  Spatial birth-and-death processes with time dependent rates are obtained as
solutions to certain stochastic equations. The existence, uniqueness,
uniqueness in law and the strong Markov property of unique solutions are proven
when the integral of the birth rate over $\mathbb{R} ^ \mathrm{d}$ grows not
faster than linearly with the number of particles of the system. Martingale
properties of the constructed process provide a rigorous connection to the
heuristic generator. We also study pathwise behavior of an aggregation model.
The probability of extinction and the growth rate of the number of particles
conditioning on non-extinction are estimated.
