The question, in the language of quantitative genetics
Take a large population with a heritable quantitative trait x (body size, flowering time, a behavioral strategy). Individuals with trait value x leave descendants at rate f(x). Selection reshapes the trait distribution: it can shift it, narrow it, skew it, or split it in two.
Most models do not follow the full distribution. They follow its mean and variance and assume it remains normal (the Gaussian closure behind Lande’s equation). That is exact for some fitness landscapes and wrong for others, and the debate about when it is admissible goes back to Turelli (1984). The paper gives a number for the cost: the invariance defect.
| Term used here | Reading for an evolutionary biologist |
|---|---|
| Density ρ(x) | The distribution of the trait in the population. One point in “the space of all densities” is one whole population. |
| Fitness f(x) | Per-capita growth rate of individuals with trait x (Malthusian fitness). Its graph is the individual fitness landscape. |
| Replicator equation | Pure selection: types copy themselves faithfully in an infinite population, with no mutation, recombination, or drift. |
| Parametric family Pθ, statistical manifold | The assumed form of the trait distribution. Here it is the normal family, so θ = (μ, σ²): the set of all populations the model is able to describe. |
| Scores, tangent space | The ways the model lets a population change in the next instant. For a normal distribution there are two: shift the mean, change the variance. |
| Fisher projection, natural gradient | The best mean-and-variance description of what selection is doing right now. The mean equation is Lande’s equation (Robertson’s secondary theorem); the variance equation is its second-moment companion. |
| Var(f) | Fisher’s fundamental theorem in this setting: the rate at which selection raises mean fitness. |
| Invariance defect δ² | The part of that rate carried by changes of shape (skew, extra modes) that a mean and a variance cannot register. It is the cost of the Gaussian closure, per unit time. |
| R² | The fraction of the response to selection that the closure retains. |
The side panel sets the fitness landscape and the current trait distribution. Every figure below reads from those settings.
1. Selection changes the whole trait distribution
A population is a density ρ(x) over a trait x. Each trait replicates at rate f(x). The replicator equation says a trait gains share when its fitness is above the population mean:
Dividing by ρ gives the relative velocity w = ∂tρ / ρ = f − Eρ[f]. It has zero mean under ρ, which is how the equation conserves total mass. Everything below is geometry applied to w.
Drag the peak sideways to move μ. With a mouse, drag up or down to change σ. The lower strip is the replicator velocity ρ·w at this instant.
The gold curve is the fitness landscape and the gray curve is the population sitting on it. Trait values whose fitness exceeds the population mean (above the dashed line) are increasing in frequency; the others are decreasing. The lower strip shows that turnover: how fast each part of the distribution is growing or shrinking. Nothing in this equation says the distribution must stay normal.
2. A mean-and-variance model can change in only two ways
Tangent vectors at ρ are zero-mass perturbations. The Fisher-Rao metric measures them as relative velocities weighted by the density: ⟨g, h⟩ρ = ∫ g h ρ dx, so ‖g‖² = Eρ[g²]. In this geometry, and for fitness that does not depend on ρ, the replicator equation is natural-gradient ascent of mean fitness on the space of all densities.
A parametric family Pθ cannot move in every direction. A parameter velocity v produces the relative velocity v·uθ(x), where uθ = ∇θ log Pθ is the score. The reachable directions form the tangent space Vθ, the span of the scores. For a Gaussian that span has two functions: z = (x − μ)/σ, which shifts the mean, and z² − 1, which changes the variance.
The best the family can do is the least-squares fit of w by score functions under the current density:
This is the natural-gradient update used in information-geometric optimization and natural evolution strategies. For the Gaussian it reads dμ/dt = Cov(X, f) and dσ²/dt = Cov((X − μ)², f), which is Lande’s equation with its second-moment companion.
The fit is weighted by the density drawn along the bottom edge, so mismatch far in the tails counts for little. The current fit gives .
The teal curve is the quadratic regression of fitness on phenotype, weighted by the current population. It is the same kind of regression used to estimate selection gradients. For a normal trait distribution its two coefficients are the continuous-time analogs of the directional gradient β = E[f′] and the quadratic gradient γ = E[f″], and the model’s equations are dμ/dt = σ²β and dσ²/dt = σ⁴γ. At the current state . In this model all trait variance is heritable, so σ² plays the role of the additive genetic variance.
Whatever the regression leaves unexplained (the shaded residual) is selection that acts on the population but has no place in a mean-and-variance description.
What selection asks of the distribution right now, in three parts
Shift the mean
Change the variance
Change the shape
Each panel shows the current population (gray) and the population a short moment later if only that part of selection acted (colored line). The same time step is used in all three, so sizes are comparable. The first two are the tangent directions of the normal family. The third is the residual: a change of shape, such as skew or a second mode, that no normal distribution can make. Percentages are shares of the variance in fitness.
3. The defect: adaptation the model cannot register
The invariance defect is the length of the residual, the distance from w to the tangent space:
Fisher’s fundamental theorem, in continuous-trait form, says the full replicator flow raises mean fitness at a rate equal to the fitness variance, d/dt E[f] = Var(f). The projected flow raises it at rate c⊤F−1c. The projection is orthogonal, so the two rates, taken at the same state, differ by the Pythagorean remainder:
full rate = rate the family can express + rate lost to the restriction
So δ² is more than a fitting error. It is the adaptation rate, in fitness per unit time, that a population forfeits at this state by being confined to the family. R² = 1 − δ²/Var(f) is the fraction of the selection response that the family retains.
