Cannings Models in Population Genetics

• Cannings models are discrete-generation, exchangeable population frameworks that generalize classical genetic models like Wright–Fisher and Moran.
• They enable precise analysis of genealogical structures by converging to coalescent limits such as Kingman’s or multiple merger coalescents under suitable scaling.
• Recent advancements extend these models to incorporate mutation, selection, spatial structure, and seed banks, broadening their applications in population genetics.

Cannings models form a fundamental class of discrete-generation, fixed-size (or more generally, size-varying), exchangeable population models used to describe the genealogical structure, allele frequency dynamics, and evolutionary outcomes in populations under neutrality or selection. Their defining property is that the joint distribution of offspring counts per individual at each generation is exchangeable, allowing precise mathematical analysis and connections to various coalescent processes, spatial models, and branching structures. Cannings models unify and generalize the Wright–Fisher and Moran models, provide a flexible platform for studying multiple-merger coalescents, and are critically involved in the mathematical description of complex biological effects such as selection, mutation, spatial structure, seed banks, and hierarchical block-resampling.

1. Formal Structure and General Formalism

A classical Cannings model for a haploid population consists of discrete generations of fixed (or deterministically/varying) size $N$. The offspring distribution in each generation is determined by an exchangeable vector $\nu = (\nu_1, ..., \nu_N)$ of non-negative integers, summing to $N$ (or $N_{r+1}$ if the generation sizes are allowed to vary in time), and the vectors across generations are independent and identically distributed (i.i.d.). Exchangeability means the law of $(\nu_1,\dots,\nu_N)$ is invariant under any permutation of indices, ensuring neutrality of the reproduction mechanism (Siri-Jégousse et al., 2022, Li et al., 12 Nov 2025).

Many variants exist:

• Wright–Fisher model: $\nu \sim \mathrm{Multinomial}(N; 1/N, ..., 1/N)$.
• Dirichlet-mixed (paintbox) model: Offspring weights $W$ are random on the simplex, $\nu \sim \mathrm{Multinomial}(N; W_1, ..., W_N)$ (Huillet et al., 2021).
• Generalized (inhomogeneous) Cannings: Allows generation sizes $q(s)$ to vary, with appropriate exchangeable offspring vectors per generation (Li et al., 12 Nov 2025).

The key technical object is the pairwise coalescence probability,

$c_N = \mathbb{E}[\nu_1(\nu_1 - 1)] / (N(N-1)),$

which dictates the rate of coalescence in the associated ancestral process (Siri-Jégousse et al., 2022).

2. Genealogy, Coalescent Limits, and Scaling Regimes

Under a suitable time and population scaling, forward- and backward-time processes derived from Cannings models converge to classical and generalized coalescent limits.

• Kingman's coalescent arises universally if the variance in offspring number is finite and higher moments vanish as $N \to \infty$: merges are almost always binary (Gufler, 2016, Huillet et al., 2021).
• Lambda ($\Lambda$)-coalescents and Xi ($\Xi$)-coalescents: If the model allows very large family sizes with non-negligible probability (heavy-tailed parental fecundities), simultaneous or multiple mergers in ancestral process persist, characterized by a finite measure $\Lambda$ (for simple multiple mergers) or $\Xi$ (for simultaneous multiple mergers; (Huillet et al., 2021, Siri-Jégousse et al., 2022, Birkner et al., 2017)).

The Möhle–Sagitov criterion provides necessary and sufficient moment conditions for convergence to Kingman’s (only binary mergers) vs. $\Lambda$/$\Xi$-coalescent (multiple merger) limits (Gufler, 2016).

A scaling theorem for inhomogeneous models describes convergence (after rescaling time by $1/c_N$ and population by $N$) of rescaled genealogical trees to continuum random trees encoded by time-changed Brownian motion conditioned on local time profiles, under suitably regular moment assumptions (Li et al., 12 Nov 2025).

3. Extensions: Mutation, Selection, and Multi-Type Structure

Mutation. In models with $K$ allelic types and neutral or parent-dependent mutation matrix $P = (p_{ij})$, offspring types are drawn according to $P$ after reproduction. Under parent-independent mutation and near-Kingman genealogy, the stationary distribution of type frequencies is close to the Dirichlet distribution, with explicit error bounds given in terms of higher moments of offspring counts (Gan et al., 2016).

Selection. Selective effects are built in by modifying the parental weights, e.g., giving a beneficial type an advantage $s_N$ (often scaling with population size) in the weight vector, in neutral or paintbox-based Cannings. Dualities with ancestral selection graph (ASG) structures and coupling to Moran model ASG under moderately weak/strong selection establish that the asymptotic fixation probability is always $2s_N/\rho^2$ in the Kingman domain, with $\rho^2$ as the limiting offspring variance (Boenkost et al., 2019, Boenkost et al., 2020).

