Why evolution proceeds in jumps

A population evolving under selection, drift and mutation is written as a stochastic process, dX(t) = μ(X(t))dt + σ(X(t))dW(t), where X is a point in phenotype or trait space, μ is a deterministic drift term encoding selection and developmental constraint, σ scales the noise from mutation and genetic drift, and W is a Wiener process. This is the same class of equation used throughout stochastic dynamics: a drift pulling the population somewhere, and noise perturbing it along the way.

What almost every treatment of this equation assumes, usually without stating it, is that μ is Lipschitz continuous: that there exists a constant L such that ‖μ(x₁) − μ(x₂)‖ ≤ L‖x₁ − x₂‖ for any two points x₁, x₂. That bounds how fast the drift can change, which in turn is exactly the condition the Picard–Lindelöf theorem needs to guarantee a unique, well-behaved trajectory from any starting point. It is a mathematically convenient assumption, and a biologically unexamined one.

The paper replaces it with the weaker condition of Hölder continuity: ‖μ(x₁) − μ(x₂)‖ ≤ C‖x₁ − x₂‖^α for some exponent α < 1. Lipschitz continuity is the special case α = 1; below it, the drift is allowed to change arbitrarily steeply at particular points, producing cusps in the fitness landscape rather than smooth slopes everywhere. That single relaxation is what the rest of the paper builds on.

Once μ is allowed to be merely Hölder continuous, the uniqueness guarantee that Lipschitz continuity provided no longer holds at the singular points, and two distinct kinds of behaviour become possible there that a smooth landscape rules out.

The first is a finite-time blow-up: near a sufficiently steep singularity, trajectories can diverge to infinity in finite time rather than merely running away slowly. Read biologically, this is extinction as a genuine dynamical event, an ejection from viable phenotype space, rather than a gradual decline asymptoting to zero.

The second is non-uniqueness at a ridge between two fitness peaks: the same starting point can admit more than one valid future trajectory. Formally this is treated as a supercritical pitchfork bifurcation, using a parametrised potential U(x; λ) = λx − x³/3, where crossing λ = 0 breaks the symmetry of a single equilibrium into two stable ones. Stochastic noise, amplified precisely at this bifurcation, is what tips an initially uniform population toward one branch or the other: speciation read as symmetry-breaking rather than as an accumulation of small differences.

The paper is explicit that these are not abstract mathematical possibilities inserted for convenience. Developmental canalisation carves valleys into a fitness landscape with genuinely steep walls; transposable elements produce discontinuous jumps in phenotype rather than continuous drift; ecological tipping points create the kind of abrupt regime change that a smooth potential cannot represent. Non-Lipschitz behaviour is the natural mathematical signature of mechanisms already in the biological literature, not an import from elsewhere.

The framework is stated to make falsifiable predictions, not merely to redescribe the punctuation already observed in the fossil record. Three are set out explicitly.

The first is that morphological disparity should increase faster than taxonomic diversity in the aftermath of a mass extinction, detectable as a distinct scaling exponent in log–log plots of disparity against diversity in fossil databases. A bifurcation-driven radiation should spread body plans apart before it multiplies species count, rather than the two tracking each other.

The second concerns the distribution of mutational effects on phenotype: it should be heavy-tailed, better fit by a power law or a Lévy distribution than by a Gaussian, which is testable directly against deep mutational scanning datasets that already exist for many organisms.

The third is genomic: lineages undergoing rapid radiation should show ancestral enrichment for transposable elements and for intrinsically disordered protein regions, relative to sister taxa that are not radiating: a signature of the same discontinuity-generating mechanisms the model requires, visible in the genome rather than only in the resulting morphology.

As a theoretical paper, this does not fit its predictions to a single new dataset; instead it sets out where each prediction could be tested and points to systems that already look like natural test cases. Viral quasispecies and cancer cell lineages are held up as fast-evolving systems where sequencing is cheap enough to test the mutational-effects prediction directly and repeatedly.

Cichlid fish radiations in the African Great Lakes are singled out as close to an ideal system: extraordinarily fast diversification, a good fossil and genomic record, and existing debate in the literature over exactly the disparity-versus-diversity question the first prediction addresses.

The paper also positions the framework against existing ideas it is not meant to replace so much as formalise: self-organised criticality, catastrophe theory, and Wright’s adaptive-landscape picture as extended within the Extended Evolutionary Synthesis. The claim is that non-Lipschitz dynamics gives these a shared mathematical account of why punctuation happens, rather than a competing description of when it happens.

The framework makes falsifiable predictions rather than merely reframing the debate: notably that disparity and diversity decouple during adaptive radiations, and that lineages evolving rapidly should carry particular genomic signatures.

It also shares its machinery with our other work. A system whose smoothness assumptions fail is exactly the case where averaged, well-behaved models mislead, which is the same problem that motivates the kinetic treatment of populations in our socio-economic paper.