Fisher’s theorem says mean fitness climbs at a rate equal to the variance in fitness. A mean-and-variance model registers only the part of that variance explained by the quadratic regression of fitness on phenotype. The R² of that regression is the fraction of adaptation the model sees. The unexplained variance δ² is adaptation that the real population achieves by changing the shape of its distribution.
Where these objects live
Drag to rotate. Three spaces are in play. (1) The surrounding volume stands for the space of all trait distributions; every point is a whole population, and most are not normal. (2) The curved surface is the statistical manifold of normal distributions, with coordinate lines of constant μ and constant σ. Moving the sliders moves the dot on it. (3) The flat sheet is the tangent space at the current population: the instantaneous changes a normal distribution can make, spanned by the mean score and the variance score. Selection pushes the population along the black arrow. Its shadow on the sheet is what the model follows; the vertical remainder δ points off the surface, toward non-normal populations. Vector components are computed from the current state (arrow length is normalized to 1, true length √Var(f)); the curvature of the surface is schematic.
Fisher’s theorem, split at the current state
- Full replicator rate, Var(f)
- Cov(X, f)² / σ²
- Cov((X−μ)², f)² / 2σ⁴
- Defect δ²
- δ
- R²
- Angle between w and the plane
- Directional + stabilizing (quadratic). δ is zero at every state. A Gaussian family is forward-invariant exactly when fitness is affine-quadratic. For an exponential family the condition is fitness affine in the sufficient statistics, the same closure as Bayesian conjugacy.
- Stabilizing, with ripples. δ grows in proportion to ε, so δ² grows as ε².
- Many optima. Widen σ. The population spans several fitness peaks, a quadratic cannot fit them, and R² falls toward 0. On the surface figure the arrow tilts up, away from the manifold.
δ depends on the family and the fitness, not on the coordinates used for θ. Under f → af + b with a > 0 it scales by a, and R² is unchanged.
4. The real population and its normal approximation, side by side
Both flows start at the state set in the side panel. When fitness does not depend on the population, the full flow has a closed form, an exponential tilt of the initial density (an e-geodesic): ρt ∝ ρ0 et f. The projected flow integrates dθ/dt = F−1c and stays Gaussian.
The same two histories, drawn on the manifold
Drag to rotate. The surface is the family of normal distributions over the range of means and standard deviations visited in this run. The teal path is the Fisher-projected flow and cannot leave the surface. The black path is the full replicator flow: its footprint is the normal distribution with the same mean and standard deviation, and its height is how far the real population is from normal, measured as the square root of the relative entropy to that best-matching normal (current value ). Blue connectors join the two populations at equal times. Positions and heights are computed; the curvature of the surface is schematic.
Mean fitness
Selection at the projected state
Distance between the two densities
Mean fitness rises along both flows, and the projected flow never decreases it, because its rate c⊤F−1c is nonnegative. The projection is also a moment closure: it uses the exact moment equations but evaluates them under the Gaussian instead of the true density. Mean and variance therefore agree with the full flow to first order in t, while density shape separates at once, with ‖ρt − Pθt‖1 ≤ t δ(θ0) + o(t).
The gray population is what selection alone produces. The teal curve is what a model that tracks only mean and variance predicts. The model uses the right equations for the mean and the variance but evaluates them as if the population were still normal, so its error starts in the shape of the distribution and only later contaminates the mean and variance. Mean fitness in the model can lag the real population, but it never goes down.
The middle chart stacks the two terms of the split along the projected path; its total height is Var(f) at the projected state. The theorem compares rates at a common state, so this chart is not the rate of the full flow along its own path. The accumulated defect ∫δ ds is a diagnostic of where omitted velocity is generated. The paper does not treat it as a calibrated error estimate.
5. Stalling: zero selection gradients under disruptive selection
The projected flow stops when the fitness-score covariance c vanishes. If δ > 0 at that state, fitness variance remains and the full flow keeps adapting at rate δ² while the Gaussian does not move. The two descriptions then disagree about whether anything is happening.
The double well f(x) = −(x² − a²)² is the standard case. A Gaussian centered between the peaks with σ² = a²/3 is stationary, with δ² = Var(f) = 8a8/27 and R² = 0. The state is not attracting. A small asymmetry μ0 grows, but slowly: near the stalling state the local escape time is t★ = 3 / (16 a² μ0²).
Parameter plane (μ, σ)
Densities at time t
Picture a population centered in the valley between two adaptive peaks. By symmetry the directional gradient is zero. At one particular variance the quadratic gradient is zero as well: disruptive selection near the center and stabilizing selection in the tails cancel on average. A mean-and-variance model then reports no response to selection, while the real population is dividing toward the two peaks. Zero estimated gradients are not evidence that selection is absent. The defect is what distinguishes this case from a population at an optimum, where the variance in fitness itself is small.
This flat chart is a map of the same manifold drawn as a surface in sections 3 and 4. the direction of the projected flow. The thinner path is the mean and standard deviation of the full density; that density is not Gaussian, so the path is only its shadow on the parameter plane. Practical reading from the paper: monitor the represented rate and δ together. A small represented rate with a large δ signals a representational stall, not convergence.