QPot: An R Package for Stochastic Differential Equation Quasi-Potential Analysis
In QPot: Quasi-Potential Analysis for Stochastic Differential Equations

Introduction to QPot

What is QPot?

QPot is an abbreviation for an R package for Quasi-Potential analysis, which is a technique used to determine relative probability and stability in 2-dimensional stochastic systems.

How does QPot work?

The quasi-potential is calculated numerically for 2-dimensional stochastic equations through an ordered upwind method developed by Sethian and Vladimirsky,

J. A. Sethian and A. Vladimirsky. Ordered upwind methods for static Hamilton-Jacobi equations. Proceedings of the National Academy of Sciences, 98(20):11069–11074, 2001.

J. A. Sethian and A. Vladimirsky. Ordered upwind methods for static Hamilton-Jacobi equations: Theory and algorithms. SIAM Journal on Numerical Analysis, 41(1):325–363, 2003.

and expanded on by Cameron

M. K. Cameron. Finding the quasipotential for nongradient SDEs. Physica D, 241(18):1532–1550, 2012.

Nolting and Abbott have recently introduced the method to the field of ecology

B. C. Nolting and K. C. Abbott. Balls, cups, and quasi-potentials: Quantifying stability in stochastic systems. Ecology, 97(4):850–864, 2016.

and we have recently published a detailed paper on how to use QPot

C. Moore, C. Stieha, B. Nolting, M. Cameron, and K. Abbott. QPot: Quasi-Potential Analysis for Stochastic Differential Equations, 2016. URL https://www.R-project.org/package=QPot. R package version 1.2.

This vignette is a condensed version of Moore et al. (2016), mostly focusing on Example 1.

How can I get QPot?

QPot can be downloaded like any other package from CRAN:

    install.packages(pkgs = "QPot")

or for the most recent working version from GitHub:

    devtools::install_github(repo = "bmarkslash7/QPot")

then simply load the library:

    library(package = "QPot")

Why all of the vignettes?

We break the vignettes into sections similar to the way we do in Moore et al. (2016). They're broken into 6 natural steps:

1. Analyzing the deterministic skeleton
2. Stochastic simulation
3. Local quasi-potential calculation
4. Global quasi-potential calculation
5. Global quasi-potential visualization
6. Vector field decomposition

Analyzing the deterministic skeleton (1) goes through the process of taking a set of ordinary (non-stochastic) differential equations and examining its dynamics. (2) Adds stochasticity to (1), and we see that we need a tool to determine how the system will behave. (3) Is the first part of quasi-potential analysis, where a local quasi-potential is calculated for each stable equilibrium. (4) Combines each local quasi-potential into a global quasi-potential surface. (5) Visualized the global quasi-potential surface. (6) Performs a vector field decomposition of the deterministic direction field, the gradient field, and remainder field.