#' Logistic kinetics
#' 
#' Function describing exponential decline from a defined starting value, with
#' an increasing rate constant, supposedly caused by microbial growth
#' 
#' @param t Time.
#' @param parent.0 Starting value for the response variable at time zero.
#' @param kmax Maximum rate constant.
#' @param k0 Minumum rate constant effective at time zero.
#' @param r Growth rate of the increase in the rate constant.
#' @return The value of the response variable at time \code{t}.
#' @note The solution of the logistic model reduces to the
#'   \code{\link{SFO.solution}} if \code{k0} is equal to \code{kmax}.
#' @references FOCUS (2014) \dQuote{Generic guidance for Estimating Persistence
#'   and Degradation Kinetics from Environmental Fate Studies on Pesticides in
#'   EU Registration} Report of the FOCUS Work Group on Degradation Kinetics,
#'   Version 1.1, 18 December 2014
#'   \url{http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics}
#' @examples
#' 
#'   # Reproduce the plot on page 57 of FOCUS (2014)
#'   plot(function(x) logistic.solution(x, 100, 0.08, 0.0001, 0.2),
#'        from = 0, to = 100, ylim = c(0, 100),
#'        xlab = "Time", ylab = "Residue")
#'   plot(function(x) logistic.solution(x, 100, 0.08, 0.0001, 0.4),
#'        from = 0, to = 100, add = TRUE, lty = 2, col = 2)
#'   plot(function(x) logistic.solution(x, 100, 0.08, 0.0001, 0.8),
#'        from = 0, to = 100, add = TRUE, lty = 3, col = 3)
#'   plot(function(x) logistic.solution(x, 100, 0.08, 0.001, 0.2),
#'        from = 0, to = 100, add = TRUE, lty = 4, col = 4)
#'   plot(function(x) logistic.solution(x, 100, 0.08, 0.08, 0.2),
#'        from = 0, to = 100, add = TRUE, lty = 5, col = 5)
#'   legend("topright", inset = 0.05,
#'          legend = paste0("k0 = ", c(0.0001, 0.0001, 0.0001, 0.001, 0.08),
#'                          ", r = ", c(0.2, 0.4, 0.8, 0.2, 0.2)),
#'          lty = 1:5, col = 1:5)
#' 
#'   # Fit with synthetic data
#'   logistic <- mkinmod(parent = mkinsub("logistic"))
#' 
#'   sampling_times = c(0, 1, 3, 7, 14, 28, 60, 90, 120)
#'   parms_logistic <- c(kmax = 0.08, k0 = 0.0001, r = 0.2)
#'   d_logistic <- mkinpredict(logistic,
#'     parms_logistic, c(parent = 100),
#'     sampling_times)
#'   d_2_1 <- add_err(d_logistic,
#'     sdfunc = function(x) sigma_twocomp(x, 0.5, 0.07),
#'     n = 1, reps = 2, digits = 5, LOD = 0.1, seed = 123456)[[1]]
#' 
#'   m <- mkinfit("logistic", d_2_1, quiet = TRUE)
#'   plot_sep(m)
#'   summary(m)$bpar
#'   endpoints(m)$distimes
#' 
#' @export
logistic.solution <- function(t, parent.0, kmax, k0, r)
{
	parent = parent.0 * (kmax / (kmax - k0 + k0 * exp (r * t))) ^(kmax/r)
}