Some theoretical results for a bisexual Galton-Watson process with population-size dependent mating (Q1781648)

The authors introduce the bisexual Galton-Watson process with population-size-dependent mating (BPSDM) as the two-type sequence \(\{(F_n, M_n)\}_n\) of the form \[ Z^*_0= N,\quad (F^*_{n+1}, M^*_{n+1})= \sum^{Z^*_n}_{i=1} (f_{ni}, m_{ni}),\quad Z^*_{n+1}= L_{Z^*_n}(F^*_{n+1}, M^*_{n+1}),\quad n= 0,1,\dots, \] where \((f_{ni}, m_{ni})\), \(i= 1,\dots,\infty\), is a sequence of i.i.d. nonnegative integer valued random variables and \(\{L_k\}\) is a sequence of nonnegative real functions, integer-valued on integers and such that \(L_k(x, y)\leq xy\). The authors consider super-additive mating functions. Stochastic monotonicity properties of the BPSDM are shown and the p.g.f. of the accumulated progeny is investigated.