Journal of Mathematical Sciences, Vol. 121, No. 5, 2004 SOME THEORETICAL RESULTS FOR A BISEXUAL GALTON–WATSON PROCESS WITH POPULATION-SIZE DEPENDENT MATING* M. Molina, M. Mota, and A. Ramos (Extremadura, Spain) UDC 519.2 1. Introduction Daley in [5] introduced the bisexual Galton–Watson branching process (BP) as a two-type branching model {(F n ,M n )} ∞ n=1 defined in the recursive form (F n+1 ,M n+1 )= Zn  i=1 (f ni ,m ni ), Z n+1 = L(F n+1 ,M n+1 ), n =0, 1,..., (1) where Z 0 = N , N being a positive integer, and the empty sum is considered to be (0, 0). Intuitively, f ni (m ni ) represents the number of females (males) produced by the ith mating unit in the nth generation with {(f ni ,m ni ),i =1, 2,... ; n = 0, 1,... } a sequence of independent and identically distributed (i.i.d.), nonnegative and integer-valued random variables, and the mating function L: R + × R + → R + is assumed to be monotonic nondecreasing in each argument and integer valued on integer arguments and such that L(x, y) ≤ xy. Consequently, F n (M n ) is the number of females(males) in the nth generation, which form Z n = L(F n ,M n ) mating units. These mating units reproduce independently through the same offspring probability distribution for each generation. A BP is said to be superadditive when its mating function L is superadditive, i.e., if for any integer n ≥ 2 the following is satisfied: L  n  i=1 x i , n  i=1 y i  ≥ n  i=1 L(x i ,y i ), x i ,y i ∈ R + , i =1,...,n. The BP has received attention in the literature. The extinction problem has been studied in [1, 2, 4–6, 15, 16]. The limit behavior has been investigated in [3, 8, 9] and some inferential questions have been considered in [7, 13, 17]. Recently, some modified bisexual Galton–Watson models for description of other possible practical situations have been introduced (see [11, 12, 14, 18]). In particular, in [18], a bisexual Galton–Watson process whose mating function depends on the population size has been defined and some results about its almost sure (a.s.) extinction established. This model will be interesting for description of bisexual populations where it is reasonable to allow an individual’s mating behavior to depend on the population size, e.g., it might seem conceivable that, by environmental or social changes or by other factors, the same number of females and males gives rise to different numbers of mating units in different generations. In this paper, we provide some probabilistic results related to this one bisexual process. In Sec. 2, the probability model is described and some basic concepts are introduced. Section 3 is devoted to looking at some stochastic monotony properties about the considered probability model and, finally, results about its accumulated progeny until a certain generation are investigated. 2. The Probability Model From model (1), Molina and others (see [18]) have introduced the bisexual Galton–Watson process with population- size dependent mating (BPSDM) as the two-type sequence {(F * n ,M * n )} ∞ n=1 defined in the form (F * n+1 ,M * n+1 )= Z * n  i=1 (f ni ,m ni ), Z * n+1 = L Z * n (F * n+1 ,M * n+1 ), n =0, 1,..., (2) where Z * 0 = N , {L k } ∞ k=1 is a sequence of mating functions, i.e., for every k, L k is monotonic nondecreasing in each argument and integer valued on integer arguments and such that L k (x, y) ≤ xy. * Partially supported by the Plan Nacional de Investigaci´ on Cient ´ ifica, Desarrollo e Innovaci´ on Tecnol´ogica (grant No. BFM2000-0356) and the Consejer ´ ia de Educaci´ on, Ciencia y Tecnolog ´ ia de la Junta de Extremadura y el Fondo Social Europeo (grant No. IPR00A056). Proceedings of the Seminar on Stability Problems for Stochastic Models, Varna, Bulgaria, 2002, Part I. 1072-3374/04/1215-2681 c  2004 Plenum Publishing Corporation 2681