# extinction_risk_2.R, ver. 2.5 2014/2/12 # $Id: extinction_risk_2.R,v 1.2 2015/05/25 12:51:56 hako Exp $ # # Author: Hiroshi Hakoyama # Copyright (c) 2013-2014 Hiroshi Hakoyama , All rights reserved. # # Redistribution and use in source and binary forms, with or without # modification, are permitted provided that the following conditions # are met: # 1. Redistributions of source code must retain the above copyright # notice, this list of conditions and the following disclaimer. # 2. Redistributions in binary form must reproduce the above copyright # notice, this list of conditions and the following disclaimer in the # documentation and/or other materials provided with the distribution. # # THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY # EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, # THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A # PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE # AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, # EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT # NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; # LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER # CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, # STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) # ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF # ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. # # Description: # Estimates the demographic parameters and the probability of extinction within # a specific time period, t, from a time series of population size based on a # density-dependent population model with environmental and demographic # stochasticity (Hakoyama and Iwasa, 2000). # # Usage: # ext2(dat, gt = 3, t = 100, INDEX = FALSE, PS = 10^6, verbose = FALSE, accuracy = 0.25) # # Arguments: # dat data.frame of 2 variables: Time and Population size # gt generation time # t a time period of interest # verbose If set to FALSE, give the ML estimate of the probability of extinction # within a specific time period t. If set to TRUE, give a more verbose output: # the ML estimate of the growth rate per generation (r), # the ML estimate of carrying capacity (K), # the ML estimate of variance per generation (s), # the ML estimate of the mean extinction time (met, year) # the ML estimate of the probability of extinction per year (lambda) # the ML estimate of the probability of extinction within a specific time period t (ext, year) # INDEX If data are an index of the population size, set to TRUE. If data are # population size, set to FALSE. If INDEX is set to TRUE, give only r and ss, # and the probability of extinction time within a specific time period t based on # Te(r, PS, ss). # PS Population size from other information. Set PS for the data of # an index of the population size. # accuracy accuracy for the integrate function # # References: # H. Hakoyama and Y. Iwasa. Extinction risk of a density-dependent population # estimated from a time series of population size. Journal of Theoretical Biology, # 204:337-359, 2000. # ext2 <- function(dat, gt = 3, t = 100, INDEX = FALSE, PS = 10^6, verbose = FALSE, accuracy = 0.25) { yr <- ts(dat[, 1], start=c(dat[, 1][1])) # Year ps <- ts(dat[, 2], start=c(dat[, 1][1])) # Population size complete <- complete.cases(yr, ps) yr <- yr[complete] ps <- ps[complete] tau <- diff(yr) / gt # time intervals, \tau_i = (t_i - t_{i-1})/gt K <- mean(ps) # Carrying capacity y <- ps - K n <- length(ps) if (sum(y[1:length(y) - 1] * y[2:length(y)]) < 0) { print("The ML estimates do not exist, because the autocorrelation with delay tau is negative (see Hakoyama & Iwasa, 2000, p-357).") return() } if (all(tau == tau[1])) { f <- function(b) { (n - 1) / n * y[1]^2 * b * (1 - b^2) - b / n * sum((y[2:length(y)] - b * y[1:length(y) - 1])^2) + (1 - b^2) * sum(y[1:length(y) - 1] * (y[2:length(y)] - b * y[1:length(y) - 1])) } beta <- uniroot(f, c(0, 1), tol = 10^-9)$root alpha <- (y[1]^2 + sum((y[2:length(y)] - beta * y[1:length(y) - 1])^2) / (1 - beta^2)) / n r <- - log(beta) / tau[1] # growth rate per generation } else { f <- function(b) { sum(- tau * b^(2 * tau - 1) / (1 - b^(2 * tau))) / n * (y[1]^2 + sum((y[2:length(y)] - b^tau * y[1:length(y) - 1])^2 / (1 - b^(2 * tau)))) + sum(b^(tau - 1) * tau * (y[2:length(y)] - b^tau * y[1:length(y) - 1]) * (b^tau * y[2:length(y)] - y[1:length(y) - 1]) / (1 - b^(2 * tau))^2) } beta <- uniroot(f, c(10^-300, 1 - 10^-9), tol = 10^-50)$root alpha <- (y[1]^2 + sum((y[2:length(y)] - beta^(tau) * y[1:length(y) - 1])^2 / (1 - beta^(2 * tau)))) / n r <- - log(beta) # growth rate per generation } if (INDEX == FALSE) { s <- (2 * alpha * r - K) / K^2 # environmental variance if (s < 0) { print("Negative variance was estimated!") return() } met <- Te(r, K, s, a = accuracy) * gt # Mean extinction time (year) lambda <- 1 / met # Extinction probability per year ext <- ifelse(1 - exp(-t * lambda) > 0, 1 - exp(-t * lambda), t * lambda) # Extinction probability within t years based on exponential distribution or approximation for small lambda } else { ss <- (2 * alpha * r) / K^2 # s for INDEX data with large population