Density-dependent Markov population processes

A sequence (XN)N≥0 of Markov processes on the k-dimensional integer lattice (Z+)k is called density dependent if XN has transition rates of the form X → X+j at rate Ngj(N-1X), where the gj are independent of N. Many phenomena of practical interest, such as epidemics, learning processes, chemical reactions and competition between species, can naturally be described in this way, with the components of XN denoting the counts of the k populations being studied. The mathematical problem is to find approximations to the processes, especially when N is large, which enable one to describe simply the way in which they evolve.

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