GillespieSSA: Implementing the Gillespie Stochastic Simulation Algorithm in R

The deterministic dynamics of populations in continuous time are traditionally described using coupled, first-order ordinary differential equations. While this approach is accurate for large systems, it is often inadequate for small systems where key species may be present in small numbers or where key reactions occur at a low rate. The Gillespie stochastic simulation algorithm (SSA) is a procedure for generating time-evolution trajectories of finite populations in continuous time and has become the standard algorithm for these types of stochastic models. This article presents a simple-to-use and flexible framework for implementing the SSA using the high-level statistical computing language R and the package GillespieSSA. Using three ecological models as examples (logistic growth, Rosenzweig-MacArthur predator-prey model, and Kermack-McKendrick SIRS metapopulation model), this paper shows how a deterministic model can be formulated as a finite-population stochastic model within the framework of SSA theory and how it can be implemented in R. Simulations of the stochastic models are performed using four different SSA Monte Carlo methods: one exact method (Gillespie's direct method); and three approximate methods (explicit, binomial, and optimized tau-leap methods). Comparison of simulation results confirms that while the time-evolution trajectories obtained from the different SSA methods are indistinguishable, the approximate methods are up to four orders of magnitude faster than the exact methods.

Symbolic trajectories in SECONDO : pattern matching and rewriting

In this paper, we introduce a novel data model for representing symbolic trajectories along with a pattern language enabling both the matching and the rewriting of trajectories. We illustrate in particular the trajectory data type and two operations for querying symbolic trajectories inside the database system Secondo. As an important application of our theory, the classi\ufb01cation and depiction of a set of real trajectories according to several criteria is demonstrated

Spectral statistics in disordered metals: a trajectories approach

We show that the perturbative expansion of the two-level correlation function, $R(\omega)$, in disordered conductors can be understood semiclassically in terms of self-intersecting particle trajectories. This requires the extension of the standard diagonal approximation to include pairs of paths which are non-identical but have almost identical action. The number of diagrams thus produced is much smaller than in a standard field-theoretical approach. We show that such a simplification occurs because $R(\omega)$ has a natural representation as the second derivative of free energy $F(\omega)$. We calculate $R(\omega)$ to 3-loop order, and verify a one-parameter scaling hypothesis for it in 2d. We discuss the possibility of applying our ``weak diagonal approximation'' to generic chaotic systems.Comment: 9 pages in REVTeX two-column format including 4 figures; submitted to Phys.Rev.