Lesson 15 — How far a population shifts when you breed from the extremes

Breed only from the largest, and the next generation slides partway toward them. How far it slides is the whole question.

A really counterintuitive thing: no matter how strong selection is on an additive trait, the population only moves by the additive variation in the population each generation. Selection sets direction. Additive variation sets the magnitude.— 202_lec19_01

A — Breeding only from one tail of a bell-shaped trait

Locked — confirm your name above to begin.

Scenario

Population of N = 2000 individuals with trait z ~ Normal(0, 1). Pick the top-fraction f as breeders. Compute the population mean z̄, the breeder mean z̄_b. S = z̄_b − z̄.

Trait distribution with selection threshold

N: 2000  |  fraction selected: 0.20  |  threshold: —  |  S: —

Prediction

  1. Q1. Selecting the top 20% on a Normal(0, 1) trait yields a selection differential S of approximately:
Try at least 4 fractions. 0/4 fractions

Controls

0.20
42

R code — truncation selection

set.seed(42)N <- 2000; z <- rnorm(N); f <- 0.2thr <- quantile(z, 1-f)S <- mean(z[z >= thr]) - mean(z)

B — What the next generation actually does

Complete Stage A.

Scenario

Only the breeders from Stage A have offspring. How much of a parent's departure from the herd shows up in its offspring is set by the carry-over dial. Run it for twenty generations and watch where the herd average goes.

Mean trait over generations

h²: 0.5  |  f selected: 0.20  |  S per gen: —  |  total response (20 gen): —

Prediction

  1. Q1. The carry-over dial sits at 0.50. Each generation the breeders average 1.4 above the herd. After twenty generations, the herd average has moved by about:
Try at least 4 (h², f) combos. 0/4 combos

Controls

0.50
0.20
42

R code — breeder's equation

set.seed(42)N <- 2000; h2 <- 0.5; f <- 0.2; gens <- 20z <- rnorm(N); means <- mean(z)for (g in 1:gens) {  thr <- quantile(z, 1-f); br <- z[z >= thr]  z <- h2 * sample(br, N, replace=TRUE) + sqrt(1-h2) * rnorm(N)  means <- c(means, mean(z))}

C — Keep pushing, generation after generation

Complete Stage B.

Scenario

Long-run selection. Plot the mean trait AND additive variance over generations. h² is rebuilt by mutational input each generation; without it, h² collapses to 0 and selection stops working.

Mean trait and h² over generations

starting h²: 0.5  |  final h² (gen 100): —  |  new variation coming in: 0.02

Prediction

  1. Q1. With μ_input = 0 (no new mutations) and starting h² = 0.5, after 100 generations of truncation selection, h² will be:
Try at least 3 (μ, f) combos. 0/3 combos

Controls

0.50
0.20
0.020
42

R code — long-run h² collapse

# Track additive variance over time; new mutations refill it.# Realistic model needs a genetic architecture; here we approximate.

D — Grant finches, year by year

Complete Stage C.

Scenario

Grant finch beak depth from data/clean/grant_finches_40y.csv. Each year: mean beak depth, breeder mean, S. Next year: mean beak depth (= response R). h² is estimated as R/S for that pair.

Year-by-year R vs S (with 200-sample bootstrap of the slope)

years: —  |  slope (= ĥ²): —  |  95% resampled range: —  |  drought years (highlighted): —

Prediction

  1. Q1. The slope of R on S across Grant finch years (heritability of beak depth) is approximately:
Refit at least 2 times. 0/2 refits

Controls

42

R code — Grant finch breeder's equation

finches <- read.csv("data/clean/grant_finches_40y.csv")# Per year: mean beak depth, mean breeder beak depth, next-year meanfit <- lm(R ~ 0 + S, data = year_pairs)coef(fit)   # slope = h²