Moment closure techniques for stochastic models in population biology

Continuous-time birth-death Markov processes serve as useful models in population biology. In this paper, we present a procedure for constructing approximate stochastic models for these processes. This is done by representing the population of a species as the continuous state of a stochastic hybrid system (SHS). This SHS is characterized by reset maps that account for births and deaths, transition intensities that correspond to the birth-death rates, and trivial continuous dynamics. It has been shown that for this type of SHS the statistical moments of the continuous state evolve according to an infinite-dimensional linear ordinary differential equation (ODE). However, for analysis purposes it is convenient to approximate this infinite-dimensional linear ODE by a finite-dimensional nonlinear one. This procedure generally approximates some higher-order moments by a nonlinear function of lower-order moments and it is called moment closure. We obtain moment closures by matching time derivatives of the infinite-dimensional ODE with its finite-dimensional approximation at some time t₀. This guarantees a good approximation, at-least locally in time. We provide explicit formulas to construct these approximations and compare this technique with alternative moment-closure methods available in the literature. This comparison takes into account both the transient and the steady-state regimens