引用本文: 马儒刚, 王燕伟, 赵涵. 两性别分枝交互粒子系统以及相应的极限方程[J]. 应用概率统计, 2024, 40(4): 543-557.
MA R G, WANG Y W, ZHAO H. Two-sex branching interacting particle systems and related limit equation [J]. Chinese J Appl Probab Statist, 2024, 40(4): 543−557.
 Citation: MA R G, WANG Y W, ZHAO H. Two-sex branching interacting particle systems and related limit equation [J]. Chinese J Appl Probab Statist, 2024, 40(4): 543−557.

## Two-Sex Branching Interacting Particle Systems and Related Limit Equation

• 摘要: 我们通过由泊松随机测度驱动的随机方程构造了一类两性别分枝交互粒子系统, 系统中的粒子可以随机地与异性进行交配, 其后产生的后代个数也是随机的, 后代个数的分布由一个与粒子特征和整个系统有关的母函数决定. 我们证明了在适当的条件下, 这个分枝交互粒子系统的重整化极限是一个确定的测度值函数, 它满足某个特定的非线性常微分方程. 最后我们得到了一个非线性常微分方程组来刻画各性别所属的亚种群的特征分布.

Abstract: 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.

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