Spine for interacting populations and sampling

We consider some Markov jump processes which model structured populations with interactions via density dependence. We propose a Markov construction involving a distinguished individual (spine) which allows us to describe the random tree and random sample at a given time via a change of probability. This spine construction involves the extension of the type space of individuals to include the state of the population. The jump rates off the spine individual can also be modified. We exploit this approach to study issues concerning population dynamics. For single type populations, we derive the phase diagram of a growth fragmentation model with competition as well as the growth of the size of transient birth and death processes which permit multiple births. We also describe the ancestral lineages of a uniform sample in multitype populations.