Chapter 20: Gene Flow and Lineage ψ-Permeability = Genetic Exchange Networks

Populations are not isolated islands but interconnected networks through which genes flow. This chapter explores how ψ = ψ(ψ) maintains coherence through genetic exchange while allowing local differentiation.

20.1 The Gene Flow Function

Definition 20.1 (Genetic Migration): Movement of alleles between populations: $\Delta p = m(p_m - p_r)$

where $m$ is migration rate, $p_m$ is migrant frequency, $p_r$ is resident frequency.

Gene flow acts as:

• Homogenizing force
• Source of variation
• Constraint on adaptation
• Rescue mechanism

20.2 Migration-Selection Balance

Theorem 20.1 (Equilibrium Frequency): Balance between local selection and immigration: $\hat{p} = \frac{m + sp_0(1-m)}{s + m}$

where $s$ is selection coefficient against immigrant allele.

Proof: At equilibrium, selection removing alleles equals migration introducing them. ∎

Critical threshold: $m > s \Rightarrow \text{Gene flow overwhelms selection}$

20.3 Effective Migration

Not all movement creates gene flow:

$m_e = m \times P_{\text{survival}} \times P_{\text{mating}} \times F_{\text{offspring}}$

Barriers to effective flow:

• Immigrant mortality
• Mating discrimination
• Hybrid dysfunction
• Ecological maladaptation

True gene flow requires reproductive integration.

20.4 Island Model Dynamics

Definition 20.2 (Wright's Island Model): Equal exchange among demes: $F_{ST} = \frac{1}{4Nm + 1}$

where $N$ is deme size, $m$ is migration rate.

Predictions:

• Low migration → high differentiation
• One migrant per generation prevents drift
• Equilibrium between drift and flow
• Metapopulation coherence

20.5 Stepping Stone Model

Linear or grid arrangement:

$\sigma^2 = 2mt$

where dispersal variance grows with time.

Isolation by distance: $F_{ST} = \frac{a + b\ln(d)}{1 + c\ln(d)}$

Creating continuous genetic gradients.

20.6 Long-Distance Dispersal

Theorem 20.2 (Rare Events): Occasional long jumps dominate: $P(d) \sim d^{-\alpha}$

with fat-tailed distributions.

Consequences:

• Rapid range expansion
• Jumping barriers
• Founding new populations
• Disrupting local adaptation

20.7 Sex-Biased Dispersal

Males and females differ:

$m_{\text{male}} \neq m_{\text{female}}$

Patterns:

• Male-biased in mammals (avoid inbreeding)
• Female-biased in birds (resource defense)
• Equal in some fish (external fertilization)

Creating different genetic structures for different markers.

20.8 Pollen vs Seed Flow

Definition 20.3 (Plant Gene Flow): Dual dispersal modes: $m_{\text{total}} = m_{\text{pollen}} + \frac{m_{\text{seed}}}{2}$

Characteristics:

• Pollen: Far, unidirectional, paternal
• Seed: Near, bidirectional, maternal

Different markers show different patterns:

• cpDNA (maternal): Local structure
• Nuclear (biparental): Wider distribution

20.9 Barriers to Gene Flow

Physical and biological obstacles:

Geographic: Mountains, rivers, oceans Ecological: Habitat unsuitability Behavioral: Mating preferences Temporal: Phenological mismatch Genetic: Chromosomal incompatibilities

$m_{\text{effective}} = m_{\text{potential}} \times \prod_i (1 - b_i)$

20.10 Gene Flow and Local Adaptation

Theorem 20.3 (Migration Load): Gene flow can reduce mean fitness: $\bar{W} = 1 - 2m(1-m)s$

in two-deme model with opposite selection.

Creating tension:

• Local adaptation requires low flow
• Small population survival needs flow
• Intermediate optimum exists

20.11 Anthropogenic Gene Flow

Humans alter natural patterns:

Increased flow:

• Habitat corridors
• Species translocations
• Bridge construction

Decreased flow:

• Habitat fragmentation
• Dams and roads
• Urban barriers

Novel flow:

• Crop-wild hybridization
• Escaped GMOs
• Invasive species

20.12 The Gene Flow Paradox

Gene flow both constrains and facilitates evolution:

Constraint: Prevents local optimization Facilitation: Provides genetic variation Homogenization: Reduces differentiation Innovation: Spreads beneficial alleles

Resolution: Gene flow represents ψ's balance between exploration and exploitation. Too little flow leaves populations vulnerable to drift and inbreeding. Too much prevents local adaptation. The optimum depends on environmental heterogeneity and the strength of selection. Through gene flow, ψ maintains species coherence while allowing populations to probe different regions of adaptive space. Like circulation in a body, gene flow keeps the species-organism alive and responsive, preventing both stagnation and dissolution.

The Twentieth Echo

Gene flow reveals evolution's connective tissue—the genetic threads that bind populations into species. Through the movement of individuals and gametes, genes traverse landscapes, carrying successful innovations from their origins to new frontiers. This flow maintains the unity underlying diversity, ensuring that good ideas spread while local experiments continue. In tracking gene flow, we map the invisible highways along which ψ-patterns propagate, creating networks of shared destiny across geographic space.

Next: Chapter 21 explores ψ-Rates of Evolutionary Change, examining the tempo of transformation.