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#' The expected number of endemics and non-endemics under the DAISIE model
#'
#' This function calculates the expected number of endemics, non-endemics and
#' the sum of these for a given set of parameter values, a given mainland
#' species pool size and a given time
#'
#' @inheritParams default_params_doc
#'
#' @return \item{out}{The output is a list with three elements: \cr \cr
#' \code{ExpE} The number of endemic species \cr \code{ExpI} The number of
#' non-endemic species \cr \code{ExpN} The sum of the number of endemics and
#' non-endemics }
#' @author Rampal S. Etienne
#' @references Valente, L.M., A.B. Phillimore and R.S. Etienne (2015).
#' Equilibrium and non-equilibrium dynamics simultaneously operate in the
#' Galapagos islands. Ecology Letters 18: 844-852.
#' @keywords models
#' @examples
#'
#' ### Compute the expected values at t = 4, for a mainland pool size of 1000 potential
#' # colonists and a vector of 5 parameters (cladogenesis, extinction, clade-level carrying
#' # capacity, immigration, anagenesis)
#'
#' DAISIE_ExpEIN(
#' t = 4,
#' pars = c(0.5,0.1,Inf,0.01,0.4),
#' M = 1000
#' )
#'
#' @export DAISIE_ExpEIN
DAISIE_ExpEIN <- function(t, pars, M, initEI = c(0, 0)) {
pars1 <- pars
lac <- pars1[1]
mu <- pars1[2]
ga <- pars1[4]
laa <- pars1[5]
if (!is.na(pars1[11])) {
M2 <- M - DDD::roundn(pars1[11] * M)
} else {
M2 <- M
}
A <- mu - lac
B <- lac + mu + ga + laa
C <- laa + 2 * lac + ga
DD <- laa + 2 * lac
E0 <- initEI[1]
I0 <- initEI[2]
if (t[1] == Inf) {
Imm <- ga * M2 / B
End <- DD / A * Imm
} else {
#Imm = M2 * ga / B * (1 - exp(-B * t))
#End = M2 * ga * (laa + 2 * lac) * (1/(A * B) -
#exp(-A*t) / (A * C) + exp(-B*t)/(B * C))
Imm <- M2 * ga / B - (M2 * ga / B - I0) * exp(-B * t)
End <- DD / C * (M2 * ga / A - M2 * ga / B +
(C / DD * E0 - M2 * ga / A + I0) *
exp(-A * t) + (M2 * ga / B - I0) * exp(-B * t))
}
All <- End + Imm
expEIN <- list(End, Imm, All)
names(expEIN) <- c("ExpE", "ExpI", "ExpN")
return(expEIN)
}
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