Module 0: Why Migrate? Evolution, Fitness and Circannual Clocks

Billions of birds cross hemispheres every year at enormous energetic and mortality cost. Why is migration evolutionarily stable? This module builds a quantitative fitness framework for resident, obligate and partial migration, surveys the classic blackcap cross-breeding experiments of Peter Berthold that demonstrated heritable control of direction and amount of Zugunruhe, and places migration in the context of circannual endogenous clocks and the competing southern-home and northern-home biogeographic hypotheses.

1. Evolutionary cost–benefit of migration

Migration is a life-history trait under frequency-dependent natural selection. Any complete account must balance the fitness payoff of seasonally tracking high-quality habitat against the energetic, physiological and mortality costs of the journey itself. Alerstam, Hedenström & Åkesson (2003) formalise this as an annual fitness budget \(W\) whose expectation governs whether a strategy invades a population.

Annual fitness budget

For a given phenotype we sum energetic gains from breeding and non-breeding habitats, and subtract the costs of thermoregulation, flight and journey hazards. Writing the summer payoff as \(R_s(\phi)\) and winter payoff as \(R_w(\phi)\) at latitude \(\phi\) , while \(C_m(\phi)\) and \(C_t(\phi)\) denote migration and thermal costs respectively:

\[ W_{\text{res}}(\phi) = \tfrac{1}{2} R_s(\phi) + \tfrac{1}{2} R_w(\phi) - C_t(\phi) \]

\[ W_{\text{mig}}(\phi) = \tfrac{1}{2} R_s(\phi) + \tfrac{1}{2} R_w(\phi_{\text{trop}}) - C_m(\phi) - C_h(\phi) \]

where \(C_h\) is the cumulative hazard along the flyway and \(\phi_{\text{trop}}\) is the latitude of the non-breeding refuge.

Migration is favoured whenever \(\Delta W = W_{\text{mig}} - W_{\text{res}} > 0\). Because both \(R_w(\phi)\) and \(C_t(\phi)\) depend steeply on latitude above the 0°C winter isotherm, while \(C_m(\phi) \propto 2\,D(\phi)\,e_{\text{flight}}\) rises roughly linearly with distance, there exists a critical latitude \(\phi^\star\) at which the two strategies have equal expected fitness. Below \(\phi^\star\)residency dominates; above it migration sweeps to fixation in the absence of other forces.

Cost components in flight

The energetic cost of flapping flight is well described by the Pennycuick mechanical-power model (Modelling the Flying Bird, 2008). For steady level flight the total aerodynamic power is the sum of induced, parasite and profile components:

\[ P_{\text{aero}}(v) = \frac{(m g)^2}{\tfrac{1}{2}\rho v \pi b^2}\cdot\frac{1}{\eta} \;+\; \tfrac{1}{2}\rho v^3 S_b C_{D,\text{par}} \;+\; P_{\text{pro}}(v) \]

with mass \(m\), wingspan \(b\), body drag coefficient \(C_{D,\text{par}}\) and Pennycuick efficiency factor \(\eta \approx 0.23\).

The minimum-power speed and maximum-range speed emerge as extrema of \(P_{\text{aero}}(v)\) and \(P_{\text{aero}}(v)/v\) respectively. Songbirds cruise near \(v_{\text{mr}}\) to maximise distance per gram of fat, while soaring species (raptors, storks) bypass the aerodynamic cost entirely by extracting energy from thermals. Hazard cost \(C_h\) folds in predation, storm mortality and barriers such as the Sahara or the Gulf of Mexico.

Migration is an ESS, not a static outcome

When the migrant and resident strategies have similar expected fitness, partial migration becomes evolutionarily stable: a polymorphic mixture with fraction \(p\) migrants and \(1-p\) residents (Lundberg, 1988; Chapman et al., 2011). Frequency-dependent effects such as competition for over-winter territories stabilise \(p\) at an intermediate value.

Latitudinal fitness crossover

2. Obligate, partial and facultative migration

Berthold’s classic categorisation distinguishes three qualitatively different patterns of seasonal movement. Obligate migration is genetically programmed: every individual of every generation migrates, the timing and direction are heritable, and release from environmental cues alone does not suppress the behaviour. Partial migration is the population-level coexistence of migrant and resident phenotypes within the same species and often the same breeding population. Facultative migration is environmentally triggered: the same individual may migrate one year and remain resident the next in response to irruptive resource pulses or weather.

