Basics of phase plane analysis with Grind

This tutorial describes an R-script, grind.R, that allows students and investigators whom are not very familiar with the R language to perform phase plane analysis and parameter fitting. Phase plane analysis is powerful graphical method to analyze the steady states of low-dimensional mathematical models formulated as ordinary differential eqautions (ODEs). Fitting ODE models to data has become a very common practice in biology.

Grind is a “wrapper” around the commonly-used R-packages deSolve, FME and rootSolve developed by Karline Soetaert and colleagues [Soetaert et al, 2009a, 2009b, 2010a, 2010b]. These packages have to be installed beforehand (see the section on Installation below). Our aim with developing grind.R was to define five easy-to-use functions:

Here is a link to Grind’s homepage.

The Lotka Volterra model in Grind

Consider the Lotka Volterra model \(R'=rR(1-R/K)-aRN\) and \(N'=caRN-dN\), with parameter values, \(r=K=a=c=1\), and \(d=0.5\), and the initial condition \(R=1\) and \(N=0.01\). Using Grind’s default names for the model, parameters, and state, i.e., model(), p, and s, this should be written as

model <- function(t, state, parms) {
  with(as.list(c(state,parms)), {  
    dR <- r*R*(1 - R/K) - a*R*N
    dN <- c*a*R*N - d*N  
    return(list(c(dR, dN)))  
  }) 
}  
p <- c(r=1,K=1,a=1,c=1,d=0.5)
s <- c(R=1,N=0.01)

where the arguments (t, state, parms) provide the time, the state \((R,N)\), and the parameters (as defined by the vector p) to the equations. The with(as.list(c(state,parms)),..) function allows one to identify parameters and variables by their name (instead of having to write parms["K"] and state["R"], and so on; see the webpage on with if you want to know more about with()). Note that the derivatives are returned as a list, and that the order of the variables in this list has to be the same as their order in the state!

Having defined a model, its initial state, and its parameters one can just call run() to start a numerical integration, plane() to plot a phase plane, or newton() to find a steady state close to the state s. Here is an example (starting with sourcing the grind.R script):

source("grind.R")
## grind.R (15-03-2024) was sourced
plane(xmin=-0.01,ymin=-0.01)
s <- run(traject=TRUE)
s <- newton(plot=TRUE)

##   R   N 
## 0.5 0.5 
## Stable point, eigenvalues:
## [1] -0.25+0.4330127i -0.25-0.4330127i

where the option xmin=ymin=-0.01 improves the plotting of nullclines on the axes, the option traject=TRUE tells run() to sketch a trajecotory (instead of making a timeplot), and the option plot=TRUE adds a bullet to the phase where the steady state was found. The call to newton() prints the eigenvalues to the screen (and returns the state that was found). If this fails, please check that the R-packages were properly installed.

Examples

The best way to learn Grind is to try a few of the following examples:

Finally, the manual explains all options for all functions.

Installation

Since Grind is a wrapper around the deSolve, FME and rootSolve R-packages, and because FME requires the coda package, these four packages need to be installed the first time you use Grind. This can be done by Install Packages in the Tools menu of RStudio.

Download the the R-codes grind.R and lotka.R, save them in a local directory, and open them in RStudio. It is probably convenient to set the working directory to the folder the files were stored (Set working directory in the Session menu of RStudio). To get started type source("grind.R"), or click the sourcebutton in RStudio, and begin executing the lotka.R script.

References

Websites

Check Grind’s homepage for links and more info. All files can also be downloaded from the Grind’s directory.