论文标题
多型双性恋Galton-Watson分支过程
The Multi-type Bisexual Galton-Watson Branching Process
论文作者
论文摘要
在这项工作中，我们研究了具有有限数量类型的双性恋Galton-Watson工艺，其中女性和男性根据“交配函数”与不同类型的夫妇相交。我们假设此功能是超级增强性的，这简单地暗示了两组女性和男性将组建大量的夫妻在一起而不是分开。在没有线性繁殖操作员的情况下，这是了解无性疾病中模型行为的关键，我们构建了凹面复制操作员并使用凹入的perron-frobenius理论来确保特征的存在。使用此工具，我们找到了几乎确定灭绝以及大量定律的必要条件。最后，我们通过识别超级马特宁格（Supermartingale）来研究重新恢复过程的长期融合，并提供足够的条件，以确保$ l^1 $的收敛到非分类限制。
In this work we study the bisexual Galton-Watson process with a finite number of types, where females and males mate according to a ''mating function'' and form couples of different types. We assume that this function is superadditive, which in simple words implies that two groups of females and males will form a larger number of couples together rather than separate. In the absence of a linear reproduction operator which is the key to understand the behaviour of the model in the asexual case, we build a concave reproduction operator and use a concave Perron-Frobenius theory to ensure the existence of eigenelements. Using this tool, we find a necessary and sufficient condition for almost sure extinction as well as a law of large numbers. Finally, we study the almost sure long-time convergence of the rescaled process through the identification of a supermartingale, and we give sufficient conditions to ensure a convergence in $L^1$ to a non-degenerate limit.
