How adaptation to food resources and death rates shape oscillatory dynamics in a microbial population

TL;DR

Microbes interacting with replenishing food sources can exhibit damped oscillations around a steady state, a phenomenon not captured by Lotka-Volterra models. The authors derive a minimal consumer-resource model with abiotic substrate inflow, show that oscillations occur if and only if the timescale to adapt to food changes, $\tau_{food}$, is not shorter than the death timescale, $\tau_{death}$, quantified by $\beta = \tau_{death}/\tau_{food} \le 4$, with a more stringent regime for observable oscillations when $\beta \le 2$. They extend the framework to include necromass recycling, multiple substrates (complementary or necessary), and environmental factors (multi-species and substrate removal), revealing that a single, unifying timescale balance governs oscillatory behavior; recycling and multiple resources tend to shrink the oscillatory region, while necessary resources broaden it. Importantly, in noisy inflows, damped deterministic oscillations map to persistent oscillations, linking environmental variability to sustained cycling and offering a general lens to describe nutrient-driven microbial dynamics in complex habitats.

Abstract

Microbes constantly interact with their environment by depleting and transforming food sources. Theoretical studies have mainly focused on Lotka-Volterra models, which do not account for food source dynamics. In contrast, consumer-resource models, which consider food source dynamics, are less explored. In particular, it is still unclear what physical mechanisms control oscillatory dynamics at a single population level, a phenomenon which can only be captured by a consumer-resource model. Here, we present a minimalistic consumer-resource model of a single microbial population with growth and death dynamics, consuming a continuously replenishing substrate. Our model reveals that decaying oscillations can occur around steady state if and only if the timescale of microbial adaptation to food supply changes exceeds the death timescale. This interplay of timescales allows us to rationalize the emergence of oscillatory dynamics when adding various biophysical ingredients to the model. We find that microbial necromass recycling or complementary use of multiple food sources reduces the parameter range for oscillations and increases the decay rate of oscillations. Requiring multiple simultaneous food sources has the opposite effect. Essentially, facilitating growth reduces the likelihood of oscillations around a fixed point. We further demonstrate that such damped oscillatory behavior is correlated with persistent oscillatory behavior in a noisy environment. We hope our work will motivate further investigations of consumer-resource models to improve descriptions of environments where food source distributions vary in space and time.

Figures (13)

• Figure 1: Schematic of the minimal consumer-resource (CR) model considered in this study involving a single abiotic resource -- the substrate in blue -- and a single consumer -- the microbial population in yellow. Black arrows indicate the main physical processes at play. Here, we study the emergence of damped oscillatory dynamics in this minimal model and several model extensions corresponding to further bio-physical ingredients in gray.
• Figure 2: Schematic of the minimal reference model in Eqs. (1-2).
• Figure 3: Possible dynamical behaviors in the reference model. (A) phase diagram representing the possible dynamics in the inflow rate versus growth rate space. (B) Same diagram but with colors indicating the magnitude of the damping rate. In both (A,B) the full line indicates the limit $\beta = 4$ and the dashed line the limit $\beta = 2$. (C) Typical oscillating solution, using $r = 2 \ \tau_{\rm death}^{-1}$ and $n = 0.1 \ \tau_{\rm death}^{-1}$; (D) damped solution using $r = 20 \ \tau_{\rm death}^{-1}$ and $n = 0.1 \ \tau_{\rm death}^{-1}$ and (E) overdamped solution, using $r = 100 \ \tau_{\rm death}^{-1}$ and $n = 0.1 \ \tau_{\rm death}^{-1}$. The locations of (C, D, E) are indicated in (A-B) with colored dots. The legend is shared across (C, D, E).
• Figure 4: Two timescales govern the dynamics of the reference model in Eqs. (1-2). (A) Schematic: in the overdamped scenario, a fast regulation of food sources prevents excess death, resulting in a stable approach of equilibrium. (B) In the oscillating case, the regulation of food sources is slow, resulting in excess death response and creating oscillatory response towards equilibrium.
• Figure 5: Emergence of persistent oscillations in the presence of noise. (A, B, C) Typical solutions in the presence of $10\%$ noise source on food inflow for the same parameter values as in Fig. \ref{['fig:figure4']} (C, D, E) indicated by the colored dots. (D) Corresponding calculated noise spectra of the food source, with lines indicating theory curves from Eq. \ref{['eq:snoise']}. For all this figure $n = 0.1\ /\tau_{\rm death}$.