Checking a life-history analysis

ecology tutorial

life history

model diagnostics

evolutionary ecology

Three checks for a life-history study in R: matching the fitness currency, the acquisition-allocation trap that hides trade-offs, and detecting bet-hedging.

Author

Tidy Ecology

Published

2026-07-31

Life-history theory turns natural selection into an optimisation problem, and optimisation problems have a way of giving confident wrong answers when the objective is mis-specified. Three checks catch the usual ways a life-history analysis misleads: the wrong fitness currency, a trade-off hidden by variation between individuals, and a bet-hedging signal that lives in the variance rather than the mean.

Check 1: is the fitness currency right for the environment?

“Fitness” is not one number. In a stationary population it is lifetime reproductive output, R0. In a growing population it is the intrinsic rate of increase, r, which discounts offspring produced late because the population they join is already larger. In a fluctuating population it is the geometric mean growth rate. These currencies rank life histories differently, and the difference is not subtle.

euler_r <- function(ages, m) uniroot(function(r) sum(m*exp(-r*ages)) - 1, c(-1, 5))$root
c(R0_both = 6, r_reproduce_early = euler_r(1, 6), r_reproduce_late = euler_r(3, 6))

          R0_both r_reproduce_early  r_reproduce_late 
        6.0000000         1.7917589         0.5972552

Two schedules that produce the same six offspring over a lifetime have identical R0. But produced early, at age one, they buy an r of 1.79; produced late, at age three, only 0.60. Feed the reproductive-effort trade-off from the first post into each currency and the optimal effort itself moves: R0 is maximised at an effort of 0.43, but r is maximised near 0.97, pushing almost everything into reproducing now. Pick the currency that matches how the population is actually growing, or the optimum you compute is the answer to a different question.

dc <- rbind(data.frame(E = Es, y = sapply(Es, R0)/max(sapply(Es, R0)),   currency = "lifetime output R0"),
            data.frame(E = Es, y = sapply(Es, r_of)/max(sapply(Es, r_of)), currency = "intrinsic rate r"))
ggplot(dc, aes(E, y, colour = currency)) +
  geom_line(linewidth = 1) +
  geom_vline(xintercept = E_R0, colour = col_r0, linetype = "dashed") +
  geom_vline(xintercept = E_r, colour = col_r, linetype = "dashed") +
  scale_colour_manual(values = c("lifetime output R0" = col_r0, "intrinsic rate r" = col_r)) +
  labs(x = "Reproductive effort", y = "Fitness (scaled to each maximum)", colour = NULL,
       title = "The currency moves the optimum")

Two scaled fitness curves against reproductive effort; the lifetime-output curve peaks at intermediate effort while the intrinsic-rate curve peaks near maximum effort. — Figure 1: The same trade-off scored by two currencies, each scaled to its own maximum. Lifetime output (green) peaks at moderate effort; the intrinsic rate (ochre), which rewards early reproduction, peaks near maximal effort.

Check 2: is the trade-off hidden by individual quality?

The core trade-offs, reproduction against survival, are trade-offs within an individual’s budget: spend on one and less is left for the other. But we usually measure them across individuals, and individuals differ in how much they have to spend. A high-quality individual with a large budget can afford both high fecundity and high survival, while a poor one manages neither, so across a population the two can be positively correlated even though the allocation trade-off is strict. Van Noordwijk and de Jong 1986 The American Naturalist 128(1):137-142 made this precise: whether the trade-off shows depends on whether individuals vary more in acquisition or in allocation.

set.seed(406); n <- 400
demo <- function(sdA, sdP) {
  A <- pmax(rnorm(n, 10, sdA), 1)          # resource acquisition (budget)
  P <- plogis(rnorm(n, 0, sdP))            # allocation fraction to reproduction (a strict trade-off)
  data.frame(repro = P*A, surv = (1-P)*A)
}
c(variation_in_acquisition = cor(demo(4, 0.3)$repro, demo(4, 0.3)$surv),
  variation_in_allocation  = cor(demo(1, 1.5)$repro, demo(1, 1.5)$surv))

variation_in_acquisition  variation_in_allocation 
              0.02585146              -0.09798322

