Pith. sign in

REVIEW 3 major objections 4 minor 52 references

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

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Periodic environmental forcing can rescue phages and bacteria from collapse by locking their population oscillations to the forcing frequency, keeping bacterial densities above the extinction threshold.

desk verdict A competent nonlinear-dynamics study whose central 'resonance rescue' claim likely rests on an extinction threshold set 10^8 times too high. read the letter →

arxiv 2512.08224 v2 pith:PFFRYSMJ submitted 2025-12-09 physics.bio-ph nlin.CDq-bio.PE

Hao-Neng Luo , Zhi-Xi Wu , Jian-Yue Guan This is my paper · ORCID
classification physics.bio-phnlin.CDq-bio.PE MSC 92D2534C1537N2592D40
keywords phage-bacteriainteractionsfrequencylockingenvironmentalforcingresonancecoexistenceextinctionthresholdArnoldtonguepopulationoscillations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Most phage-bacteria models assume a static environment, so large-amplitude oscillations can drive one or both populations to extinction. This paper argues that when the environment varies periodically, the bacterial population can synchronize to that rhythm, and this frequency locking reduces oscillation amplitude enough to prevent the deadly crashes. The result would turn environmental variability from a threat into a possible stabilizer, and it gives a concrete mechanism for the coexistence of lytic phages with susceptible bacteria observed in nature. The same framework also explains why slower bacterial growth can be an advantage under strong phage pressure: it damps the oscillations that would otherwise drive extinction.

What carries the argument

The central object is a minimal three-species ODE model (susceptible bacteria B, infected bacteria I, free phage P) with logistic bacterial growth, phage adsorption, and lysis. The mechanism is the coupling between the intrinsic limit-cycle oscillator (created by a Hopf bifurcation) and the periodic external forcing K(t) through the growth term r(1−N/K(t)). The key identity is the frequency-locking condition f≈fc·n, where fc is the natural limit-cycle frequency, which produces Arnold tongues in the amplitude–frequency plane. The extinction threshold ε=1, imposed as an absorbing boundary in the rescaled variables, is what converts the mathematical limit cycle into biological extinction and wh

What would settle it

Rerun the same simulations with the extinction threshold expressed in absolute abundance, e.g., ε=10^-8 in the rescaled variables (corresponding to 1 cell/mL), and check whether the unforced (A=0) limit cycle's minimum bacterial density B_min ever falls below that threshold. If it does not, then the oscillation-induced extinction that the frequency locking is said to prevent does not occur in a biologically grounded model, and the Arnold-tongue rescue is an artifact of the overly high threshold.

Watch

Extended reading notes

Core claim

The central claim is that sinusoidal variation of the carrying capacity, K(t)=K0+AK0 sin(2πft), can suppress the large-amplitude limit-cycle oscillations of a lytic phage-bacteria system and rescue the bacteria from extinction. In the unforced system, when the adsorption rate is high enough, the coexistence equilibrium loses stability via a Hopf bifurcation and the bacteria exhibit deep population crashes that fall below the extinction threshold, leading to collapse. When the forcing frequency is near a harmonic of the intrinsic limit-cycle frequency, the system enters an Arnold tongue of 1:1 (or 1:n) frequency locking; the oscillation amplitude shrinks, the minimum bacterial density rises a

Load-bearing premise

The rescue effect hinges on the extinction threshold being ε=1 in units where the reference carrying capacity is 10^8 cells/mL—meaning populations are declared extinct only when their density falls below a far larger value than a true minimum viable population; if the realistic threshold is orders of magnitude lower (say 1 cell/mL), the unforced limit cycle likely never dips below it, so there is no collapse to rescue.

