisexual Galton-Watsonprocesses, based on recent techniques developed by Bansaye, Caballero andMéléard. We also identify the possible limits of these rescaledprocesses as solutions of a stochastic system, coupling two equations throughsingular coefficients in Poisson terms added to square roots as coefficients ofBrownian motions. Under some additional integrability assumptions, pathwiseuniqueness of this limiting system of stochastic differential equations andconvergence of the rescaled processes are obtained. Two examples correspondingto mutual fidelity are considered.