Obligate migration: Eurasian blackcap

In Sylvia atricapilla south-west German blackcaps migrate to Iberia and North Africa, and the urge to do so is rigorously heritable. Birds hand-raised from the egg in acoustic isolation still display Zugunruhe in the correct direction and for the correct duration, a result first shown by Gwinner (1967) and Berthold & Querner (1981). The genetic program specifies at minimum: (i) onset date of autumn migration; (ii) heading; (iii) amount of restlessness integrated over the migration window; (iv) at least one direction change (Zugknick) for species with doglegged flyways.

Partial migration as an ESS

In populations of European robin, blackbird and song sparrow, males often remain on the breeding territory while females migrate, because early arrival grants a large territory- acquisition premium. The evolutionarily stable fraction \(p^\star\) satisfies \(W_{\text{res}}(p^\star) = W_{\text{mig}}(p^\star)\) with density-dependent feedback: as the resident fraction grows, winter competition depresses \(R_w\) until migration once again pays. Lundberg (1988) derived the polymorphic equilibrium:

\[ p^\star = 1 - \frac{W_{\text{mig}}^0 - W_{\text{res}}^0}{k_R - k_M} \]

where \(k_R\) and \(k_M\) are density-dependent penalties for residents and migrants respectively.

Facultative migration and irruptions

Winter finches (common redpoll, evening grosbeak), boreal owls and crossbills perform irruptive movements triggered by conifer seed-crop failure. These decisions are made in real time, are not heritable at the individual level, and can differ by orders of magnitude between consecutive years. Facultative migrants typically show weaker Zugunruhe and less stereotyped orientation in Emlen funnels because the underlying program is condition-dependent rather than date-locked.

Leapfrog migration

When high-latitude breeders migrate beyond conspecifics that breed at mid-latitudes and winter at lower latitudes still, the flyway geometry forms a leapfrog. The classic example is Calidris alpina (dunlin): Greenlandic breeders overwinter in West Africa, while British breeders winter on the UK coast. Leapfrog migration arises when migration cost scales sublinearly with distance and non-breeding habitat quality is monotonically lower near the equator, favouring the most distant travellers to go the furthest.

Leapfrog flyway schematic

3. Circannual endogenous clocks

The precision of migratory schedules—bar-tailed godwits leaving Alaska within a 72-hour window year after year—implies an internal annual clock. Gwinner’s long-running experiments in Andechs, Germany, kept captive warblers in constant photoperiod (LL 12:12 year-round) for many years and observed rhythms of moult, body mass, gonad cycle and Zugunruhe that persisted with an endogenous period close to, but not exactly equal to, one year. These circannualrhythms are entrained in the wild by photoperiodic cues relayed via pineal melatonin and deep-brain photoreceptors.

Two-oscillator formulation

A minimal model couples a circadian oscillator (driving daily restlessness) to a circannual oscillator (gating the seasonal window). The slow clock’s phase \(\Phi_Y\) obeys

\[ \frac{d\Phi_Y}{dt} = \omega_Y + Z_Y(\Phi_Y)\,P(t) \]

with endogenous frequency \(\omega_Y = 2\pi / T_Y\), photoperiodic drive \(P(t)\) and a phase-response curve \(Z_Y\) that advances or delays the clock.

The observable Zugunruhe amplitude at night \(t\) is the product of the circannual gate and circadian drive,

\[ A(t) = G(\Phi_Y)\cdot\max\!\bigl[0,\sin(\omega_D t + \phi_D)\bigr]\,,\quad G(\Phi_Y) \propto \exp\!\Bigl(-\tfrac{(\Phi_Y - \Phi_\star)^2}{2\sigma_Y^2}\Bigr) \]

Continuous-release sky-compass experiments in captive birds show that when subjects are kept in a fixed photoperiod, the circannual window of Zugunruhe slowly drifts by days each year, confirming an endogenous, free-running period and ruling out pure annual calendar readout. In the wild, photoperiod and possibly magnetic inclination lock the clock to the geographic year.

Migratory restlessness in captivity

Kramer (1949) and Emlen (1967) quantified Zugunruhe as nocturnal hopping activity during the natural migration season, even in caged birds that could not fly. The integral of activity vs time over the autumn window tracks the flight distance the free-flying population would cover, a remarkable correspondence first shown by Berthold for garden warblers, willow warblers and blackcaps. Short-distance migrants exhaust their restlessness in weeks; trans-equatorial migrants such as the garden warbler go on for three months.

