Dynamics of fluctuating populations in multi-state switching environments

Technical Summary: Dynamics of Fluctuating Populations in Multi-State Switching Environments

Problem Statement
Microbial populations typically evolve in volatile environments where resource availability fluctuates over time. Traditionally, these "feast–famine" cycles have been modeled using binary (two-state) switching models, where the carrying capacity abruptly switches between abundant and scarce conditions. While mathematically tractable, recent experimental evidence suggests that real-world feast–famine environments often exhibit more complex temporal dynamics, involving gradual transitions through intermediate states rather than abrupt jumps. These intermediate conditions significantly influence population bottlenecks, strain competition, and the evolution of ecological strategies (e.g., generalist vs. specialist). The paper addresses the gap in theoretical understanding regarding how populations compete in environments characterized by a finite number of intermediate states, each with distinct carrying capacities, and how these multi-state dynamics differ from their binary counterparts.

Methodology
The author investigates the competition between two strains: a faster-growing type ($F$) with baseline fitness $f_F=1$ and a slower-growing type ($S$) with fitness $f_S=1-s$ (where $0 < s \ll 1$). The population size $N(t)$ is time-fluctuating and limited by a carrying capacity $K(t)$ that evolves as a continuous-time Markov chain.

1. Model Formulation: The environment is modeled as a $(2n+1)$-state system, ordered from the harshest state ($i=-n$, famine) to the mildest ($i=n$, feast). Each state $i$ possesses a specific carrying capacity $K_i$, ordered such that $K_{-n} < \dots < K_0 < \dots < K_n$. Transitions occur between nearest-neighbor states with rates $\nu_{2n+1}^\pm$ (for harsh states) and $\omega_{2n+1}^\pm$ (for mild states), allowing for asymmetric switching dynamics controlled by a bias parameter $\delta$ and a rate asymmetry parameter $\epsilon$.
2. Stochastic Simulation: The individual-based dynamics are simulated using the Gillespie algorithm. The system is defined by birth-death processes where birth rates depend on fitness and death rates depend on the logistic term $N/K(t)$.
3. Analytical Approaches:
   • Mean-Field Analysis: Used to establish timescale separations between ecological dynamics (population size $N$) and evolutionary dynamics (strain fraction $x$).
   • Moran Approximation: Applied in static environments to derive fixation probabilities and mean fixation times.
   • Piecewise Deterministic Markov Process (N-PDMP): A hybrid approximation where $N$ evolves deterministically between environmental switches. This is used to approximate the population size distribution (PSD) and fixation probabilities in fluctuating environments.
   • Limit Regimes: Analytical expressions are derived for "slow switching" (where environmental changes are slower than selection) and "fast switching" (where environmental fluctuations self-average), utilizing "quenched" and "annealed" approximations respectively.

Key Contributions and Results

• Population Size Distribution (PSD):

  • The study characterizes the PSD across three regimes: slow, intermediate, and fast switching.
  • In the slow switching regime ($\nu \ll 1$), the PSD is multimodal, exhibiting $2n+1$ peaks corresponding to the carrying capacities of the individual environmental states. This contrasts with binary models, which show only two peaks.
  • In the fast switching regime ($\nu \gg 1$), the PSD becomes unimodal, centered around an effective carrying capacity $K_{2n+1}$ (the harmonic mean of $K_i$ weighted by the stationary distribution).
  • The PSD is generally right-tailed due to the asymmetry of logistic growth and decay. The presence of intermediate states broadens the distribution compared to binary models.

• Fixation Probability of the Slow Strain ($\phi$):

  • The paper quantifies how the probability of the slower strain ($S$) fixing (taking over the population) depends on the number of states ($n$), the switching rate, and the distribution of carrying capacities.
  • Slow Switching: The fixation probability is approximated by a weighted average of Moran fixation probabilities over the stationary distribution of states. Multi-state environments can either enhance or suppress $\phi$ relative to binary models, depending on the position of the median carrying capacity ($K_0$) and the environmental bias ($\delta$). Specifically, if the environment spends significant time in harsh states (low $K$), the fixation probability of the slow strain can be higher than in binary models.
  • Fast Switching: The fixation probability is determined by the effective carrying capacity $K_{2n+1}$. The paper derives conditions under which $K_{2n+1}$ in multi-state models is smaller than the effective capacity in binary models, thereby increasing the fixation probability of the slow strain.
  • Intermediate Switching: An N-PDMP-based approximation (Eq. 19) is proposed, which rescales the switching rate to account for the number of switches occurring on the evolutionary timescale. This approximation accurately captures the non-monotonic behavior of fixation probability in the intermediate regime, including weak peaks observed in simulations.

• Comparison with Binary Models:

  • The study demonstrates that multi-state models capture dynamical features absent in binary models, such as the existence of intermediate population bottlenecks and complex fixation scenarios where the outcome depends non-trivially on the switching frequency and amplitude.
  • The environmental bias ($\delta$) significantly influences fixation probabilities; for instance, a bias toward mild states ($\delta > 0$) generally reduces the fixation probability of the slow strain. Furthermore, increasing the number of environmental states generally reduces the fixation probability of the slow strain, an effect that is strongly amplified when the environmental bias favors mild conditions.

Significance and Claims
The paper claims that multi-state stochastic switching models provide a necessary coarse-grained framework for understanding feast–famine cycles that cannot be captured by binary approximations. By explicitly modeling intermediate environmental states, the author shows that the frequency and amplitude of environmental fluctuations jointly shape eco-evolutionary dynamics in complex ways.

Specifically, the work highlights that:

1. Intermediate states matter: The presence of gradual transitions between feast and famine alters population size statistics and fixation probabilities, leading to scenarios where the slower strain is either more or less likely to fixate compared to abrupt switching scenarios.
2. Analytical tractability: Despite the complexity, the dynamics can be effectively characterized using N-PDMP approximations and limit regimes, providing a bridge between microscopic stochastic processes and macroscopic population statistics.
3. Experimental relevance: The models align with recent experimental findings suggesting that the temporal statistics of environmental fluctuations (frequency and amplitude) are critical determinants of microbial adaptation and competition outcomes.

The author concludes that this framework offers a versatile tool for investigating how statistical properties of environmental variability influence evolutionary outcomes, without claiming to be a mechanistic description of specific nutrient uptake processes. The work is presented as a step toward understanding the more difficult problem of eco-evolutionary dynamics under continuous environmental noise.