## Without Population Structure

For a start consider the case without any population structure and with time running in steps, that is, generations do not overlap and newborns are stochastically equal (i.e., parents do not bequeath any properties other than long-term heritable traits). Then

R(E(t)) = (individual offsprmg numbers)individuals|E(t) [6]

Formula [4] also applies for unstructured populations in continuous time described by dn - r(E(t))n [7]

(Note that here r(E(t)) is itself an average over the stochastic behavior of individuals, which themselves just give birth and die.)

By letting dE} denote the fraction of time E(t) spends in a small set {dE} , the time average in [4] can be re-expressed as p =y r (E)p{dE} [8]

Formula [8] is particularly useful in the context of stochastic population models where it may be relatively easy to calculate p.

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