Editorial extensions

If this is right

  • If correct, environmental periodicity becomes a controllable stabilizing factor: phage-bacteria persistence can be predicted by comparing environmental frequency with the intrinsic oscillation frequency of the host-phage pair.
  • The model offers a mechanistic explanation for synchronized population rhythms observed in microbial communities: entrainment to external cycles may be adaptive because it prevents catastrophic crashes.
  • Slow bacterial growth is not merely a cost of resistance; under high phage pressure it is a survival trait that reduces oscillation amplitude, consistent with observed post-infection growth reduction.
  • Environmental variability has a dual role—destabilizing at low-to-moderate infection pressure but stabilizing at high infection pressure—so management strategies for phage therapy or microbiome engineering must account for the infective regime.
  • The Arnold-tongue geometry implies that the rescue effect is non-monotonic in both amplitude and frequency, so not all fluctuations are equal: only those near resonance protect the population.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The resonance mechanism is generic: any predator-prey or host-parasite system that exhibits Hopf-induced limit cycles and experiences periodic environmental modulation should show similar Arnold-tongue stabilization, not just phages and bacteria.
  • Because the stabilization is confined to narrow frequency windows, real-world environmental noise (which spans many frequencies) may rarely hit the resonance condition, so the protective effect might be weaker in nature than in this idealized single-frequency forcing.
  • A testable extension: in a chemostat with periodic nutrient pulses, the minimum bacterial density should peak when the pulse frequency is an integer multiple of the intrinsic oscillation frequency; measuring this curve would directly verify the predicted resonance windows.
  • The model treats the extinction threshold as an absorbing boundary; replacing it with demographic stochasticity or an Allee-effect term would likely produce the same qualitative behavior but might blur the Arnold-tongue boundaries, suggesting that the sharp rescue windows are a deterministic idealization.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript studies a minimal ODE model of lytic phage-bacteria interaction with logistic bacterial growth, infection, lysis, phage decay, and a periodically forced carrying capacity K(t)=K0+AK0 sin(2πft). The authors perform a linear stability analysis, identify phage-free, coexistence, and limit-cycle regimes, and then introduce an extinction threshold to study initial-condition-dependent bistability. With the threshold in place, they report that periodic forcing can reduce oscillation amplitude via frequency locking (Arnold tongues), elevating the local minima of the bacterial population above the extinction threshold and 'rescuing' a population that would otherwise collapse in a static environment. They also report a counterintuitive trade-off in which lower bacterial growth rates improve survival under high phage adsorption pressure. The main ecological claim is that environmental fluctuations can suppress destructive oscillations and promote coexistence where static models predict collapse.

Significance. The frequency-locking mechanism itself is a well-motivated nonlinear-dynamics observation: forced limit-cycle oscillators can synchronize to external periodic driving, and the paper provides a clear demonstration of Arnold tongues with amplitude reduction in a phage-bacteria context. If robust, this could be an interesting contribution to microbial ecology and phage-therapy modeling. Strengths include the explicit analytical expression for the phage-invasion threshold acrit, the systematic basin-of-attraction scans, and the openly available MATLAB code. The significance of the central claim, however, hinges entirely on the extinction threshold used in the simulations. Because that threshold is shown here to be dimensionally inconsistent and is never sensitivity-tested, the ecological 'rescue' narrative is not yet established.

major comments (3)
  1. [II.B, Table I, Figs. 3 and 9] The extinction threshold is load-bearing but dimensionally inconsistent. The simulations use rescaled variables \hat B=B/K_r with K_r=10^8 cells/mL, so setting ε=1 truncates any density below 10^8 cells/mL, not '1 individual per unit volume' as Sec. II.B states. In Figs. 9(a,b) the unforced system has B_min<ε, so the 'collapse' is just the normal fluctuation of a population below its carrying capacity. Since ε is never sensitivity-tested, the rescue effect and the entire extinction narrative may be artifacts of this threshold. Please repeat the key simulations with ε=10^-8 (1 cell/mL) and report whether the unforced limit cycle actually crosses that threshold; if it does not, the central claim as stated must be revised.
  2. [III.E, Fig. 9(c)] The claimed correspondence between survival regions and Arnold tongues is threshold-dependent. In Fig. 9(c), survival is defined by B_min>ε; lowering ε expands the survival regions and could make them cover most of the (A,f) plane, decoupling them from the tongue structure. The paper should quantify the amplitude-reduction effect directly (e.g., oscillation amplitude or peak-to-trough range before and after locking) independently of ε, and show that the tongue structure itself, rather than the arbitrary cutoff, drives any 'rescue'.
  3. [III.C, Fig. 5(a)] The competitive reversal (slower growth improves survival under high adsorption) is measured as the fraction of initial conditions with B_min>ε. This metric conflates genuine persistence with avoiding an arbitrary density cutoff. With a biologically plausible ε of 1 cell/mL, the extinction basin may shrink dramatically and the reversal may vanish. Please recompute the survival heatmaps with at least two threshold values (e.g., 1 cell/mL and 10^-2 cells/mL) and report whether the qualitative pattern persists.
minor comments (4)
  1. [III.C, Fig. 5] The y-axis label 'Bacterial Density (mL-1)' is inconsistent with initial conditions such as B0=0.1, which indicate rescaled dimensionless variables. Please use consistent units throughout all figures.
  2. [III.B] The statement that 'phase trajectories do not cross' is not automatically obvious for a non-autonomous system. Specify that the no-crossing property holds in the extended state space including time.
  3. [Throughout] Typos: 'bactrial' in Sec. III.C, 'lefting' in the Fig. 7 caption, and 'arise' in Sec. III.E. These should be corrected.
  4. [Sec. II.B / Code availability] The code repository [35] would benefit from a version stamp or DOI to aid reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the frequency-locking rescue is an emergent simulation result, not a fitted or self-referential construct.

