  We consider a hierarchically structured population in which the amount of
resources an individual has access to is affected by individuals that are
larger, and that the intake of resources by an individual only affects directly
the growth rate of the individual. We formulate a deterministic model, which
takes the form of a delay equation for the population birth rate. We also
formulate an individual based stochastic model, and study the relationship
between the two models. In particular the stationary birth rate of the
deterministic model is compared to that of the quasi-stationary birth rate of
the stochastic model. Since the quasi-stationary birth rate cannot be obtained
explicitly, we derive a formula to approximate it. We show that the stationary
birth rate of the deterministic model can be obtained as the large population
limit of the quasi-stationary birth rate of the stochastic model. This relation
suggests that the deterministic model is a good approximation of the stochastic
model when the number of individuals is sufficiently large.
