Browsing by Author "Anderson, DF"

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    A weak trapezoidal method for a class of stochastic differential equations
    (Communications in Mathematical Sciences, 2011-03-01) Anderson, DF; Mattingly, JC
    We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the weak sense. The method is simpler than many second order methods in that it neither requires the construction of iterated It̂o integrals nor the evaluation of any derivatives. The method consists of two steps. In the first an explicit Euler step is used to take a fractional step. The resulting fractional point is then combined with the initial point to obtain a higher order, trapezoidal like, approximation. The higher order of accuracy stems from the fact that both the drift and the quadratic variation of the underlying SDE are approximated to second order. © 2011 International Press.
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    Propagation of Fluctuations in Biochemical Systems, II: Nonlinear Chains
    Anderson, DF; Mattingly, Jonathan Christopher
    We consider biochemical reaction chains and investigate how random external fluctuations, as characterized by variance and coefficient of variation, propagate down the chains. We perform such a study under the assumption that the number of molecules is high enough so that the behavior of the concentrations of the system is well approximated by differential equations. We conclude that the variances and coefficients of variation of the fluxes will decrease as one moves down the chain and, through an example, show that there is no corresponding result for the variances of the chemical species. We also prove that the fluctuations of the fluxes as characterized by their time averages decrease down reaction chains. The results presented give insight into how biochemical reaction systems are buffered against external perturbations solely by their underlying graphical structure and point out the benefits of studying the out-of-equilibrium dynamics of systems.