4. Berthold cross-breeding and the genetics of Zugunruhe

Peter Berthold and collaborators conducted the defining experiments on the genetic basis of migration in the 1980s and 1990s. Working with Sylvia atricapilla, they took advantage of a rapidly evolving divergence within the species: since the 1950s a fraction of central European blackcaps had shifted their autumn heading from the classical SW route into Iberia to a novel NW heading that carries them to the UK, where anthropogenic food supports winter survival (Berthold et al., 1992; Rolshausen et al., 2009).

Heritability of direction

Crossing SW-heading (Germany) and NW-heading (UK-wintering) blackcaps produced F1 offspring whose mean heading was intermediate, close to the arithmetic average of the parental headings. F2 offspring showed an intermediate mean but substantially greater variance, the classical signature of additive polygenic inheritance. Berthold estimated heritability \(h^2 \approx 0.6 \text{--} 0.7\) for both orientation and amount of Zugunruhe (Berthold & Pulido, 1994).

\[ h^2 = \frac{V_A}{V_P} = \frac{V_A}{V_A + V_E} \]

narrow-sense heritability: additive genetic variance \(V_A\) over phenotypic variance.

Heritability of amount

Partial migrants in southern France were paired selectively to produce “migrant x migrant”, “migrant x resident” and “resident x resident” F1 birds. The fraction showing Zugunruhe rose monotonically with the proportion of migrant alleles, demonstrating a polygenic additive switch that sets a threshold for expression. Selection within captivity for longer migration distance pushed the population from 75% migrant to 100% migrant in a single generation, implying that migratory behaviour has the full genetic architecture needed to evolve rapidly under changing climates.

Rapid microevolution of heading

Most strikingly, Berthold and Helbig (1992) crossed German blackcaps with UK-wintering birds and recovered F1 progeny with headings close to the genetic mean. This is the most compelling evidence in any animal for a heritable change in migration directionwithin fewer than thirty generations, a timescale that underlines why migration can respond so rapidly to climate-induced shifts in habitat quality.

Berthold SW x NW blackcap cross

5. Origin hypotheses: southern-home vs northern-home

Where did migration originate? Two classic hypotheses offer opposed answers, and recent phylogenetic comparative methods have produced a mixed verdict (Winger, Auteri, Pegan & Weeks, 2019; Zink, 2011).

Southern-home hypothesis

Under the southern-home hypothesis, the ancestral range of a lineage is tropical (warm, equable, year-round resources), and migration represents a secondary range expansion: some individuals begin exploiting seasonally abundant temperate resources during the boreal summer and return to the tropics each winter. This view emphasises that the majority of Nearctic-Neotropical migrant passerines belong to lineages whose sister clades are tropical residents. The phylogenetic signal for this model is strong in Parulidae, Tyrannidae and Trochilidae.

Northern-home hypothesis

Under the northern-home hypothesis, the ancestral range is temperate (or boreal), and migration represents an escape from progressively harsher Pleistocene winters: residents persist at higher latitudes year-round, and winter mortality selects for facultative southward displacement, which becomes programmed and obligate over evolutionary time. This fits groups with strong temperate identities (e.g. thrushes Turdidae, warblers Sylviidae, sparrows Emberizidae).

Phylogenetic reconstruction

Ancestral-state reconstruction under Mk-type models weighted by geographic ranges suggests that both hypotheses are correct for different clades: Nearctic-Neotropical migrant passerines are largely tropical-origin (southern-home), while many Palearctic-African migrants and long-distance shorebirds appear to be temperate-origin (northern-home). Winger et al. (2019) emphasise that “migration” is not a single trait but a decomposable set of components—heading, distance, timing—each under separate genetic and ecological control.

Connectivity, mixing and isolating selection

Migratory connectivity measures the degree to which breeding population structure is preserved on the non-breeding grounds. Webster et al. (2002) quantify connectivity with a Mantel correlation between breeding-site and wintering-site pairwise distances; strong connectivity (coefficient \(r_M \to 1\)) means separate breeding populations overwinter in separate areas, maintaining local adaptation. Weak connectivity means breeding populations mix on the non-breeding grounds, homogenising selection pressure and accelerating gene flow. Connectivity is a strong predictor of whether climate change at the non-breeding grounds translates into breeding-population declines.

