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A branching process model for dormancy and seed banks in randomly fluctuating environments

Jochen Blath, Felix Hermann, Martin Slowik

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' Bienaymé-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. [...]

A branching process model for dormancy and seed banks in randomly fluctuating environments

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' Bienaymé-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. [...]

Paper Structure

This paper contains 16 sections, 11 theorems, 65 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Biological motivation
  3. Deterministic vs. stochastic population dynamic modeling and known results
  4. Modeling approach and organization of the present paper
  5. Models and main results
  6. Branching processes with dormancy in constant environment
  7. Branching processes with dormancy in randomly fluctuating environment
  8. Asymptotic growth of BGWPDREs and Lyapunov exponents
  9. Fair comparison of BGWPDREs with different switching strategies
  10. Technical results for rank-1 and rank-2 switching strategies
  11. Rank-1-matrices and exact results
  12. Rank-2-matrices and bounds
  13. Discussion and outlook
  14. Discussion
  15. Outlook
  16. ...and 1 more sections

Key Result

Proposition 2.2

Let $X = (X_n)$ be a 1-type BGWP, and $Z = (Z_n)$ a BGWPD with $X_0 = 1$ and $Z_0 = (1,0)$. Assume that the offspring distributions $Q_X$ and $Q_Z$, respectively, are of finite variance with $Q^2_Z(0,0) > 0$ and $\mathop{\mathrm{\mathbb{P}}}\nolimits[Z^2_1>0] > 0$. Set and denote by the survival probabilities and extinction times of $Z$ and $X$, respectively. If for all $k \geq 1$, then the fol

Figures (9)

  • Figure 1: Offspring distribution of $Z$ for active (white) individuals on the left and dormant (gray) individuals on the right.
  • Figure 2: Parameter regimes of Example \ref{['exps:strong-advantages']} (1) and (3) respectively, Lyapunov exponents taken as functions of $\alpha$. black: $\varphi_X$, red: $\varphi_{\mathrm{res}}$, blue: $\varphi_{\mathrm{sto}}$, green: $\varphi_{\mathrm{pre}}$.
  • Figure 3: Phase diagram of the maximal Lyapunov exponents $\varphi_X$, $\varphi_{\mathrm{res}}$, $\varphi_{\mathrm{sto}}$, $\varphi_{\mathrm{pre}}$ of Example \ref{['exps:strong-advantages']} with $\alpha = 1/20$ (left) and $\alpha = 1/10$ (right). Strong advantage of $\varphi_{\mathrm{res}}$ (red), $\varphi_{\mathrm{sto}}$ (blue), and $\varphi_{\mathrm{pre}}$ (green). Advantegous and supercritical: light red, light blue, light green resp.
  • Figure 4: Phase diagram of the maximal Lyapunov exponents $\varphi_X$, $\varphi_{\mathrm{res}}$, $\varphi_{\mathrm{sto}}$, $\varphi_{\mathrm{pre}}$, $\varphi_{\mathrm{cc}}(1/6)$ and $\varphi_{\mathrm{cc}}(5/6)$ of Example \ref{['exps:strong-advantages']} with $\alpha = 1/20$ (left) and $\alpha = 1/10$ (right), and Remark \ref{['rem:convex-combination']} with $q=1/6$ and $q=5/6$. Strong advantage of $\varphi_{\mathrm{res}}$ (red), $\varphi_{\mathrm{sto}}$ (blue), $\varphi_{\mathrm{pre}}$ (green), $\varphi_{\mathrm{cc}}(1/6)$ (purple), and $\varphi_{\mathrm{cc}}(5/6)$ (cyan).
  • Figure 5: Phase diagram of the maximal Lyapunov exponents $\varphi_X$, $\varphi_{\mathrm{res}}$, $\varphi_{\mathrm{sto}}$, $\varphi_{\mathrm{pre}}$ of Remark \ref{['rem:weighted-fair-comp']} with $\alpha = 1/20$, $\gamma = 1/2$ (left) and $\gamma = 2$ (right). Strong advantage of $\varphi_{\mathrm{res}}$ (red), $\varphi_{\mathrm{sto}}$ (blue), $\varphi_{\mathrm{pre}}$ (green), and $\varphi_{X}$ (black).
  • ...and 4 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3: Binary random environment
  • Remark 2.4
  • Example 2.5: Switching strategies
  • Remark 2.6: Comparison to multi-type branching process models considered in DMB11 and JW14
  • Remark 2.7: Comparison to switching strategies employed in MS08
  • Remark 2.8: Exact computation of Lyapunov exponents
  • Remark 2.9: Approximation of Lyapunov exponents
  • Remark 2.10: Lyapunov exponent, fitness and survival-probability
  • ...and 29 more