Bisexual Galton-Watson Branching Processes in Random Environments

Ma, Shibin

doi:10.1007/s10255-006-0317-4

Cited by 15 publications

(1 citation statement)

References 18 publications

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“…Since Daley's work, extinction conditions have been studied for models with a more general family of superadditive mating functions (see for instance [Hul82], [Bru84], [DHT86]) and, in the last decades, results on the limit behaviour of this kind of processes were obtained (see for example [AR96], [AR02], [GM96], [GM97]). From these works, new models of two-sex populations have been developed, such as processes in random or varying environment ( [Ma06], [MM09], [MMR03]), processes with immigration ([GMM00], [GM96], [MX06]), processes with mating function depending on the number of couples ( [MMR02], [MMR06], [XW05]) and more recently, processes with random mating ([JMM17], [MM19]). The interested reader can also consult the surveys of Hull [Hul03], Alsmeyer [Als02] and Molina [Mol10] for a wide description of the work accomplished on this family of processes.…”

Section: Example 1 Some Examples Of Mating Functions Arementioning

confidence: 99%

The Multi-type Bisexual Galton-Watson Branching Process

Fritsch,

Villemonais,

Zalduendo

2022

Preprint

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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.

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