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

TL;DR

The paper develops a rigorous diffusion-approximation framework for the evolution of seed banks in plants within a Wright–Fisher setting, spanning constant, slowly varying, and fast-changing environments. Central to the analysis is a stochastic dynamical system driven by a large drift that is projected onto a one-dimensional attractor Γ via a projection map Φ, yielding diffusion limits on Γ. By coupling Katzenberger’s dimension-reduction theory with Parsons–Rogers derivatives and Lyapunov–Schmidt reduction, the authors obtain explicit formulas for limiting diffusion coefficients and second-order terms, enabling precise comparisons of fixation probabilities for seed-bank mutations against neutral non-dormant mutations. Across environmental regimes, the results consistently show seed banks are favored under population decline or fluctuation and disfavored under growth, with fast environmental fluctuations strongly buffering against adverse conditions. These findings provide a principled, general framework for diffusion approximations in models with strong drift and nonlinear constraints, with implications for understanding dormancy as an adaptive strategy in plants under environmental change.

Abstract

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.

Figures (5)

• Figure 1: For $B \in \{0, 0.1, 0.5, 1, 2\}$, we plot the function $\psi_B$ that determines, for sufficiently small $y$, the initial condition for our diffusion approximation with seed bank that we argue in the main text provides an appropriate comparison to the case without seed bank started from initial condition $y$. Observe that, as expected (c.f. Lemma \ref{['Paper03_lemma_correspondence_value_branching_phase']}), $\psi_B$ is a decreasing function of the mean germination time $B$.
• Figure 2: Comparison of the fixation probability $\mathbb P(T_1 < T_0 )$ of the SDE \ref{['Paper03_SDE_WF_constant_environment']} in (a) with the upper bound provided by the map $\Psi(B,0.01)$ in (b) for the case $K = 2$, as a function of the parameters $b_0$ and $b_1/(1 - b_0)$. The fixation probability was computed via the scale function of the the SDE using the formula \ref{['Paper03_drift_K_2']}, and starting the diffusion phase from $x_0 = \psi_B(0.01)$, while $\Psi(B,0.01)$ was computed using the identity \ref{['Paper03_identity_fixation_probability_bound']}. Plot (c) shows the difference $\Psi(B,0.01) - \mathbb P(T_1 < T_0)$. Note that the vertical axis in (c) has a different scale from those in (a) and (b).
• Figure 3: Fixation probability of $(\rho_0(t))_{t \geq 0}$ for different values of $\xi_{\infty} \in \{0.7, 0.8, 0.9, 1.0, 1.1, 1.2\}$ and $b_0 \in (0,1]$. We assume the dynamics of $(\xi(t))_{t \geq 0}$ is described by \ref{['Paper03:logistic_growth']} with $r = 20$, and that $(\rho_0(t))_{t \geq 0}$ solves the SDE \ref{['Paper03_SDE_number_mut_deterministic_env']} with $K = 1$, so that $\frac{\partial^2 \Phi_0}{\partial \rho^2}(\boldsymbol{\rho})$ is given by \ref{['Paper03_drift_K_1']}. To take into account the impact of dormancy on the branching phase (see Lemma \ref{['Paper03_lemma_correspondence_value_branching_phase']}), we computed the fixation probability assuming that $\rho_0(0) = \psi_B(0.01)$ for all $B = 1 - b_0$, where $\psi_B$ is the function defined in Lemma \ref{['Paper03_lemma_correspondence_value_branching_phase']}. Observe that in the absence of dormancy, i.e. when $b_0 = 1$, the fixation probability of a neutral trait is not affected by the changes in population size. Our numerical simulations suggest that for sufficiently small values of $\xi_{\infty}$, i.e. for sufficiently large declines in population size, the evolution of the seed bank trait is favoured with respect to the evolution of a neutral trait without dormancy.
• Figure 4: Plot of the function $g$ defined in \ref{['Paper03_integrand_understand_impact_slow_fluctuations_environment']} for $K = 1$ and $B = 1 - b_0 \in \{0.1,\, 0.5,\, 1.0\}$. The values of $g$ were computed by applying \ref{['Paper03_drift_K_1']} to \ref{['Paper03_integrand_understand_impact_slow_fluctuations_environment']}, fixing different values of $\xi$ in panels (a) ($\xi = 0.8$) and (b) ($\xi = 1.2$). For large values of $B$, the function $g$ becomes negative when the proportion of mutants $\rho_0$ is sufficiently high. The threshold value of $B$ for which this occurs depends on the population size (parameter $\xi$); indeed, in contrast to panel (a), in panel (b) the function $g$ is negative for $B = 0.5$ and $\rho_0 > 0.9$.
• Figure 5: Contour plot of the function $h(x_0,b_0)$ of \ref{['Paper03_functionh_fast_env']} on the domain $[0,1]^2$. Darker colours are associated with larger values of $h$, while lighter colours are associated with smaller values. For $K = 1$, $h$ is strictly non-negative, its maximum value is $4/3$ and is achieved at $(0,0)$. The plot suggests that in the fast-changing environment, the seed bank trait is favoured, as expected.

Theorems & Definitions (70)

• Remark
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Proposition 2.4
• Theorem 2.5
• Remark
• Theorem 2.6
• Corollary 2.7
• Proposition 2.8