R/logistic.R

In mjg211/phaseR: Phase Plane Analysis of One- And Two-Dimensional Autonomous ODE Systems

#' The logistic growth model
#'
#' The derivative function of the logistic growth model, an example of a
#' two-dimensional autonomous ODE system.
#'
#' \code{logistic} evaluates the derivative of the following ODE at the point
#' \ifelse{html}{\out{(<i>t</i>, <i>y</i>)}}{\eqn{(t, y)}}:
#'
#' \ifelse{html}{\out{<i>dy</i>/<i>dt</i> = <i>&beta;y</i>(1 -
#' <i>y</i>/<i>K</i>).}}{\deqn{\frac{dy}{dt} = \beta y(1 - y/K).}}
#'
#' Its format is designed to be compatible with \code{\link[deSolve]{ode}} from
#' the \code{\link[deSolve]{deSolve}} package.
#'
#' @param t The value of \ifelse{html}{\out{<i>t</i>}}{\eqn{t}}, the independent
#' variable, to evaluate the derivative at. Should be a
#' \code{\link[base]{numeric}} \code{\link[base]{vector}} of
#' \code{\link[base]{length}} one.
#' @param y The value of \ifelse{html}{\out{<i>y</i>}}{\eqn{y}}, the dependent
#' variable, to evaluate the derivative at. Should be a
#' \code{\link[base]{numeric}} \code{\link[base]{vector}} of
#' \code{\link[base]{length}} one.
#' @param parameters The values of the parameters of the system. Should be a
#' \code{\link[base]{numeric}} \code{\link[base]{vector}} with parameters
#' specified in the following order:
#' \ifelse{html}{\out{<i>&beta;</i>}}{\eqn{\beta}},
#' \ifelse{html}{\out{<i>K</i>}}{\eqn{K}}.
#' @return Returns a \code{\link[base]{list}} containing the value of the
#' derivative at \ifelse{html}{\out{(<i>t</i>, <i>y</i>)}}{\eqn{(t, y)}}.
#' @author Michael J Grayling
#' @seealso \code{\link[deSolve]{ode}}
#' @examples
#' # Plot the velocity field, nullclines and several trajectories
#' logistic_flowField     <- flowField(logistic,
#'                                     xlim       = c(0, 5),
#'                                     ylim       = c(-1, 3),
#'                                     parameters = c(1, 2),
#'                                     points     = 21,
#'                                     system     = "one.dim",
#'                                     add        = FALSE)
#' logistic_nullclines    <- nullclines(logistic,
#'                                      xlim       = c(0, 5),
#'                                      ylim       = c(-1, 3),
#'                                      parameters = c(1, 2),
#'                                      system     = "one.dim")
#' logistic_trajectory    <- trajectory(logistic,
#'                                      y0         = c(-0.5, 0.5, 1.5, 2.5),
#'                                      tlim       = c(0, 5),
#'                                      parameters = c(1, 2),
#'                                      system     = "one.dim")
#' # Plot the phase portrait
#' logistic_phasePortrait <- phasePortrait(logistic,
#'                                         ylim       = c(-0.5, 2.5),
#'                                         parameters = c(1, 2),
#'                                         points     = 10,
#'                                         frac       = 0.5)
#' # Determine the stability of the equilibrium points
#' logistic_stability_1   <- stability(logistic,
#'                                     ystar      = 0,
#'                                     parameters = c(1, 2),
#'                                     system     = "one.dim")
#' logistic_stability_2   <- stability(logistic,
#'                                     ystar      = 2,
#'                                     parameters = c(1, 2),
#'                                     system     = "one.dim")
#' @export
logistic <- function(t, y, parameters) {
  list(parameters[1]*y*(1 - y/parameters[2]))
}