Relationships between stochastic and deterministic population models

The infinitesimal parameters of a variety of Markov population models can be written as q k (N) ,k+l=Nℝl(N-1k) where k,l ∈ ℤdand N is a parameter which is of the same order of magnitude as the population size. Under appropriate conditions, a family of Markov processes {XN} with these parameters satisfies \( \mathop{{\lim }}\limits_{{N \to \infty }} {N^{{ - 1}}}{X_N}(t) = X(t) \), in probability where X(t) is a solution of the differential equation \( \mathop{{X = F(X) \equiv \sum\limits_l {l{\mathbb{R}_l}} (X)}}\limits^o \).

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