When individuals vary mostly in acquisition, the observed correlation between reproduction and survival investment is +0.72, positive: the data say “more reproduction goes with more survival”, the opposite of a trade-off. When they vary mostly in allocation, the correlation is -0.95, the real negative trade-off. The underlying allocation rule is identical in both; only the source of variation differs. A raw phenotypic correlation across individuals is therefore no test of a trade-off. What tests it is manipulation, add offspring to a brood experimentally and measure the survival cost, or a design that holds acquisition constant.

dd <- rbind(cbind(d_acq, case = "variation in acquisition"), cbind(d_all, case = "variation in allocation"))
ggplot(dd, aes(repro, surv, colour = case)) +
  geom_point(size = 1.6, alpha = 0.5) +
  geom_smooth(method = "lm", se = FALSE, linewidth = 0.9) +
  scale_colour_manual(values = c("variation in acquisition" = col_r, "variation in allocation" = col_r0)) +
  labs(x = "Investment in reproduction", y = "Investment in survival", colour = NULL,
       title = "The same trade-off, hidden or revealed by where the variation sits")

Two scatter clouds of reproduction against survival investment; the ochre cloud from variable acquisition slopes upward, the green cloud from variable allocation slopes downward. — Figure 2: Reproduction against survival investment for the same allocation trade-off. When acquisition varies (ochre) the cloud slopes up and hides the trade-off; when allocation varies (green) it slopes down and reveals it.

Check 3: is the signal in the mean or the variance?

Bet-hedging is invisible to the usual analysis because it does not optimise the mean. A bet-hedging phenotype deliberately accepts a lower arithmetic-mean fitness to cut variance, so measured against the mean it looks suboptimal, and an analysis that scores strategies by average success will reject it. Detecting it requires the thing that is hardest to measure: variance in fitness across generations, which needs long runs of data on marked individuals or genotypes, not a single well-sampled year. This is why the empirical evidence for bet-hedging remains thin despite strong theory, and why claims of it demand a demonstration that the variance, not just the mean, is doing the work (Simons 2011 Proceedings of the Royal Society B 278(1712):1601-1609).

The theory is only as good as its objective

These three checks share a theme: the machinery of life-history optimisation is sound, and the danger is all in what you feed it. The wrong currency, a correlation mistaken for a trade-off, or a mean where the variance is the point will each produce a clean, confident, wrong optimum. State the fitness measure and justify it, test trade-offs by manipulation rather than correlation, and when variance is the hypothesis, measure variance. The optimum is only as trustworthy as the objective it maximised.

References

Simons AM 2011. Proceedings of the Royal Society B 278(1712):1601-1609 (10.1098/rspb.2011.0176).
van Noordwijk AJ, de Jong G 1986. The American Naturalist 128(1):137-142 (10.1086/284547).

