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#' 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)
}
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