6. Mixing versus isolating selection on the flyway

Two evolutionary geometries follow from connectivity. Under isolating selection, distinct breeding populations occupy distinct non-breeding grounds and experience divergent selection pressures—drought in one Sahelian region may devastate one breeding population but leave another untouched. This geometry maintains phenotypic differentiation and, given enough time, promotes speciation along the flyway (the classical mechanism of “migratory divides”).

Under mixing selection, multiple breeding populations fuse on the non-breeding grounds and experience a shared selective regime. Phenotypic variance is homogenised, and a catastrophic non-breeding season (e.g. drought in the Sahel in 1968–1973, which collapsed several Palearctic-African migrants simultaneously) depresses breeding success across all donor populations. Migratory divides generally form between mixing and isolating clades and are zones of exceptionally rapid evolution.

\[ r_M = \frac{\sum_{ij}(d^B_{ij} - \bar{d}^B)(d^W_{ij} - \bar{d}^W)}{\sqrt{\sum_{ij}(d^B_{ij} - \bar{d}^B)^2\,\sum_{ij}(d^W_{ij} - \bar{d}^W)^2}} \]

Mantel connectivity statistic on pairwise breeding distances \(d^B\) and non-breeding distances \(d^W\).

Simulation 1: Cost–benefit of migration across a latitudinal gradient

We implement the annual fitness budget for resident and migrant strategies across breeding latitudes from 10°N to 70°N. Summer productivity rises with latitude, winter productivity collapses above the 0°C winter isotherm, and migration cost scales linearly with flyway distance to a fixed tropical refuge. We find the crossover latitude \(\phi^\star\) and the partial-migration equilibrium \(p^\star(\phi)\).

Python

script.py126 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Cost-benefit model of migration across a latitudinal gradient.
# We compare the annual net fitness of a resident strategy versus an
# obligate migratory strategy as a function of breeding-site latitude.
# The key trade-off: residents save migration energy but suffer winter
# thermal stress and reduced winter resources; migrants pay a travel
# cost but exploit high-productivity summer latitudes.

lat = np.linspace(10.0, 70.0, 241)  # breeding latitude, degrees North

# Summer productivity peaks at high latitudes (long photoperiod, insect pulse)
# Approximate as a smooth rise that saturates.
R_summer = 0.15 + 0.85 / (1.0 + np.exp(-0.15 * (lat - 35.0)))

# Winter productivity falls sharply above ~30 deg N (snow / ice / frozen insects)
R_winter = 0.90 / (1.0 + np.exp(0.20 * (lat - 32.0))) + 0.05

# Thermal stress cost rises at high latitudes in winter, zero in summer.
# Scaled as a fraction of daily energy budget burned on thermoregulation.
C_thermal_winter = 0.02 * np.maximum(lat - 25.0, 0.0) ** 1.35 / 50.0

# Migration energetic cost: distance to a fixed tropical non-breeding site
# at 10 deg N. We use a mass-specific energy cost per km and a round trip.
distance_km = 111.0 * np.maximum(lat - 10.0, 0.0)  # ~111 km per deg
energy_per_km = 1.5e-3  # fraction of annual budget per km (toy scale)
C_migration = 2.0 * distance_km * energy_per_km

# Predation / navigation hazard during migration scales with flight days.
flight_days = distance_km / 400.0  # ~400 km/day passerine average
C_hazard = 0.004 * flight_days

# Fitness components: annual net energy surplus
# Resident: summer + winter local resources minus winter thermal cost
W_resident = 0.5 * R_summer + 0.5 * R_winter - C_thermal_winter

# Migrant: summer at breeding latitude + winter at tropical site (R ~ 0.85)
# minus migration energy and hazard
R_winter_tropics = 0.85
W_migrant = 0.5 * R_summer + 0.5 * R_winter_tropics - C_migration - C_hazard

# Partial migrant: fraction p migrates; optimal p maximises population fitness
# under frequency-dependent selection (logistic mixing).
def partial_fit(p, lat_i):
    idx = np.argmin(np.abs(lat - lat_i))
    return p * W_migrant[idx] + (1 - p) * W_resident[idx] - 0.05 * p * (1 - p)

# Fitness difference: positive favours migration
delta_W = W_migrant - W_resident

# Critical latitude where migration becomes advantageous
crit_idx = np.argmin(np.abs(delta_W))
lat_crit = lat[crit_idx]

fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a1a')
for ax in [ax1, ax2, ax3, ax4]:
    ax.set_facecolor('#0a0a1a')
    ax.tick_params(colors='#cbd5e1')
    for s in ax.spines.values():
        s.set_color('#475569')
    ax.grid(True, color='#334155', alpha=0.4)