full rationale

The central claim is that sinusoidally varying K(t) can lock the limit-cycle oscillation to the forcing frequency and thereby raise B_min above the extinction threshold. This is not circular: the model (Eq. 1) contains no frequency-locking term or survival criterion; the Arnold-tongue structure in Fig. 8 is diagnosed by the dominant-frequency ratio, while the survival regions in Fig. 9(c) are diagnosed by B_min > ε. These are independent observables computed from the same ODEs, and their overlap is a nontrivial result of the dynamics, not an identity. No parameters are fitted to data, and the parameter set is adopted from an external reference ('The parameter values used in our studies are mainly adopted from Ref. [34]'); the only self-citation is the Github code link (Ref. [35]), which is not load-bearing. The extinction threshold ε=1 is a modeling assumption, not a fitted target; the biological calibration of this threshold in rescaled variables (B̂=B/K_r, ε=1 corresponds to 10^8 cells/mL) is questionable and is not sensitivity-tested, so the 'static collapse' result carries a robustness risk. But that is a correctness concern, not circularity: changing ε would change which regimes are labeled survival/extinction, but it would not turn the B_min response, the frequency-locking region, or their correspondence into inputs of the derivation.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The model uses standard ecological building blocks and no invented entities. The main hand-chosen input that materially shapes the conclusions is the extinction threshold ε, whose scaling is inconsistent between text and simulations; initial-condition scan ranges are also chosen by hand and affect the reported 'survival probabilities'.

free parameters (2)
  • Extinction threshold ε = 1 (rescaled; equivalent to 10^8 cells/mL if B_hat=B/K_r)
    Hand-chosen and load-bearing: defines 'extinction' when B_min<ε. No sensitivity analysis is performed, and the stated biological justification ('1 individual per unit volume') contradicts the rescaled value used in simulations.
  • Initial-condition scan ranges for survival basins = B0,I0 ∈ [10^6, 5×10^7], P0 ∈ [10^6, 10^8] (Fig. 4); B0=P0=0.1 (rescaled) for Fig. 5
    The 'survival probability' is the fraction of initial conditions in these manually chosen grids that avoid extinction; the area of the basin depends on the chosen ranges, yet the result is reported as a probability.
assumptions (5)
  • domain assumption Bacterial growth is logistic and phage infection follows mass-action adsorption to both uninfected and infected bacteria are explicitly tracked (Eq. 1).
    Standard ecological modeling assumptions, not derived from first principles; they define the model structure.
  • domain assumption Infected bacteria do not reproduce; they lyse at rate α releasing Ω phage particles.
    Classic lytic phage modeling assumption; excludes lysogeny, immune response, and spatial structure.
  • domain assumption Environmental fluctuations enter only through carrying capacity K(t)=K0+AK0 sin(2πft).
    The paper restricts environmental variation to this single periodic channel; other parameters (r, a, α, δ) are held constant.
  • domain assumption Deterministic ODE densities plus a hard extinction threshold ε substitute for demographic stochasticity and Allee effects.
    The extinction threshold is the mechanism that converts low-amplitude oscillations into collapse; its biological calibration is not established.
  • standard math Routh-Hurwitz conditions and local linearization determine global stability of E1 and the Hopf boundary.
    Standard stability theory; the Hopf boundary itself is located numerically.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Frequency Locking to Environmental Forcing Suppresses Oscillatory Extinction in Phage-Bacteria Interactions." pith.science (2026). https://pith.science/paper/PFFRYSMJ

@misc{pith2026251208224,
  author       = {Pith},
  title        = {Pith review of: Frequency Locking to Environmental Forcing Suppresses Oscillatory Extinction in Phage-Bacteria Interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFFRYSMJ}},
  note         = {Machine review of arXiv:2512.08224}
}
read the original 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 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 highlight bacterial hosts as active builders of ecological dynamics and environmental variation as a potential stabilizing force. Our findings thus bridge a theory-experiment gap and provide a framework for predicting microbial responses to environmental stress, which might have potential implications for phage therapy, microbiome management, and climate-impacted community resilience as well.

