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#' Expression of null response coefficient
#'
#' \emph{(Utilitary function)}. Gives result of a transformed expression of the response coefficient
#'
#'
#' @details
#' Null this expression corresponds to search the effective equilibrium in case of regulation, with or without competition.
#'
#'
#' @usage sol_eqeff(tau_fun,E_ini_fun,A_fun,B_fun,correl_fun)
#'
#' @param tau_fun Numeric. Driving variable of the system
#' @param A_fun Numeric vector of activities
#' @inheritParams range_tau
#' @param correl_fun Character string indicating the constraint abbreviation. Three constraints are allowed here: \code{"RegNeg"}, \code{"CRPos"} and \code{"CRNeg"}.
#'
#'
#'
#' @return A numeric value
#'
#' @references Coton et al. (2021)
#'
#'
#' @seealso
#' See function \code{\link{predict_eff}} to compute the effective equilibrium.
#'
#'
#'
#'
# Solution to null response coefficient to search the effective equilibrium in case of regulation
sol_eqeff <- function(tau_fun,E_ini_fun,A_fun,B_fun,correl_fun) {
e0_fun <- E_ini_fun/sum(E_ini_fun)
if (correl_fun=="RegNeg") {
deriv_fun <- sum(1/(B_fun*A_fun*(tau_fun*(1/B_fun-e0_fun)+e0_fun)^2))
}
if (correl_fun=="CRPos"|correl_fun=="CRNeg") {
deriv_fun <- sum((1/B_fun-e0_fun)/(A_fun*(tau_fun*(1/B_fun-e0_fun)+e0_fun)^2))
}
return(deriv_fun)
}
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