On stochastic differential equations and equilibrium distribution: a conditional moment approach

Abstract

In the recent past there have been several attempts to obtain the equilibrium distribution of multiple populations and their moments in the context of some biological or ecological processes (e.g., Matis and Kiffe in Biometrics 52:155, 1996; Matis and Kiffe in Environ Ecol Stat 9:237, 2002; Renshaw in J Math Appl Med Biol, 15:1, 1998). In particular, the method of cumulant truncation (Matis and Kiffe in Biometrics 52:155, 1996) is a pioneering work in this field. However it requires solving a large number of cumulant functions even in the case of two simultaneous differential equations. Besides the solutions are approximate and depend on the precision of the software. Renshaw (Math Biosci 168:57, 2000) provided a nice extension of the univariate truncated saddle point procedure to multivariate scenarios. But this approach involves a multivariate Newton-Raphson type iterative algorithm whose performance and convergence are critically dependent on the choice of the initial values. In the present paper we propose a new and simple approach to obtain the equilibrium distribution of populations and their conditional moments in a system of differential equations of any dimension. Our proposed method, which is a natural extension of the classical variational matrix approach, has several advantages which are discussed in detail in the paper; among other things it includes the derivation of additional conditions which can be interpreted as environmental surrogates.