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.
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.
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")
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.
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