Figures

Figures reproduced from arXiv: 2512.08224 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic illustration of the lytic phage infection [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Dynamics and stability of the bacteria-phage system in the absence of an extinction threshold. (a-c) Time series [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Typical time series of the population dynamics with [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Basin of attractions and extinction boundary. (a) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Dependence of the survival phase on the bacterial growth rate [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Typical time series of different types of popula [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Response of the phage-bacteria system under periodic environmental forcing with different amplitude. Left column: [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Frequency locking and the emergence of Arnold [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Environmental forcing modulates the risk of ex [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Bacterial growth rate [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 7 canonical work pages

  1. [1]

    C. A. Suttle, Marine viruses–major players in the global ecosystem, Nat Rev Microbiol5, 801 (2007)

    2007

  2. [2]

    R. W. Hendrix, M. C. M. Smith, R. N. Burns, M. E. Ford, and G. F. Hatfull, Evolutionary relationships among di- verse bacteriophages and prophages: All the world’s a phage, Proceedings of the National Academy of Sciences 96, 2192 (1999)

    1999

  3. [3]

    Batinovic, F

    S. Batinovic, F. Wassef, S. A. Knowler, D. T. Rice, C. R. Stanton, J. Rose, J. Tucci, T. Nittami, A. Vinh, G. R. Drummond, C. G. Sobey, H. T. Chan, R. J. Se- viour, S. Petrovski, and A. E. Franks, Bacteriophages in natural and artificial environments, Pathogens8, 10.3390/pathogens8030100 (2019)

    work page doi:10.3390/pathogens8030100 2019

  4. [4]

    A. R. Hall, P. D. Scanlan, A. D. Morgan, and A. Buck- ling, Host–parasite coevolutionary arms races give way to fluctuating selection, Ecology Letters14, 635 (2011)

    2011

  5. [5]

    Bondy-Denomy, B

    J. Bondy-Denomy, B. Garcia, S. Strum, M. Du, M. F. Rollins, Y. Hidalgo-Reyes, B. Wiedenheft, K. L. Maxwell, and A. R. Davidson, Multiple mechanisms for crispr–cas inhibition by anti-crispr proteins, Nature526, 136 (2015)

    2015

  6. [6]

    Wang and S

    X. Wang and S. Leptihn, Defense and anti-defense mech- anisms of bacteria and bacteriophages, Journal of Zhe- jiang University-SCIENCE B25, 181 (2024)

    2024

  7. [7]

    Koskella and M

    B. Koskella and M. A. Brockhurst, Bacteria–phage co- evolution as a driver of ecological and evolutionary pro- cesses in microbial communities, FEMS Microbiology Re- views38, 916 (2014)

    2014

  8. [8]

    Bertozzi Silva, Z

    J. Bertozzi Silva, Z. Storms, and D. Sauvageau, Host re- ceptors for bacteriophage adsorption, FEMS Microbiol- ogy Letters363, 10.1093/femsle/fnw002 (2016)

    work page doi:10.1093/femsle/fnw002 2016

Show all 52 references
  1. [9]

    Y. Wang, J. Li, Y. Zhang, Y. Li, X. Chen, J. Cui, F. Xue, J. Ren, J. Dai, and F. Tang, Multisite binding of bacterio- phages on lipopolysaccharides in escherichia coli o157:h7 and the adaptive costs of phage resistance, Microbiology Spectrum13, e00067 (2025)

    2025

  2. [10]

    L. Chen, X. Zhao, S. Wongso, Z. Lin, and S. Wang, Trade-offs between receptor modification and fitness drive host-bacteriophage co-evolution leading to phage extinction or co-existence, The ISME Journal18, 10.1093/ismejo/wrae214 (2024)

    2024 doi

  3. [11]

    Avrani, O

    S. Avrani, O. Wurtzel, I. Sharon, R. Sorek, and D. Lin- dell, Genomic island variability facilitates prochlorococ- cus–virus coexistence, Nature474, 604 (2011)

    2011

  4. [13]

    L. You, F. Suthers Patrick, and J. Yin, Effects of es- cherichia coli physiology on growth of phage t7 in vivo and in silico, Journal of Bacteriology184, 1888 (2002)