Multi-Type Structure. When individuals have types in $E$, the process structure generalizes to multi-type Cannings, with exchangeable reproduction within types and mutation. The limiting genealogy, under appropriate scaling, is a multi-type exchangeable coalescent with integral representations for the merger rates incorporating type structure (Flamm et al., 14 Apr 2025, Möhle, 2023).

4. Spatial, Hierarchical, and Seed Bank Cannings Models

Spatial/hierarchical structure. Cannings models can be extended to populations on (ultra)metric lattices or hierarchical groups, with local and block-level resampling events, migration, and possible random environments (quenched $\Lambda$-measures per block). The genealogy is then described by interacting spatial coalescents with block-level multiple mergers and duality relations (Greven et al., 2017, Greven, 2019, Greven et al., 2012). Sound scaling of migration/resampling parameters yields dichotomies between "clustering" (fixation/monomorphism) and "local coexistence" (persistent diversity), determined by infinite sums involving block-wise volatility and resampling rates.

Seed banks. Models with dormancy (seed banks) introduce additional lags in the genealogical graph, effectively lengthening coalescent times and reducing the probability of rapid coalescence. Scaling limits show that the coalescence clock is slowed by the average seed bank time, genealogical process is stretched, and site-frequency spectra and genetic drift are smoothed (Casanova et al., 2022).

5. Markovian and Tree-Valued Process Representations

Cannings models can be recast as Markov chains on tree-space, giving a process-valued genealogy perspective. The state encapsulates the ultrametric matrix of pairwise coalescence times or as an equivalence class in the space of marked metric-measure spaces. Invariance principles assert that under suitable rescaling and conditions, these chains converge to tree-valued Fleming–Viot processes encoding the full genealogy of the population (Gufler, 2016). The limiting generator involves both tree growth and resampling (coalescence), with rates given by the governing $\Xi$ (or $\Lambda$) measure.

A recent result establishes that the scaling limit of contour/height functions associated to genealogical trees in (possibly inhomogeneous) Cannings models converges to time-changed Brownian motion conditioned on prescribed local time profiles, providing a continuum random tree limit (Li et al., 12 Nov 2025).

6. Applications and Biological Significance

Cannings models serve as a mathematical foundation for analyzing virtually all neutral population genetics scenarios:

• Population genetics inference: Predicting site-frequency spectra, genealogical distances, and effect of demography on diversity (Casanova et al., 2022, Greven et al., 2017).
• Experimental evolution and microbial adaptation: Modeling of adaptation, fitness trajectories, and mutation fixation dynamics in large experimental populations, e.g., Lenski’s LTEE, is facilitated by the Cannings structure (Baake et al., 2018).
• Marine and high-fecundity species: Heavy-tailed offspring distributions in the Cannings framework explain multiple merger coalescent events observed in such species.
• Spatial population structure and block catastrophes: Hierarchical Cannings with block-resampling capture the effects of environmental or catastrophic events on genetic diversity (Greven et al., 2012).
• Theoretical genomics: Tree-valued and metric-measure representation clarifies statistical properties of genetic data under Cannings genealogies (Gufler, 2016).

7. Limitations, Generalizations, and Recent Directions

The Cannings framework is general but presumes exchangeable offspring numbers for each generation. Generalizations to non-exchangeable but non-heritable offspring distributions (e.g., asymmetric Cannings) yield convergence to $\Xi$-coalescents under weaker symmetry, encompassing models with highly asymmetric reproduction and recurrent bottlenecks (Siri-Jégousse et al., 2022). Conditional (quenched) coalescent processes for fixed pedigrees in diploid Cannings distinguish sharply from the annealed (averaged) coalescent, with significant implications for multi-locus genealogical statistics (Alberti et al., 21 May 2025).

Recent research is focused on:

• Quantitative error bounds in finite-$N$ settings for allele frequency distributions (Gan et al., 2016).
• Full characterization of genealogical scaling limits under complex, time-inhomogeneous, or random environments (Freund, 2019, Greven et al., 2017).
• Multi-type, selection, recombination, and seed-bank extensions (Möhle, 2023, Casanova et al., 2022).
• Tree-valued or marked metric-measure process frameworks, which provide powerful invariance principles and universality classes (Gufler, 2016, Greven et al., 2017, Greven et al., 2012).

The Cannings framework thus underpins much of modern mathematical population genetics, with current research spanning from rigorous probabilistic limit theorems to concrete biological applications spanning a spectrum of evolutionary phenomena.