A total of 152 shark specimens of the genus Etmopterus were collected in southern Brazil between Chuí (34°20′S) and the Santa Marta Grande Cape (28°30′S). The specimens were caught by bottom-trawl cruises at depths ranging between 100 and 600 m from August to September 2001 and March to April 2002. The species, identified as Etmopterus gracilispinis, E. lucifer and E. bigelowi, occurred only in the south of 31°S at different depth layers. Etmopterus bigelowi occurred in the deep strata from 400–599 m, whereas E. gracilispinis occurred from 300–599 m. Moreover, two sampled E. lucifer males occurred at a depth of 540 m. The hydrographic conditions of temperature and salinity revealed no seasonal differences. The fact that most of the collected specimens were immature is strong evidence of size segregation, suggesting that the adults of these species occur at depths greater than 600 m. Sexual maturity in E. bigelowi occurred at 63–65 cm total length (TL) in females and 60 cm TL in males. All E. gracilispinis specimens were immature. One E. bigelowi female caught in August was pregnant. Birth was confirmed only in March for E. bigelowi and E. gracilispinis with TL at birth of about 17 and 13 cm, respectively.

We consider d-dimensional stochastic processes which take values in (R+)d These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some

Here τ:(R+)d→R +, |x| = σ1d |x(i)|, A {x ∈(R+)d: |x| 1} and T: A→A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Zn|ρnwp. This result is a generalization of the main limit theorem for supercritical branching processes; note, however, that in the present situation both ρ and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

We consider d-dimensional stochastic processes which take values in (R+)d. These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some

Here τ: (R+)d → R+, |x| = Σ1d |x(i)| A = {x ∈ (R+)d: |x| = 1} and T: A → A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Znρ−n = wp. This result is a generalization of the main limit theorem for super-critical branching processes; note, however, that in the present situation both p and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.