  There is studied an infinite system of point entities in $\mathbb{R}^d$ which
reproduce themselves and die, also due to competition. The system's states are
probability measures on the space of configurations of entities. Their
evolution is described by means of a BBGKY-type equation for the corresponding
correlation (moment) functions. It is proved that: (a) these functions evolve
on a bounded time interval and remain sub-Poissonian due to the competition;
(b) in the Vlasov scaling limit they converge to the correlation functions of
the time-dependent Poisson point field the density of which solves the kinetic
equation obtained in the scaling limit from the equation for the correlation
functions. A number of properties of the solutions of the kinetic equation are
also established.
