[Paper Review] A branching process model for dormancy and seed banks in randomly fluctuating environments

Jochen Blath, Felix Hermann+1 more|arXiv (Cornell University)|Jul 13, 2020

Evolution and Genetic Dynamics51 references24 citations

TL;DR

This paper develops a 2-type branching process model to study the evolutionary fitness benefits of dormancy in microbial populations facing randomly fluctuating environments. By comparing active-only (1-type) and active-dormant (2-type) populations under resource constraints and various switching regimes—spontaneous, responsive, or hybrid—it demonstrates that certain dormancy strategies can be super-critical (i.e., lead to population growth) even when active-only populations are sub-critical, under fair resource comparisons. The key result is that dormancy can provide a robust selective advantage, especially under responsive switching, even with reproductive trade-offs.

ABSTRACT

The goal of this article is to contribute towards the conceptual and quantitative understanding of the evolutionary benefits for (microbial) populations to maintain a seed bank (consisting of dormant individuals) when facing fluctuating environmental conditions. To this end, we compare the long term behaviour of `1-type' Bienaym\'e-Galton-Watson branching processes (describing populations consisting of `active' individuals only) with that of a class of `2-type' branching processes, describing populations consisting of `active' and `dormant' individuals. All processes are embedded in an environment changing randomly between `harsh' and `healthy' conditions, affecting the reproductive behaviour of the populations accordingly. For the 2-type branching processes, we consider several different switching regimes between active and dormant states. We also impose overall resource limitations which incorporate the potentially different `production costs' of active and dormant offspring, leading to the notion of `fair comparison' between different populations, and allow for a reproductive trade-off due to the maintenance of the dormancy trait. Our switching regimes include the case where switches from active to dormant states and vice versa happen randomly, irrespective of the state of the environment (`spontaneous switching'), but also the case where switches are triggered by the environment (`responsive switching'), as well as combined strategies. It turns out that there are rather natural scenarios under which either switching strategy can be super-critical, while the others, as well as complete absence of a seed bank, are strictly sub-critical, even under `fair comparison' wrt. available resources. In such a case, we see a clear selective advantage of the super-critical strategy, which is retained even under the presence of a (potentially small) reproductive trade-off. [...]

Motivation & Objective

• To understand the evolutionary advantages of seed banks (dormant individuals) in fluctuating environments, particularly in microbial populations.
• To model and compare the long-term survival and growth dynamics of populations with and without dormancy under random environmental fluctuations.
• To incorporate realistic constraints such as resource limitations and reproductive trade-offs between active and dormant offspring.
• To evaluate the fitness of different dormancy strategies—spontaneous, responsive, and hybrid switching—under fair resource allocation.
• To rigorously analyze the conditions under which dormancy strategies achieve super-critical growth (i.e., long-term survival and expansion) despite environmental stochasticity.

Proposed method

• Formulates a 2-type branching process where individuals are either active (Type 1) or dormant (Type 2), with distinct offspring distributions.
• Embeds the process in a randomly switching environment alternating between 'harsh' and 'healthy' states, affecting reproductive rates.
• Introduces three switching regimes: spontaneous (independent of environment), responsive (triggered by environmental state), and hybrid strategies.
• Imposes overall resource limitations to ensure fair comparison across populations, modeling the cost of maintaining dormancy.
• Uses Lyapunov exponents of random matrix products to analyze long-term growth rates, with the largest eigenvalue of the mean matrix determining criticality.
• Applies results from random matrix theory and branching process theory (e.g., [6, Theorem V.4.4]) to derive asymptotic survival and extinction probabilities.

Experimental results

Research questions

• RQ1 Under what conditions does a dormancy strategy lead to super-critical growth (i.e., positive Lyapunov exponent) in a randomly fluctuating environment?
• RQ2 How does responsive switching (environment-triggered dormancy) compare to spontaneous switching in terms of long-term population fitness?
• RQ3 Can dormancy provide a selective advantage even when there is a reproductive trade-off (i.e., cost of producing dormant offspring)?
• RQ4 Under what conditions does a 2-type branching process (with dormancy) outperform a 1-type process (active-only) in terms of survival probability and growth rate?
• RQ5 How do resource limitations and fair comparison protocols affect the relative fitness of different dormancy strategies?

Key findings

• There exist natural scenarios in which responsive switching leads to super-critical growth (i.e., ϱ > 1), while the same environment renders the 1-type process sub-critical, even under fair resource allocation.
• Spontaneous switching can also be super-critical in certain parameter regimes, but responsive switching often provides a stronger and more robust fitness advantage.
• The selective advantage of dormancy is preserved even when there is a non-zero reproductive trade-off (e.g., reduced fecundity of dormant offspring), as long as the cost is not too high.
• The survival probability σZ of the 2-type process is strictly less than that of the 1-type process (σZ < σX) when the latter is non-extinct, but the 2-type process can still grow faster due to higher Lyapunov exponent.
• The asymptotic survival rate of the 2-type process satisfies limₙ→∞ ϱ⁻ⁿP(Z₁ₙ + Z₂ₙ > 0) ∈ (0, ∞), confirming non-trivial long-term growth when ϱ > 1.
• The largest eigenvalue ϱ of the mean matrix M determines criticality: if ϱ > 1, the 2-type process survives with positive probability; if ϱ ≤ 1, extinction is almost sure.