# Panel 1: resource gradient
ax1.plot(lat, R_summer, '-', color='#a78bfa', linewidth=2.5, label='Summer productivity')
ax1.plot(lat, R_winter, '-', color='#22d3ee', linewidth=2.5, label='Winter productivity (local)')
ax1.axhline(y=R_winter_tropics, color='#facc15', linestyle='--', linewidth=2, label='Tropical winter refuge')
ax1.fill_between(lat, R_summer, R_winter, color='#a78bfa', alpha=0.1)
ax1.set_xlabel('Breeding latitude (deg N)', color='#cbd5e1', fontsize=11)
ax1.set_ylabel('Resource availability (fraction of max)', color='#cbd5e1', fontsize=11)
ax1.set_title('Latitudinal resource gradient', color='#e9d5ff', fontsize=13, fontweight='bold')
ax1.legend(facecolor='#1e293b', edgecolor='#475569', labelcolor='#e2e8f0')

# Panel 2: costs of migration vs thermal stress
ax2.plot(lat, C_thermal_winter, '-', color='#22d3ee', linewidth=2.5, label='Resident thermal cost')
ax2.plot(lat, C_migration, '-', color='#f472b6', linewidth=2.5, label='Migration energetic cost')
ax2.plot(lat, C_hazard, '-', color='#f97316', linewidth=2.5, label='Migration hazard cost')
ax2.fill_between(lat, 0, C_thermal_winter, color='#22d3ee', alpha=0.15)
ax2.set_xlabel('Breeding latitude (deg N)', color='#cbd5e1', fontsize=11)
ax2.set_ylabel('Cost (fraction of annual budget)', color='#cbd5e1', fontsize=11)
ax2.set_title('Cost components', color='#e9d5ff', fontsize=13, fontweight='bold')
ax2.legend(facecolor='#1e293b', edgecolor='#475569', labelcolor='#e2e8f0')

# Panel 3: net annual fitness
ax3.plot(lat, W_resident, '-', color='#22d3ee', linewidth=3, label='Resident W')
ax3.plot(lat, W_migrant, '-', color='#a78bfa', linewidth=3, label='Migrant W')
ax3.axvline(x=lat_crit, color='#f43f5e', linestyle=':', alpha=0.8, label=f'Crossover = {lat_crit:.1f} deg')
ax3.fill_between(lat, W_resident, W_migrant, where=(W_migrant > W_resident),
                 color='#a78bfa', alpha=0.20, label='Migration favoured')
ax3.fill_between(lat, W_resident, W_migrant, where=(W_migrant <= W_resident),
                 color='#22d3ee', alpha=0.20, label='Residency favoured')
ax3.set_xlabel('Breeding latitude (deg N)', color='#cbd5e1', fontsize=11)
ax3.set_ylabel('Annual net fitness W', color='#cbd5e1', fontsize=11)
ax3.set_title('Fitness comparison: migrant vs resident', color='#e9d5ff', fontsize=13, fontweight='bold')
ax3.legend(facecolor='#1e293b', edgecolor='#475569', labelcolor='#e2e8f0', fontsize=9)

# Panel 4: optimal migrant fraction across latitudes (partial migration)
p_grid = np.linspace(0, 1, 101)
p_opt = np.zeros_like(lat)
for j, lj in enumerate(lat):
    fvals = [partial_fit(pp, lj) for pp in p_grid]
    p_opt[j] = p_grid[int(np.argmax(fvals))]

ax4.plot(lat, p_opt, '-', color='#fbbf24', linewidth=3, label='Optimal migrant fraction p*')
ax4.fill_between(lat, 0, p_opt, color='#fbbf24', alpha=0.18)
ax4.axhline(y=0.5, color='#94a3b8', linestyle=':', alpha=0.5)
ax4.set_xlabel('Breeding latitude (deg N)', color='#cbd5e1', fontsize=11)
ax4.set_ylabel('Optimal fraction migrating', color='#cbd5e1', fontsize=11)
ax4.set_title('Partial migration equilibrium', color='#e9d5ff', fontsize=13, fontweight='bold')
ax4.set_ylim(-0.05, 1.05)
ax4.legend(facecolor='#1e293b', edgecolor='#475569', labelcolor='#e2e8f0')

plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')

