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Effective growth rates in a periodically changing environment: From mutation to invasion

Manuel Esser, Anna Kraut

TL;DR

This work develops a stochastic adaptive-dynamics framework for an asexual population in a time-periodic environment, where birth, death, and competition rates vary on an intermediate time-scale across $\ell$ phases. By letting the carrying capacity $K$ grow and mutations vanish as $\mu_K=K^{-1/\alpha}$ with $1\ll \lambda_K \ll \ln K$, the authors prove that the mutant-exponent vector $\beta^K_v(t)$ converges to a deterministic, piecewise affine function $\bar{\beta}_v(t)$ driven by time-averaged invasion fitness $f^{\mathrm{av}}_{w,v}$, yielding a mesoscopic averaging of invasion dynamics. This leads to a recursive algorithm that generates a macroscopic trait-substitution sequence $\nu$, describing successive resident traits as mutants fixate and replace incumbents through phase-aligned invasions. The results extend adaptive dynamics to periodically fluctuating environments and provide a rigorous mechanism for invasion, fixation, and substitution under environmental cycles, with implications for understanding evolution in seasonal or pulsed-treatment settings. All key quantities, including the phase-wise equilibria $\bar{n}^i_v$, invasion fitness $f^i_{w,v}$, and time-averaged fitness $f^{\mathrm{av}}_{w,v}$, are incorporated into a mathematically tractable framework for mesoscopic population dynamics.

Abstract

We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation, with linear birth and death rates, as well as a density-dependent competition. To depict repeating changes of the environment, all of these parameters vary over time as piecewise constant and periodic functions, on an intermediate time-scale between those of stabilization of the resident population (fast) and exponential growth of mutants (slow). Studying the growth of emergent mutants and their invasion of the resident population in the limit of small mutation rates for a simultaneously diverging population size, we are able to determine their effective growth rates. We describe this growth as a mesoscopic scaling-limit of the orders of population sizes, where we observe an averaging effect of the invasion fitness. Moreover, we prove a limit result for the sequence of consecutive macroscopic resident traits that is similar to the so-called trait-substitution-sequence.

Effective growth rates in a periodically changing environment: From mutation to invasion

TL;DR

This work develops a stochastic adaptive-dynamics framework for an asexual population in a time-periodic environment, where birth, death, and competition rates vary on an intermediate time-scale across phases. By letting the carrying capacity grow and mutations vanish as with , the authors prove that the mutant-exponent vector converges to a deterministic, piecewise affine function driven by time-averaged invasion fitness , yielding a mesoscopic averaging of invasion dynamics. This leads to a recursive algorithm that generates a macroscopic trait-substitution sequence , describing successive resident traits as mutants fixate and replace incumbents through phase-aligned invasions. The results extend adaptive dynamics to periodically fluctuating environments and provide a rigorous mechanism for invasion, fixation, and substitution under environmental cycles, with implications for understanding evolution in seasonal or pulsed-treatment settings. All key quantities, including the phase-wise equilibria , invasion fitness , and time-averaged fitness , are incorporated into a mathematically tractable framework for mesoscopic population dynamics.

Abstract

We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation, with linear birth and death rates, as well as a density-dependent competition. To depict repeating changes of the environment, all of these parameters vary over time as piecewise constant and periodic functions, on an intermediate time-scale between those of stabilization of the resident population (fast) and exponential growth of mutants (slow). Studying the growth of emergent mutants and their invasion of the resident population in the limit of small mutation rates for a simultaneously diverging population size, we are able to determine their effective growth rates. We describe this growth as a mesoscopic scaling-limit of the orders of population sizes, where we observe an averaging effect of the invasion fitness. Moreover, we prove a limit result for the sequence of consecutive macroscopic resident traits that is similar to the so-called trait-substitution-sequence.
Paper Structure (24 sections, 26 theorems, 176 equations, 9 figures)

