  An individual-based model of an infinite system of point particles in
$\mathbb{R}^d$ is proposed and studied. In this model, each particle at random
produces a finite number of new particles and disappears afterwards. The phase
space for this model is the set $\Gamma$ of all locally finite subsets of
$\mathbb{R}^d$. The system's states are probability measures on $\Gamma$ the
Markov evolution of which is described in terms of their correlation functions
in a scale of Banach spaces. The existence and uniqueness of solutions of the
corresponding evolution equation are proved.