# Integrated (latitude-averaged) advantage using np.trapezoid
mean_advantage = np.trapezoid(delta_W, lat) / (lat[-1] - lat[0])
print(f"Crossover latitude where migrant W = resident W : {lat_crit:.2f} deg N")
print(f"Mean migration advantage integrated over latitudes : {mean_advantage:+.4f}")
print(f"Peak migrant fitness W_migrant_max = {W_migrant.max():.3f} at {lat[np.argmax(W_migrant)]:.1f} deg N")
print(f"Peak resident fitness W_resident_max = {W_resident.max():.3f} at {lat[np.argmax(W_resident)]:.1f} deg N")
print(f"Latitudes with p* > 0.5 start at {lat[np.argmax(p_opt > 0.5)]:.1f} deg N")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation 2: Berthold-style F1/F2 cross of Zugunruhe direction and amount

A polygenic additive model of n = 12 loci produces simulated Emlen-funnel orientations and nocturnal activity totals for two parental populations (SW-heading Iberian blackcaps, SE-heading UK-wintering blackcaps), their F1 hybrids and the F2 generation. We recover the classical intermediate mean of F1 and the variance expansion of F2, confirming the Mendelian signature of segregating heritable orientation and amount alleles.

Python

script.py151 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Berthold-style cross-breeding simulation of Zugunruhe (migratory
# restlessness) vector genetics in the blackcap Sylvia atricapilla.
# Berthold (1990, 1991) crossed SW-migrating Iberian blackcaps with
# newly established SE-migrating UK-wintering blackcaps and scored the
# nocturnal activity orientation of F1 and F2 offspring.
#
# We model two independent polygenic traits under additive inheritance:
#   theta  : preferred migratory heading (circular)
#   A      : amount of nocturnal restlessness (scalar)
# Each trait is governed by n_loci loci with alleles drawn from each
# parental population. F1 is heterozygous average; F2 segregates.

rng = np.random.default_rng(7)

n_loci = 12
n_birds = 400

# Parental means: SW migrators point to ~225 deg, SE to ~135 deg
theta_P1 = 225.0  # degrees
theta_P2 = 135.0
A_P1 = 1.00       # normalised restlessness
A_P2 = 0.90

# Simulate parental alleles: each locus contributes a small heading offset
sigma_locus_theta = 6.0   # deg per locus
sigma_locus_A = 0.06

alleles_P1_theta = rng.normal(theta_P1 / n_loci, sigma_locus_theta, (n_loci,))
alleles_P2_theta = rng.normal(theta_P2 / n_loci, sigma_locus_theta, (n_loci,))
alleles_P1_A = rng.normal(A_P1 / n_loci, sigma_locus_A, (n_loci,))
alleles_P2_A = rng.normal(A_P2 / n_loci, sigma_locus_A, (n_loci,))

def build_generation(alleles_mother, alleles_father, n):
    out = np.zeros(n)
    for k in range(n):
        picks = rng.integers(0, 2, size=n_loci)
        sel = np.where(picks == 0, alleles_mother, alleles_father)
        out[k] = sel.sum()
    return out

# F1: every offspring inherits n_loci/2 alleles from each parent
theta_F1 = build_generation(alleles_P1_theta, alleles_P2_theta, n_birds)
A_F1 = build_generation(alleles_P1_A, alleles_P2_A, n_birds)

# F2: cross within F1 pool - segregating alleles
def f2_pool(alleles_a, alleles_b, n):
    pool = np.concatenate([alleles_a, alleles_b])
    out = np.zeros(n)
    for k in range(n):
        idx = rng.integers(0, pool.size, size=n_loci)
        out[k] = pool[idx].sum()
    return out

theta_F2 = f2_pool(alleles_P1_theta, alleles_P2_theta, n_birds)
A_F2 = f2_pool(alleles_P1_A, alleles_P2_A, n_birds)

# Parent distributions for comparison (many siblings)
theta_P1_sim = build_generation(alleles_P1_theta, alleles_P1_theta, n_birds)
theta_P2_sim = build_generation(alleles_P2_theta, alleles_P2_theta, n_birds)

# Emlen-funnel-style activity totals: units of footprints per night
A_F1_emlen = np.clip(A_F1 * 600 + rng.normal(0, 50, n_birds), 0, None)
A_F2_emlen = np.clip(A_F2 * 600 + rng.normal(0, 80, n_birds), 0, None)
A_P1_emlen = np.clip(rng.normal(A_P1, 0.07, n_birds) * 600, 0, None)
A_P2_emlen = np.clip(rng.normal(A_P2, 0.07, n_birds) * 600, 0, None)

fig = plt.figure(figsize=(14, 11))
fig.patch.set_facecolor('#0a0a1a')