--- title: "Checking a life-history analysis" description: "Three checks for a life-history study in R: matching the fitness currency, the acquisition-allocation trap that hides trade-offs, and detecting bet-hedging." date: "2026-07-31 10:00" categories: [ecology tutorial, R, life history, model diagnostics, evolutionary ecology] image: thumbnail.png image-alt: "A scatter of reproduction against survival investment that is positive when acquisition varies and negative when allocation varies, hiding or revealing a trade-off." --- ```{r setup} #| include: false knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE, dev = "png", dpi = 120, fig.path = "figures/", fig.width = 7, fig.height = 4.4) library(ggplot2) theme_te <- theme_minimal(base_size = 12) + theme(panel.background = element_rect(fill = "white", colour = NA), plot.background = element_rect(fill = "white", colour = NA), panel.grid.minor = element_blank(), plot.title = element_text(face = "bold", size = 12)) theme_set(theme_te) col_r0 <- "#3a6b52"; col_r <- "#b5651d"; col_ref <- "#555555" ## (a) currency euler_r <- function(ages, m) uniroot(function(r) sum(m*exp(-r*ages)) - 1, c(-1, 5))$root r_early <- euler_r(1, 6); r_late <- euler_r(3, 6) fmax <- 6; smax <- 0.85; fec <- function(E) fmax*E^0.7; sur <- function(E) smax*(1-E)^0.6 R0 <- function(E) fec(E)/(1 - sur(E)) r_of <- function(E) { s <- sur(E); m <- fec(E); uniroot(function(r) m*sum(s^(0:80)*exp(-r*(1:81))) - 1, c(-1, 5))$root } Es <- seq(0.05, 0.97, 0.01) E_R0 <- Es[which.max(sapply(Es, R0))]; E_r <- Es[which.max(sapply(Es, r_of))] ## (b) acquisition-allocation set.seed(406); n <- 400 demo <- function(sdA, sdP) { A <- pmax(rnorm(n, 10, sdA), 1); P <- plogis(rnorm(n, 0, sdP)) data.frame(repro = P*A, surv = (1-P)*A) } d_acq <- demo(4, 0.3); d_all <- demo(1, 1.5) r_acq <- cor(d_acq$repro, d_acq$surv); r_all <- cor(d_all$repro, d_all$surv) ``` Life-history theory turns natural selection into an optimisation problem, and optimisation problems have a way of giving confident wrong answers when the objective is mis-specified. Three checks catch the usual ways a life-history analysis misleads: the wrong fitness currency, a trade-off hidden by variation between individuals, and a bet-hedging signal that lives in the variance rather than the mean. ## Check 1: is the fitness currency right for the environment? "Fitness" is not one number. In a stationary population it is lifetime reproductive output, `R0`. In a growing population it is the intrinsic rate of increase, `r`, which discounts offspring produced late because the population they join is already larger. In a fluctuating population it is the geometric mean growth rate. These currencies rank life histories differently, and the difference is not subtle. ```{r currency} euler_r <- function(ages, m) uniroot(function(r) sum(m*exp(-r*ages)) - 1, c(-1, 5))$root c(R0_both = 6, r_reproduce_early = euler_r(1, 6), r_reproduce_late = euler_r(3, 6)) ``` Two schedules that produce the same six offspring over a lifetime have identical `R0`. But produced early, at age one, they buy an `r` of `r sprintf("%.2f", r_early)`; produced late, at age three, only `r sprintf("%.2f", r_late)`. Feed the reproductive-effort trade-off from the [first post](../life-history-trade-offs/) into each currency and the optimal effort itself moves: `R0` is maximised at an effort of `r sprintf("%.2f", E_R0)`, but `r` is maximised near `r sprintf("%.2f", E_r)`, pushing almost everything into reproducing now. Pick the currency that matches how the population is actually growing, or the optimum you compute is the answer to a different question. ```{r fig-currency} #| fig-cap: "The same trade-off scored by two currencies, each scaled to its own maximum. Lifetime output (green) peaks at moderate effort; the intrinsic rate (ochre), which rewards early reproduction, peaks near maximal effort." #| fig-alt: "Two scaled fitness curves against reproductive effort; the lifetime-output curve peaks at intermediate effort while the intrinsic-rate curve peaks near maximum effort." dc <- rbind(data.frame(E = Es, y = sapply(Es, R0)/max(sapply(Es, R0)), currency = "lifetime output R0"), data.frame(E = Es, y = sapply(Es, r_of)/max(sapply(Es, r_of)), currency = "intrinsic rate r")) ggplot(dc, aes(E, y, colour = currency)) + geom_line(linewidth = 1) + geom_vline(xintercept = E_R0, colour = col_r0, linetype = "dashed") + geom_vline(xintercept = E_r, colour = col_r, linetype = "dashed") + scale_colour_manual(values = c("lifetime output R0" = col_r0, "intrinsic rate r" = col_r)) + labs(x = "Reproductive effort", y = "Fitness (scaled to each maximum)", colour = NULL, title = "The currency moves the