  We study a system of particles in a two-dimensional geometry that move
according to a reinforced random walk with transition probabilities dependent
on the solutions of reaction-diffusion equations for the underlying fields. A
birth process and a history-dependent killing process are also considered. This
system models tumor-induced angiogenesis, the process of formation of blood
vessels induced by a growth factor released by a tumor. Particles represent
vessel tip cells, whose trajectories constitute the growing vessel network. New
vessels appear and may fuse with existing ones during their evolution. Thus,
the system is described by tracking the density of active tips, calculated as
an ensemble average over many realizations of the stochastic process. Such
density satisfies a novel discrete master equation with source and sink terms.
The sink term is proportional to a space-dependent and suitably fitted killing
coefficient. Results are illustrated studying two influential angiogenesis
models.
