Two-Sex Branching Interacting Particle Systems and Related Limit Equation

We construct a two-sex branching interacting particle systems as the solutions of jump-type stochastic integral equations, in which a particle can mate with a heterosexual particle randomly. The number of their offspring is a random variable determined by a generating function which is depends on the particles' traits and the current system. We prove that under appropriate conditions, the renormalization of this branching interacting particle system converges to a measure valued function which satisfies a specific nonlinear ordinary differential equation. Finally, we obtain a nonlinear ordinary differential equation system to describe the distribution of the traits of the two subpopulation.