How seed banks evolve in plants: a stochastic dynamical system subject to a strong drift

We study how changes in population size and fluctuating environmental conditions influence the establishment of seed banks in plants. Our model is a modification of the Wright-Fisher model with seed bank, introduced by Kaj, Krone and Lascoux. We distinguish between wild type individuals, producing only nondormant seeds, and mutants, producing seeds with dormancy. To understand how changing population size shapes the establishment of seed banks, we analyse the process under a diffusive scaling. The results support the biological insight that seed banks are favoured in a declining population, and disfavoured if population size is constant or increasing. The surprise is that this is true even when population sizes are changing very slowly – over evolutionary timescales. We also investigate the influence of short-term fluctuations, such as annual variations in rainfall or temperature. Mathematically, our analysis reduces to a stochastic dynamical system forced onto a manifold by a large drift, which converges under scaling to a diffusion on the manifold. Inspired by the Lyapunov–Schmidt reduction, we derive an explicit formula for the limiting diffusion coefficients by projecting the system onto its linear counterpart. This provides a general framework for deriving diffusion approximations in models with strong drift and nonlinear constraints.

💡 Research Summary

This paper investigates the evolutionary dynamics of seed banks in plants by extending the classical Wright–Fisher model to include a dormancy trait. Two genotypes are considered: a wild type that produces only non‑dormant seeds and a mutant that can produce dormant seeds. The authors ask under which demographic and environmental conditions a mutation conferring dormancy can invade and become fixed, and how its fixation probability compares with that of a neutral allele lacking a seed bank.

The mathematical core of the work is a stochastic dynamical system with a very large drift term that forces the process onto a low‑dimensional manifold. After a suitable diffusive scaling, the dynamics separate into a fast deterministic component (the strong drift) that rapidly projects the state onto the manifold, and a slow stochastic component that evolves along the manifold. By adapting ideas from Lyapunov–Schmidt reduction, the authors derive a general quadratic approximation for the flow near an attracting manifold and obtain explicit formulas for the diffusion coefficients of the limiting stochastic differential equation (SDE). This provides a systematic method for reducing high‑dimensional, multi‑timescale population models to tractable one‑dimensional diffusions.

Three ecological scenarios are analysed:

1. Constant population size. Theorem 2.5 shows that, when the total population remains fixed, the seed‑bank mutation is not selectively advantageous: its fixation probability is of the same order as that of a neutral non‑dormant allele. However, once the seed‑bank trait is common, loss of the trait is unlikely (Theorem 2.3, Proposition 2.4), matching empirical observations in stable environments.

2. Slowly varying population size (evolutionary timescale). Theorem 2.6 and Corollary 2.7 demonstrate that a declining population favours the dormancy trait, whereas a growing population favours the non‑dormant type. Random fluctuations in population size on the evolutionary timescale tend to produce a stable coexistence equilibrium where both genotypes persist. These results explain why seed banks are more prevalent in habitats subject to long‑term stress (e.g., high‑latitude or arid regions).

3. Fast environmental fluctuations (single‑generation timescale). By modelling annual variations in seed production while keeping the total adult population constant, Theorem 2.10 shows that short‑term adverse conditions (temporary reductions in seed output) can increase the fixation probability of the dormancy allele for a broad class of germination‑time distributions. This captures the intuition that “bad years” make a dormant seed bank a valuable insurance policy.

The paper situates its contributions within a broad literature on seed banks, highlighting that earlier models (e.g., Cohen 1966, Blath & Tóbiás 2016) either assume an already established seed bank, impose geometric germination times, or neglect genetic drift. By incorporating both drift and demographic stochasticity, the present work bridges population genetics and ecological theory.

Methodologically, Sections 3.1–3.4 develop the dynamical‑systems framework, while Sections 4–5 combine it with stochastic analysis to prove the diffusion approximations and the fixation theorems. The authors also provide a detailed proof of the quadratic manifold approximation (Section 6.2) and illustrate it on a concrete ODE example.

In conclusion, the study provides a rigorous mathematical justification for the long‑standing biological insight that seed banks are advantageous in declining or highly variable environments, even when the dormancy period spans only a few generations. Moreover, the reduction technique for systems with strong drift offers a versatile tool for future work on other life‑history traits involving multiple timescales. Potential extensions include more realistic germination‑time distributions, explicit costs of dormancy, and empirical validation using genetic data from plant populations across climatic gradients.