Lesson 17 — Helping relatives when cooperation can spread

A habit that costs the one doing it and helps others — how could that ever spread? Set up a population that never helps, drop in a helping type, and find where it tips.

Hamilton's rule. Kin selection happens when r×b > c. r is relatedness. b is the benefit to the recipient. c is the cost to the giver. A big benefit to a distant relative can be okay. A small benefit to a close relative can be okay. Random stranger needs a kidney — most won't give. Family member — many will.— 461_lec22_05

A — What happens without the rule

Locked — confirm your name above to begin.

Scenario

An altruism allele is introduced at frequency 0.1 in a population where no one is related to anyone else (r = 0). The allele pays the cost c and gets nothing back from a random partner. With no kin structure, there is nothing for Hamilton's rule to bite on. Each replicate is one possible future.

Five replicate trajectories — pure drift, no Hamilton math

N: 500  |  gens: 100  |  replicates shown: 5  |  mean ending freq: —

Prediction

  1. Q1. In a population with no kin structure and no altruism, an "altruism" allele introduced at frequency 0.1 will:
Try at least 3 seeds. 0/3 seeds

Controls

42

R code — neutral baseline

set.seed(42)N <- 500; gens <- 100; p <- 0.1for (g in 1:gens) p <- rbinom(1,2*N,p)/(2*N)

B — Add altruism — sliders for r, b, c

Complete Stage A.

Scenario

Pairs of individuals interact. With probability r, each pair shares the altruism allele (they're related). The altruist pays c to give benefit b to its partner. Track allele frequency over generations.

Altruist frequency over generations

r: 0.50  |  b: 0.30  |  c: 0.10  |  r·b − c: —  |  altruist freq at gen 200: —

Prediction

  1. Q1. With r = 0.5 (siblings), b = 0.3, c = 0.1, the altruist allele will:
Try at least 5 (r, b, c) combos straddling the boundary. 0/5 combos

Controls

0.50
0.30
0.10
42

R code — altruism dynamics

set.seed(42)N <- 500; r <- 0.5; b <- 0.3; cc <- 0.1; p <- 0.2; gens <- 200# Mean fitness of altruist: 1 - c + r*b (recipient has altruist allele w.p. r)# Mean fitness of selfish: 1 + p*r*b (gets benefit from altruistic relatives at rate p*r... simplified)for (g in 1:gens) {  w_alt <- 1 - cc + r * b  w_sel <- 1  p_new <- p * w_alt / (p * w_alt + (1-p) * w_sel)  p <- rbinom(1,2*N,p_new)/(2*N)}

C — Where helping starts to pay

Complete Stage B.

Scenario

The line r·b = c divides the (b, c) plane into two regions. With r = 0.5, the line is c = 0.5·b: altruism spreads only when the benefit is at least twice the cost. With r = 1 (clones), any net benefit spreads. With r = 0 (strangers), nothing spreads.

(b, c) phase plane with r·b = c boundary

r: 0.50  |  your point (b, c): (—)  |  spreads? —

Prediction

  1. Q1. As r increases (more relatedness), the region of (b, c) space where altruism spreads:
Try at least 3 r values. 0/3 r values

Controls

0.50
0.60
0.20

R code — phase plane

r <- 0.5; b <- 0.6; cc <- 0.2r*b > cc   # Hamilton's rule satisfied?

D — Same rule, three biologies — who should you help?

Complete Stage C.

Scenario

Same Hamilton's rule for all three. Different relatedness values for each because of how they inherit DNA:

  • Diploid mammal (mouse, human): two parents each contribute half. r to sister, daughter, brother all = 0.5.
  • Haplodiploid bee (female worker): female is diploid, brothers are haploid from unfertilized eggs. r to full sister = 0.75 (mom's recombined 0.5 + dad's clonal 1.0, halved). r to own daughter = 0.5. r to brother = 0.25.
  • Clonal bacterium (or a Wolbachia-style parthenogen): every offspring of every cell in your colony is your clone. r = 1 to all "kin".

For each, ask: with a fixed (b, c), which kin is it most worth helping?

r to each kin, by inheritance system

organism: all three  |  highest r: —

Prediction

  1. Q1. A female haplodiploid worker (e.g. a honey bee) compares helping her mother make more sisters vs. raising her own daughters. Hamilton's rule favors:
  2. Q2. A clonal bacterium considers an act that saves a neighboring cell at some cost to itself. Why does Hamilton's rule trivially favor helping in this case?
Toggle through all three organism views. 0/3 toggles

Controls

R code — relatedness across systems

# Three inheritance systems; one rule.diploid <- c(sister=0.5, daughter=0.5, brother=0.5)haplo   <- c(sister=0.75, daughter=0.5, brother=0.25)clonal  <- c(sister=1, daughter=1, brother=1)# Hamilton's rule: helping spreads if r * b > c, regardless of which row you're in.