Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purpose, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, positive and real, while the derivative matching guarantees a good approximation, at-least locally in time. Moreover, the accuracy of the approximation can be improved by increasing the order of the approximate model.To the best of our knowledge, this paper is the first to propose a systematic procedure to construct moment closure functions of arbitrary order that guarantee biologically meaningful equilibria. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching. However, for certain ranges of parameter models, moment closure functions that lack the separability property lead to biologically meaningless scenarios of imaginary and even stable negative steady-states of the closed moment equations.