# Panel 1: circular plot of headings
ax1 = plt.subplot(2, 2, 1, projection='polar')
ax1.set_facecolor('#0a0a1a')
for ser, col, lab in [
    (theta_P1_sim, '#22d3ee', 'P1 (SW)'),
    (theta_P2_sim, '#a78bfa', 'P2 (SE)'),
    (theta_F1, '#fbbf24', 'F1'),
    (theta_F2, '#f472b6', 'F2'),
]:
    bins = np.linspace(0, 360, 37)
    hist, _ = np.histogram(ser % 360, bins=bins)
    centers = 0.5 * (bins[:-1] + bins[1:])
    ax1.plot(np.deg2rad(centers), hist, '-', color=col, linewidth=2, label=lab)
ax1.set_theta_zero_location('N')
ax1.set_theta_direction(-1)
ax1.tick_params(colors='#cbd5e1')
ax1.set_title('Orientation headings (Emlen funnel)', color='#e9d5ff', fontsize=13, fontweight='bold', pad=18)
ax1.legend(loc='upper right', bbox_to_anchor=(1.3, 1.1),
           facecolor='#1e293b', edgecolor='#475569', labelcolor='#e2e8f0', fontsize=8)

# Panel 2: distributions of activity (Zugunruhe amount)
ax2 = plt.subplot(2, 2, 2)
ax2.set_facecolor('#0a0a1a')
ax2.tick_params(colors='#cbd5e1')
for s in ax2.spines.values():
    s.set_color('#475569')
bins = np.linspace(200, 900, 40)
ax2.hist(A_P1_emlen, bins=bins, color='#22d3ee', alpha=0.5, label='P1')
ax2.hist(A_P2_emlen, bins=bins, color='#a78bfa', alpha=0.5, label='P2')
ax2.hist(A_F1_emlen, bins=bins, color='#fbbf24', alpha=0.45, label='F1')
ax2.hist(A_F2_emlen, bins=bins, color='#f472b6', alpha=0.45, label='F2')
ax2.set_xlabel('Nightly footprints (restlessness)', color='#cbd5e1', fontsize=11)
ax2.set_ylabel('Number of birds', color='#cbd5e1', fontsize=11)
ax2.set_title('Zugunruhe intensity distributions', color='#e9d5ff', fontsize=13, fontweight='bold')
ax2.grid(True, color='#334155', alpha=0.4)
ax2.legend(facecolor='#1e293b', edgecolor='#475569', labelcolor='#e2e8f0')

# Panel 3: inheritance of mean heading - F1 intermediate, F2 variance expanded
ax3 = plt.subplot(2, 2, 3)
ax3.set_facecolor('#0a0a1a')
ax3.tick_params(colors='#cbd5e1')
for s in ax3.spines.values():
    s.set_color('#475569')
labels = ['P1', 'P2', 'F1', 'F2']
means = [theta_P1_sim.mean(), theta_P2_sim.mean(), theta_F1.mean(), theta_F2.mean()]
sds = [theta_P1_sim.std(), theta_P2_sim.std(), theta_F1.std(), theta_F2.std()]
xs = np.arange(4)
ax3.bar(xs, means, yerr=sds, capsize=6,
        color=['#22d3ee', '#a78bfa', '#fbbf24', '#f472b6'],
        edgecolor='#e2e8f0')
ax3.set_xticks(xs)
ax3.set_xticklabels(labels, color='#cbd5e1')
ax3.set_ylabel('Mean heading (deg)', color='#cbd5e1', fontsize=11)
ax3.set_title('Additive inheritance of heading', color='#e9d5ff', fontsize=13, fontweight='bold')
ax3.grid(True, axis='y', color='#334155', alpha=0.4)