optimum") ``` ## Check 2: is the trade-off hidden by individual quality? The core trade-offs, reproduction against survival, are trade-offs *within* an individual's budget: spend on one and less is left for the other. But we usually measure them *across* individuals, and individuals differ in how much they have to spend. A high-quality individual with a large budget can afford both high fecundity and high survival, while a poor one manages neither, so across a population the two can be *positively* correlated even though the allocation trade-off is strict. Van Noordwijk and de Jong 1986 The American Naturalist 128(1):137-142 made this precise: whether the trade-off shows depends on whether individuals vary more in acquisition or in allocation. ```{r acq-alloc} set.seed(406); n <- 400 demo <- function(sdA, sdP) { A <- pmax(rnorm(n, 10, sdA), 1) # resource acquisition (budget) P <- plogis(rnorm(n, 0, sdP)) # allocation fraction to reproduction (a strict trade-off) data.frame(repro = P*A, surv = (1-P)*A) } c(variation_in_acquisition = cor(demo(4, 0.3)$repro, demo(4, 0.3)$surv), variation_in_allocation = cor(demo(1, 1.5)$repro, demo(1, 1.5)$surv)) ``` When individuals vary mostly in acquisition, the observed correlation between reproduction and survival investment is `r sprintf("%+.2f", r_acq)`, positive: the data say "more reproduction goes with more survival", the opposite of a trade-off. When they vary mostly in allocation, the correlation is `r sprintf("%+.2f", r_all)`, the real negative trade-off. The underlying allocation rule is identical in both; only the source of variation differs. A raw phenotypic correlation across individuals is therefore no test of a trade-off. What tests it is manipulation, add offspring to a brood experimentally and measure the survival cost, or a design that holds acquisition constant. ```{r fig-acq} #| fig-cap: "Reproduction against survival investment for the same allocation trade-off. When acquisition varies (ochre) the cloud slopes up and hides the trade-off; when allocation varies (green) it slopes down and reveals it." #| fig-alt: "Two scatter clouds of reproduction against survival investment; the ochre cloud from variable acquisition slopes upward, the green cloud from variable allocation slopes downward." dd <- rbind(cbind(d_acq, case = "variation in acquisition"), cbind(d_all, case = "variation in allocation")) ggplot(dd, aes(repro, surv, colour = case)) + geom_point(size = 1.6, alpha = 0.5) + geom_smooth(method = "lm", se = FALSE, linewidth = 0.9) + scale_colour_manual(values = c("variation in acquisition" = col_r, "variation in allocation" = col_r0)) + labs(x = "Investment in reproduction", y = "Investment in survival", colour = NULL, title = "The same trade-off, hidden or revealed by where the variation sits") ``` ## Check 3: is the signal in the mean or the variance? Bet-hedging is invisible to the usual analysis because it does not optimise the mean. A bet-hedging phenotype deliberately accepts a lower arithmetic-mean fitness to cut variance, so measured against the mean it looks *suboptimal*, and an analysis that scores strategies by average success will reject it. Detecting it requires the thing that is hardest to measure: variance in fitness across generations, which needs long runs of data on marked individuals or genotypes, not a single well-sampled year. This is why the empirical evidence for bet-hedging remains thin despite strong theory, and why claims of it demand a demonstration that the variance, not just the mean, is doing the work (Simons 2011 Proceedings of the Royal Society B 278(1712):1601-1609). ## The theory is only as good as its objective These three checks share a theme: the machinery of life-history optimisation is sound, and the danger is all in what you feed it. The wrong currency, a correlation mistaken for a trade-off, or a mean where the variance is the point will each produce a clean, confident, wrong optimum. State the fitness measure and justify it, test trade-offs by manipulation rather than correlation, and when variance is the hypothesis, measure variance. The optimum is only as trustworthy as the objective it maximised. ## Related tutorials - [Life-history trade-offs in R](../life-history-trade-offs/) - [Bet-hedging and geometric mean fitness](../bet-hedging-and-geometric-mean-fitness/) - [The optimal germination fraction](../optimal-germination-fraction/) - [The life-table response experiment](../life-table-response-experiment/) ## References - Simons AM 2011. Proceedings of the Royal Society B 278(1712):1601-1609 (10.1098/rspb.2011.0176). - van Noordwijk AJ, de Jong G 1986. The American Naturalist 128(1):137-142 (10.1086/284547).