The aim of this paper is twofold. First we analyze the sequence of intensity measures of a spatial branching point process arising in a multiple-target tracking context. We study its stability properties, characterize its long time behavior, and provide a series of weak Lipschitz-type functional contraction inequalities. Second we design and analyze an original particle scheme to approximate numerically these intensity measures. Under appropriate regularity conditions, we obtain uniform and nonasymptotic estimates and a functional central limit theorem. To the best of our knowledge, these are the first sharp theoretical results available for this class of spatial branching point processes.