This paper contains 24 sections, 26 theorems, 176 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Model and Main Results
  3. Individual-based model in a time-dependent evironment
  4. Important quantities
  5. Results
  6. Heuristics and discussion
  7. Heuristics of the proof of Theorem \ref{['thm:conv_beta']}
  8. Discussion of assumptions
  9. Mutation kernel and probability
  10. Monomorphic resident population
  11. Examples
  12. Ensured monomorphism despite temporarily fit resident trait
  13. Different possible outcomes for two-phase cycles
  14. Proofs
  15. Stability of the resident trait
  16. ...and 9 more sections

Key Result

Theorem 2.4

Let a finite graph $G=(V,E)$ and $\alpha\in{\Bbb R}_{>0}\setminus{\Bbb N}$ be given and consider the model defined by eq:time_dep_generator. Let $v_0\in V$ and assume that, for every $w\in V$, $\beta^K_w(0)\to \bar{\beta}_w(0)$ in probability, as $K\to\infty$, where the limits satisfy eq:result_init where $\bar{\beta}_w$ are the deterministic, piecewise affine, continuous functions defined in eq:d

Figures (9)

  • Figure 1: Two steps of approximation: Original process $N^{K}_v/K$ and new equilibrium $\bar{n}^i_v$ in black. Bounding birth death processes with self-competition $X^{(K,\varepsilon,-,i)}/K$ and $X^{(K,\varepsilon,+,i)}/K$ in red. Limiting deterministic solutions $x^{(\varepsilon,-,i)}$ and $x^{(\varepsilon,+,i)}$ with respective perturbed equilibrium sizes $\bar{n}^{(\varepsilon,-,i)}_v=\bar{n}^i_v-\varepsilon\underline{C}^i$ and $\bar{n}^{(\varepsilon,+,i)}_v=\bar{n}^i_v+\varepsilon\overline{C}^i$ in blue.
  • Figure 2: Concatenation of phases $i-1$, $i$ and $i+1$. Original process $N^{K}_v/K$ and corresponding equilibria in black. Bounding birth death processes with self-competition $X^{(K,\varepsilon,-,i)}/K$ and $X^{(K,\varepsilon,+,i)}/K$ in red (fast re-equilibration from step 1) and orange (long stability from step 2). Equilibrium sizes $\bar{n}^{(\varepsilon,-,i)}_v=\bar{n}^i_v-\varepsilon\underline{C}^i$ and $\bar{n}^{(\varepsilon,+,i)}_v=\bar{n}^i_v+\varepsilon\overline{C}^i$ of the corresponding (perturbed) deterministic system in blue. Bounding functions $\phi^{(K,\varepsilon,-)}_v$ and $\phi^{(K,\varepsilon,+)}_v$ in green.
  • Figure 3: Substeps of the $(k+1)^\text{st}$ invasion: Resident population $N_{v_k}^K(t)$ (blue), invading mutant $N_{v_{k+1}}^K(t)$ (red) and new emerging subpopulation $N_w^K(t)$ (green), together with the triggered stopping times and the corresponding bounds and thresholds.
  • Figure 4: Generating function $g(s)$ and the corresponding affine upper and lower bounds from \ref{['eq:bd_GenFct']}.
  • Figure 5: Graph of the limiting exponent $\beta+r_{\mathrm{av}} s$ and the time $T_\varepsilon$ that separates steps 1 and 2 of the proof of Lemma \ref{['lem:bd_conv_beta_r_neg']}.
  • ...and 4 more figures

Theorems & Definitions (57)

  • Definition 2.1: Lotka-Volterra system
  • Definition 2.2: Invasion fitness
  • Definition 2.3: Order of the population size
  • Remark 1
  • Remark 2
  • Theorem 2.4: Convergence of $\beta$
  • Remark 3
  • Corollary 2.5: Sequence of resident traits
  • Theorem 4.1
  • proof
  • ...and 47 more