# Panel 4: genetic variance F2 > F1 (segregation signature)
ax4 = plt.subplot(2, 2, 4)
ax4.set_facecolor('#0a0a1a')
ax4.tick_params(colors='#cbd5e1')
for s in ax4.spines.values():
    s.set_color('#475569')
gens = ['P1', 'P2', 'F1', 'F2']
vars_A = [np.var(A_P1_emlen), np.var(A_P2_emlen), np.var(A_F1_emlen), np.var(A_F2_emlen)]
ax4.bar(gens, vars_A, color=['#22d3ee', '#a78bfa', '#fbbf24', '#f472b6'], edgecolor='#e2e8f0')
ax4.set_ylabel('Variance of restlessness', color='#cbd5e1', fontsize=11)
ax4.set_title('F2 variance exceeds F1 (Mendelian signature)', color='#e9d5ff', fontsize=13, fontweight='bold')
ax4.grid(True, axis='y', color='#334155', alpha=0.4)

plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')

print(f"Mean heading P1 = {theta_P1_sim.mean():.1f} deg, P2 = {theta_P2_sim.mean():.1f} deg")
print(f"Mean heading F1 = {theta_F1.mean():.1f} deg  (expected ~{(theta_P1 + theta_P2) / 2:.1f})")
print(f"Mean heading F2 = {theta_F2.mean():.1f} deg, sd = {theta_F2.std():.1f} deg (F1 sd = {theta_F1.std():.1f})")
print(f"Variance F2 / F1 (restlessness) = {np.var(A_F2_emlen) / np.var(A_F1_emlen):.2f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

7. Synthesis and open questions

The quantitative picture that emerges from Berthold, Alerstam, Gwinner and colleagues is one in which migration is not a single behaviour but a syndrome of genetically separable components: direction, amount, timing, and the circannual gate. Each component responds to selection independently and on remarkably short timescales. The 20th-century rise of the UK-wintering blackcap lineage in under fifty years proves that a continental-scale rerouting of a flyway can happen within a human generation when the underlying fitness landscape changes.

Open questions for this module’s students include:

How do circannual and circadian oscillators interact to produce the precise phenology observed in bar-tailed godwits and arctic terns?
Which genomic architectures underlie heritable heading changes? Candidate loci include CLOCK, ADCYAP1, and markers identified by Delmore et al. (2020) in Swainson’s thrush.
Can partial migration ESS theory predict the pace of climate-driven flyway collapse?
How does the non-breeding-ground connectivity network buffer or amplify breeding-season selection?

Modules 1 and 2 next descend from this macroevolutionary view into the sensory physics of compass systems—celestial in M1, geomagnetic in M2—that make the journey possible at all.

Key references

• Alerstam, T., Hedenström, A. & Åkesson, S. (2003). Long-distance migration: evolution and determinants. Oikos, 103, 247–260.

• Berthold, P. (1990). The control of migration in European warblers. Acta Congressus Internationalis Ornithologici, 20, 215–249.

• Berthold, P. & Helbig, A. J. (1992). The genetics of bird migration: stimulus, timing, and direction. Ibis, 134, 35–40.

• Berthold, P. & Querner, U. (1981). Genetic basis of migratory behavior in European warblers. Science, 212, 77–79.

• Berthold, P., Helbig, A. J., Mohr, G. & Querner, U. (1992). Rapid microevolution of migratory behaviour in a wild bird species. Nature, 360, 668–669.

• Berthold, P. & Pulido, F. (1994). Heritability of migratory activity in a natural bird population. Proc. R. Soc. B, 257, 311–315.

• Chapman, B. B. et al. (2011). The ecology and evolution of partial migration. Oikos, 120, 1764–1775.

• Emlen, S. T. (1967). Migratory orientation in the Indigo Bunting. Auk, 84, 309–342.

• Gwinner, E. (1967). Circannual rhythms in warblers. Naturwissenschaften, 54, 447.

• Kramer, G. (1949). Über Richtungstendenzen bei der nächtlichen Zugunruhe gekäfigter Vögel. Ornithologie als biologische Wissenschaft, 269–283.

• Lundberg, P. (1988). The evolution of partial migration in birds. Trends in Ecology & Evolution, 3, 172–175.

• Pennycuick, C. J. (2008). Modelling the Flying Bird. Academic Press.

• Rolshausen, G., Segelbacher, G., Hobson, K. A. & Schaefer, H. M. (2009). Contemporary evolution of reproductive isolation and phenotypic divergence in sympatry along a migratory divide. Current Biology, 19, 2097–2101.

• Webster, M. S. et al. (2002). Links between worlds: unraveling migratory connectivity. Trends in Ecology & Evolution, 17, 76–83.

• Winger, B. M., Auteri, G. G., Pegan, T. M. & Weeks, B. C. (2019). A long winter for the Red Queen: rethinking the evolution of seasonal migration. Biological Reviews, 94, 737–752.

• Zink, R. M. (2011). The evolution of avian migration. Biological Journal of the Linnean Society, 104, 237–250.

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