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Math: Demographic Models

Demographic models for population genetics analysis.

Functions: exponential_growth_effective_size, bottleneck_effective_size, two_epoch_effective_size.

Overview

Demographic history (population size changes) affects genetic diversity and the distribution of polymorphisms. This module provides basic demographic models to estimate effective population size under different demographic scenarios.

Functions

exponential_growth_effective_size(current_size, growth_rate, generations)

Calculate effective population size under exponential growth.

Parameters:

  • current_size: Current population size (N_t)
  • growth_rate: Per-generation growth rate (r). Positive for growth, negative for decline
  • generations: Number of generations over which to calculate harmonic mean

Returns: Effective population size (harmonic mean)

Formula: Under exponential growth N(t) = N₀ × e^(r×t), the effective size is the harmonic mean: Ne = t / Σ(1/N_i)

Example:

from metainformant.math.demography import exponential_growth_effective_size

# Population growing from ~1000 to 10000 over 10 generations
ne = exponential_growth_effective_size(
    current_size=10000,
    growth_rate=0.23,  # ~23% per generation
    generations=10
)
# Returns: ~4342 (harmonic mean of sizes over time)

bottleneck_effective_size(pre_bottleneck_size, bottleneck_size, bottleneck_duration, recovery_generations=0)

Calculate effective population size through a bottleneck.

Parameters:

  • pre_bottleneck_size: Population size before bottleneck
  • bottleneck_size: Population size during bottleneck
  • bottleneck_duration: Number of generations at bottleneck size
  • recovery_generations: Number of generations after bottleneck (default: 0)

Returns: Effective population size (harmonic mean)

Example:

from metainformant.math.demography import bottleneck_effective_size

# Severe bottleneck: 10000 → 100 for 5 generations
ne = bottleneck_effective_size(
    pre_bottleneck_size=10000,
    bottleneck_size=100,
    bottleneck_duration=5
)
# Returns: ~109 (heavily weighted by bottleneck)

# With recovery to original size over 10 generations
ne_recovery = bottleneck_effective_size(
    pre_bottleneck_size=10000,
    bottleneck_size=100,
    bottleneck_duration=5,
    recovery_generations=10
)
# Returns: ~200 (recovery improves effective size)

two_epoch_effective_size(ancient_size, current_size, time_since_change)

Calculate effective population size for two-epoch model.

Parameters:

  • ancient_size: Population size before change
  • current_size: Population size after change
  • time_since_change: Number of generations since size change

Returns: Effective population size (harmonic mean)

Example:

from metainformant.math.demography import two_epoch_effective_size

# Population expansion: 1000 → 10000, 50 generations ago
ne = two_epoch_effective_size(
    ancient_size=1000,
    current_size=10000,
    time_since_change=50
)
# Returns: ~1960 (harmonic mean of 1000 and 10000)

Use Cases

1. Interpreting Genetic Diversity

Effective population size affects genetic diversity:

from metainformant.math.demography import exponential_growth_effective_size
from metainformant.dna.population import nucleotide_diversity

# Calculate diversity
sequences = ["AAAA", "AAAT", "AATT"]
pi = nucleotide_diversity(sequences)

# Estimate effective size from growth model
ne = exponential_growth_effective_size(
    current_size=10000,
    growth_rate=0.1,
    generations=100
)

# Compare observed diversity to expected under model
# (requires mutation rate)

2. Modeling Population History

Combine multiple demographic events:

from metainformant.math.demography import (
    bottleneck_effective_size,
    exponential_growth_effective_size,
)

# Model: Bottleneck, then recovery and growth
# Step 1: Bottleneck
ne_bottleneck = bottleneck_effective_size(
    pre_bottleneck_size=10000,
    bottleneck_size=100,
    bottleneck_duration=5,
    recovery_generations=10
)

# Step 2: Growth after recovery
ne_growth = exponential_growth_effective_size(
    current_size=10000,
    growth_rate=0.05,
    generations=50
)

# Overall effective size is dominated by bottleneck

3. Comparing Demographic Models

Compare different demographic scenarios:

from metainformant.math.demography import (
    exponential_growth_effective_size,
    two_epoch_effective_size,
)

# Model 1: Exponential growth
ne_exp = exponential_growth_effective_size(10000, 0.1, 50)

# Model 2: Two-epoch (sudden change)
ne_epoch = two_epoch_effective_size(1000, 10000, 50)

# Compare effective sizes
print(f"Exponential growth Ne: {ne_exp:.2f}")
print(f"Two-epoch Ne: {ne_epoch:.2f}")

Theory

Harmonic Mean

Effective population size under changing population size is the harmonic mean:

Ne = t / (1/N₁ + 1/N₂ + ... + 1/Nₜ)

This reflects that coalescence rates are inversely proportional to population size, so periods of small population size dominate the effective size.

Exponential Growth

Under exponential growth: N(t) = N₀ × e^(r×t)

The effective size is less than the current size because the harmonic mean is weighted by the smaller historical sizes.

Bottlenecks

Bottlenecks dramatically reduce effective population size because:

  1. Small population sizes during bottleneck dominate harmonic mean
  2. Genetic drift is much stronger during bottleneck
  3. Recovery period may not fully compensate

Limitations

  1. Deterministic models: These models assume deterministic population size changes. Stochastic fluctuations are not modeled.

  2. Simple scenarios: Complex demographic histories (multiple bottlenecks, migration, etc.) require more sophisticated models.

  3. No migration: Models assume isolated populations. Migration would require additional parameters.

  4. No selection: Models assume neutral evolution. Selection would affect genetic diversity differently.

Integration with Other Modules

With Coalescent Theory

from metainformant.math.demography import exponential_growth_effective_size
from metainformant.math.population_genetics.coalescent import expected_segregating_sites

# Calculate expected diversity under demographic model
ne = exponential_growth_effective_size(10000, 0.1, 100)
theta = 4 * ne * 0.0001  # Assuming mutation rate μ = 0.0001
S_expected = expected_segregating_sites(
    sample_size=10,
    theta=theta,
    sequence_length=1000
)

With Effective Size Estimators

from metainformant.math.demography import bottleneck_effective_size
from metainformant.math.popgen import effective_population_size_from_heterozygosity

# Estimate Ne from observed heterozygosity
observed_H = 0.3
mutation_rate = 0.0001
ne_estimated = effective_population_size_from_heterozygosity(
    observed_heterozygosity=observed_H,
    mutation_rate=mutation_rate,
    ploidy=2
)

# Compare to demographic model
ne_model = bottleneck_effective_size(10000, 100, 5)

References

  • Nei, M., Maruyama, T., & Chakraborty, R. (1975). The bottleneck effect and genetic variability in populations. Evolution, 29(1), 1-10.

  • Maruyama, T., & Fuerst, P. A. (1984). Population bottlenecks and nonequilibrium models in population genetics. I. Allele numbers when populations evolve from zero variability. Genetics, 108(3), 745-763.

  • Slatkin, M., & Hudson, R. R. (1991). Pairwise comparisons of mitochondrial DNA sequences in stable and exponentially growing populations. Genetics, 129(2), 555-562.

See Also