size met2 <- Te(r, PS, ss, a = accuracy) * gt lambda2 <- 1 / met2 ext2 <- ifelse(1 - exp(- t * lambda2) > 0, 1 - exp(-t * lambda2), t * lambda2) } # for INDEX data if (INDEX == FALSE) { if (verbose == TRUE) { results <- list(t = t, verbose = verbose, INDEX = INDEX, Growth.rate.per.generation = r, Carrying.capacity = K, Variance.per.generation = s, Mean.extinction.time = met, Extinction.probability.per.year = lambda, Extinction.probability.within.t.years = ext) class(results) <- "ext2" return(results) } else { results <- list(t = t, verbose = verbose, INDEX = INDEX, Extinction.probability.within.t.years = ext) class(results) <- "ext2" return(results) } } else { if (verbose == TRUE) { results <- list(t = t, verbose = verbose, INDEX = INDEX, Growth.rate.per.generation = r, Variance.per.generation = ss, Extinction.probability.within.t.years = ext2) class(results) <- "ext2" return(results) } else { results <- list(t = t, verbose = verbose, INDEX = INDEX, Extinction.probability.within.t.years = ext2) class(results) <- "ext2" return(results) } } } print.ext2 <- function(obj, digits = 5) { # cat("\nExtinction risk\n") if ((obj$INDEX == FALSE) && (obj$verbose == TRUE)) { output.est <- data.frame(c( formatC(obj$Growth.rate.per.generation, digits = digits), formatC(obj$Carrying.capacity, digits = digits), formatC(obj$Variance.per.generation, digits = digits), formatC(obj$Mean.extinction.time, digits = digits), formatC(obj$Extinction.probability.per.year, digits = digits), formatC(obj$Extinction.probability.within.t.years, digits = digits))) dimnames(output.est) <- list( c("Growth rate per generation:", "Carrying capacity:", "Variance per generation:", "Mean extinction time (year):", "Extinction probability per year:", paste("Extinction probability within", obj$t, "years:")), c("Estimate")) print(output.est) } if ((obj$INDEX == TRUE) && (obj$verbose == TRUE)) { output.est <- data.frame(c( formatC(obj$Growth.rate.per.generation, digits = digits), formatC(obj$Variance.per.generation, digits = digits), formatC(obj$Extinction.probability.within.t.years, digits = digits))) dimnames(output.est) <- list( c("Growth rate per generation:", "Variance per generation:", paste("Extinction probability within", obj$t, "years:")), c("Estimate")) print(output.est) } if (obj$verbose == FALSE) { output.est <- data.frame( c(formatC(obj$Extinction.probability, digits = digits))) dimnames(output.est) <- list( c(paste("Extinction probability within", obj$t, "years:")), c("Estimate")) print(output.est) } } Te <- function(r, K, s2, x0 = K, a = 0.25, limit = 10) { # Mean extinction time, Te (generation time) R <- 2 * r / (s2 * K) DD <- 1 / s2 f1 <- function(u, v) { exp(- R * x0 * (1 / v - 0.5 * (tanh(0.5 * pi * sinh(u)) + 1)) + (R * (K + DD) + 1) * log(((x0 / v + DD) / (0.5 * x0 * (tanh(0.5 * pi * sinh(u)) + 1) + DD)))) * (0.5 * (pi * cosh(u)) / (1 + cosh(pi * sinh(u)))) / (1 + v * DD / x0) } T1 <- function() { 2 * DD * integrate(function(x) { sapply(x, function(x) { integrate(function(y) f1(x, y), 0, 1, stop.on.error = T, rel.tol = .Machine$double.eps^a)$value }) }, -limit, limit, stop.on.error = T, rel.tol = .Machine$double.eps^a)$value } f2 <- function(u, v) { exp(- R * (x0 * v - 0.5 * x0 * (tanh(0.5 * pi * sinh(u)) + 1)) + (R * (K + DD) + 1) * log(((x0 * v + DD) / (0.5 * x0 * (tanh(0.5 * pi * sinh(u)) + 1) + DD)))) * (0.5 * (pi * cosh(u)) / (1 + cosh(pi * sinh(u)))) / (v^2 + v * DD / x0) } T2 <- function() { 2 * DD * integrate(function(x) { sapply(x, function(x) { integrate(function(y) f2(x, y), 0.5 * (tanh(0.5 * pi * sinh(x)) + 1), 1, stop.on.error = T, rel.tol = .Machine$double.eps^a)$value }) }, -limit, limit, stop.on.error = T, rel.tol = .Machine$double.eps^a)$value } T1() + T2() } # # Examples # Yellowstone grizzly bears (from Dennis et al., 1991) # dat <- data.frame(Year = c(1959, 1960, 1961, 1962, 1963, 1964, 1965, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1975, 1976, 1977, 1978, 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987), # Population = c(44, 47, 46, 44, 46, 45, 46, 40, 39, 39, 42, 44, 41, 40, 33, 36, 34, 39, 35, 34, 38, 36, 37, 41, 39, 51, 47, 57, 47)) # the probability of extinction (to decline to population size 0) within 100 years # ext2(dat, gt = 10, t = 100) # ext2(dat, gt = 10, t = 100, verbose = TRUE) # CPUE data of the Japanese crucian carp in Lake Biwa (from Hakoyama and Iwasa, 2000) # dat2 <- data.frame(Year = c(1955, 1956, 1957, 1958, 1959, 1960, 1961, 1962, 1963, 1964, 1965, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1975, 1976, 1977, 1978, 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989), # CPUE = c(0.230711329, 0.287504747, 0.321044547, 0.248123271, 0.266040689, 0.276898734, 0.360330579, 0.365279529, 0.405311276, 0.415012942, 0.466610313, 0.373273942, 0.349548646, 0.245173745, 0.31368529, 0.320981211, 0.236671001, 0.263181412, 0.263037511, 0.346241458, 0.290079925, 0.250327654, 0.284950658, 0.253397633, 0.303960837, 0.35359857, 0.399908592, 0.320795504, 0.237847222, 0.260603205, 0.291603821, 0.301130524, 0.272430669, 0.221655329, 0.186635945)) # ext2(dat2, gt = 3, t = 100, INDEX = TRUE, PS = 10^6)