We are interested in modeling some two-level population dynamics, resulting
from the interplay of ecological interactions and phenotypic variation of
individuals (or hosts) and the evolution of cells (or parasites) of two types
living in these individuals. The ecological parameters of the individual
dynamics depend on the number of cells of each type contained by the individual
and the cell dynamics depends on the trait of the invaded individual. Our
models are rooted in the microscopic description of a random (discrete)
population of individuals characterized by one or several adaptive traits and
cells characterized by their type. The population is modeled as a stochastic
point process whose generator captures the probabilistic dynamics over
continuous time of birth, mutation and death for individuals and birth and
death for cells. The interaction between individuals (resp. between cells) is
described by a competition between individual traits (resp. between cell
types). We look for tractable large population approximations. By combining
various scalings on population size, birth and death rates and mutation step,
the single microscopic model is shown to lead to contrasting nonlinear
macroscopic limits of different nature: deterministic approximations, in the
form of ordinary, integro- or partial differential equations, or probabilistic
ones, like stochastic partial differential equations or superprocesses. The
study of the long time behavior of these processes seems very hard and we only
develop some simple cases enlightening the difficulties involved