Frequency Locking to Environmental Forcing Suppresses Oscillatory Extinction in Phage-Bacteria Interactions

TL;DR

This work addresses how intrinsic bacterial traits and periodic environmental forcing shape phage–bacteria dynamics beyond static environments. It introduces a minimal ODE framework coupling logistic bacterial growth with phage infection and lysis, augmented by a time-varying carrying capacity $K(t)$ to model environmental forcing; analyses reveal three possible fates—phage extinction, stable coexistence, or oscillation-driven extinction—modulated by the growth rate $r$, adsorption rate $a$, and forcing parameters. A key finding is that environmental forcing can suppress destructive oscillations through resonance, producing frequency-locked states (Arnold tongues) that stabilize populations and expand survival regions, especially under high infection pressure. These results illuminate how environmental rhythms can promote coexistence and resilience in microbial communities, with potential implications for phage therapy, microbiome management, and climate-influenced ecosystem stability, while highlighting limitations such as the focus on lytic phages and periodic, rather than irregular, forcing.

Abstract

Bacteriophage-bacteria interactions are central to microbial ecology, influencing evolution, biogeochemical cycles, and pathogen behavior. Most theoretical models assume static environments and passive bacterial hosts, neglecting the joint effects of bacterial traits and environmental fluctuations on coexistence dynamics. This limitation hinders the prediction of microbial persistence in dynamic ecosystems such as soils and oceans.Using a minimal ordinary differential equation framework, we show that the bacterial growth rate and the phage adsorption rate collectively determine three possible ecological outcomes: phage extinction, stable coexistence, or oscillation-induced extinction. Specifically, we demonstrate that environmental fluctuations can suppress destructive oscillations through resonance, promoting coexistence where static models otherwise predict collapse. Counterintuitively, we find that lower bacterial growth rates are helpful in enhancing survival under high infection pressure, elucidating the observed post-infection growth reduction.Our studies reframe bacterial hosts as active builders of ecological dynamics and environmental variation as a potential stabilizing force. Our findings thus bridge a key theory-experiment gap and provide a foundational framework for predicting microbial responses to environmental stress, which might have potential implications for phage therapy, microbiome management, and climate-impacted community resilience.

Figures (10)

• Figure 1: Schematic illustration of the lytic phage infection cycle. (1) Free phage particles are adsorbed to uninfected bacterial hosts $B$ (top left). (2) Successful adsorption converts $B$ into infected intermediates $I$ (top right). (3) Maturation of $I$ culminates in cell lysis, releasing progeny virions that re-initiate infection (bottom).
• Figure 2: Dynamics and stability of the bacteria-phage system in the absence of an extinction threshold. (a-c) Time series for three different phage adsorption rates: (a) $\hat{a} = 0.09$, (b) $\hat{a} = 0.005$, and (c) $\hat{a} = 0.0006$. The dynamics of the bacterial population (B, blue), infected bacteria (I, red), and phage (P, yellow) exhibit qualitatively distinct behaviors: a stable limit cycle (a), stable coexistence at equilibrium (b), and bacterial dominance with phage extinction (c). (d) Phase diagram in the $r-\hat{a}$ parameter space, illustrating the regions of different dynamical behavior. The solid line represents the analytical threshold $a_{\text{crit}} = \delta / [K (\Omega - 1)]$, below which phage invasion is impossible. The dashed line indicates the numerically determined boundary where the leading eigenvalue of the coexistence equilibrium acquires a positive real part, indicating the arise of a Hopf bifurcation. Gray region: bacteria persist, but with phage extinction; cyan region: stable coexistence of bacteria and phages; green region: limit cycle oscillations of the two species. Points A, B, and C denote the parameter sets inspected in panels (a), (b), and (c), respectively. Parameters: $\hat{K} = 1$, initial conditions $B(0) = 10^7$, $I(0) = 10^7$, $P(0) = 10^5$, and all other parameters as listed in Table \ref{['tab:para']}.
• Figure 3: Typical time series of the population dynamics with (a) and without (b) an extinction threshold $\epsilon=1$, where the population size of each species falling below this value is set to zero. Identical parameters and initial conditions may lead to eventual extinction of all the types of populations in panel (a), or to the occurrence of oscillations of them in panel (b), depending on whether the critical extinction threshold is imposed on or not during transient oscillations.
• Figure 4: Basin of attractions and extinction boundary. (a) Three-dimensional map of the basin of attractions generated from simulations starting from different initial conditions. Each point represents a unique initial condition $(B_0, I_0, P_0)$; green points indicate outcomes converging to stable coexistence of all the types of populations $B$, $I$ and $P$, while blue points indicate those trajectories leading to system-wide extinction. Panels (c-d) show two-dimensional slices of the 3D basin in (a) along each principal plane, providing orthogonal views of the basin structure. (e) An iso-surface plot delineating the boundary between the basins of attraction for coexistence and extinction phase. We systematically scanned the initial conditions over the ranges: $B_0 \in [10^{6}, 5\times10^7]$, $I_0 \in [10^{6}, 5\times10^7]$, and $P_0 \in [10^{6}, 10^{8}]$, logarithmically spaced with 50 points in each dimension. Simulations were performed with parameter values $\hat{K} = 1$, $\hat{a} = 0.06$, $r = 1$; all other parameters are the same as listed in Table \ref{['tab:para']}.
• Figure 5: Dependence of the survival phase on the bacterial growth rate $r$ and phage adsorption rate $a$. (a) Heatmap of the survival probability for bacteria-phage coexistence. Color intensity represents long-term survival probability for different combinations of the bacterial growth rate ($r$ , y-axis) and the phage adsorption rate ($a$ , x-axis). High growth rates maximize survival at low phage pressure (top-left), while low growth rates dominate under high phage pressure (bottom-right ), revealing a competitive reversal in defense strategies. (b) and (c) Representative time series of the simulations without an extinction threshold at a fixed adsorption rate $\hat{a} = 0.07$ and fixed initial conditions $B_0 = P_0 = 0.1$, showing the distinct population dynamics for a low initial $I_0 = 0.05$ (b) and a high initial $I_0 = 0.4$ (c), for various growth rates $r$. The black dashed line indicates the extinction threshold $\epsilon = 1$. (d)-(k) Two-dimensional slices of the basin of attraction in the $B_0$–$I_0$ plane for a fixed initial phage density $P_0 = 0.1$. Green and blue colors correspond, respectively, to the cases of initial conditions leading to coexistence and extinction. Panels (d)-(g) correspond to a fixed adsorption rate $\hat{a} = 0.038$ for different $r$, while panels (h)-(k) correspond to a fixed $\hat{a} = 0.07$ for different $r$. The geometry of the survival basin is highly sensitive to both the bacterial growth rate and the adsorption rate.