    2002

  5. [14]

    Z. J. Storms, T. Brown, D. G. Cooper, D. Sauvageau, and R. L. Leask, Impact of the cell life-cycle on bacteriophage t4 infection, FEMS Microbiology Letters353, 63 (2014)

    2014

  6. [15]

    E. L. Attrill, U. Lapi´ nska, E. R. Westra, S. V. Harding, and S. Pagliara, Slow growing bacteria survive bacterio- phage in isolation, ISME Communications3, 95 (2023)

    2023

  7. [16]

    Pearl, C

    S. Pearl, C. Gabay, R. Kishony, A. Oppenheim, and N. Q. Balaban, Nongenetic individuality in the host–phage in- teraction, PLOS Biology6, e120 (2008)

    2008

  8. [17]

    Skliros, P

    D. Skliros, P. G. Kalatzis, C. Kalloniati, F. Ko- maitis, S. Papathanasiou, E. D. Kouri, M. K. Udvardi, C. Kokkari, P. Katharios, and E. Flemetakis, The devel- opment of bacteriophage resistance in vibrio alginolyti- cus depends on a complex metabolic adaptation strategy, Viru...

    2021 doi

  9. [18]

    Fister, C

    S. Fister, C. Robben, A. K. Witte, D. Schoder, M. Wag- ner, and P. Rossmanith, Influence of environmental fac- 15 tors on phage–bacteria interaction and on the efficacy and infectivity of phage p100, Frontiers in Microbiology 7, 10.3389/fmicb.2016.01152 (2016)

    2016

  10. [19]

    Hadas, M

    H. Hadas, M. Einav, I. Fishov, and A. Zaritsky, Bacte- riophage t4 development depends on the physiology of its host escherichia coli, Microbiology143, 179 (1997)

    1997

  11. [20]

    V. H. W. Rudolf, The role of seasonal timing and pheno- logical shifts for species coexistence, Ecology Letters22, 1324 (2019)

    2019

  12. [21]

    B. E. Kendall, C. J. Briggs, W. W. Murdoch, P. Turchin, S. P. Ellner, E. McCauley, R. M. Nisbet, and S. N. Wood, Why do populations cycle? a synthesis of statistical and mechanistic modeling approaches, Ecology80, 1789 (1999)

    1999

  13. [22]

    B. R. Levin, F. M. Stewart, and L. Chao, Resource- limited growth, competition, and predation: A model and experimental studies with bacteria and bacterio- phage, The American Naturalist111, 3 (1977)

    1977

  14. [23]

    Boldin, The importance of ecological dynamics in evo- lutionary processes: A host-bacteriophage model revis- ited, Journal of Theoretical Biology539, 111057 (2022)

    B. Boldin, The importance of ecological dynamics in evo- lutionary processes: A host-bacteriophage model revis- ited, Journal of Theoretical Biology539, 111057 (2022)

    2022

  15. [24]

    Kimchi, Y

    O. Kimchi, Y. Meir, and N. S. Wingreen, Bacterial defense and phage counterdefense lead to coexistence in a modeled ecosystem, Proceedings of the National Academy of Sciences121, e2414229121 (2024)

    2024

  16. [25]

    Choua, M

    M. Choua, M. R. Heath, and J. A. Bonachela, Evolu- tionarily stable coevolution between a plastic lytic virus and its microbial host, Frontiers in Microbiology12, 10.3389/fmicb.2021.637490 (2021)

    2021

  17. [26]

    Igler, Phenotypic flux: The role of physiology in ex- plaining the conundrum of bacterial persistence amid phage attack, Virus Evolution8, 10.1093/ve/veac086 (2022)

    C. Igler, Phenotypic flux: The role of physiology in ex- plaining the conundrum of bacterial persistence amid phage attack, Virus Evolution8, 10.1093/ve/veac086 (2022)

    2022 doi

  18. [27]

    R. S. Eriksen, N. Mitarai, and K. Sneppen, Sustainability of spatially distributed bacteria-phage systems, Scientific Reports10, 3154 (2020)

    2020

  19. [28]

    J. B. Bruce, S. Lion, A. Buckling, E. R. Westra, and S. Gandon, Regulation of prophage induction and lysog- enization by phage communication systems, Current Bi- ology31, 5046 (2021)

    2021

  20. [29]

    Dahan, N

    Y. Dahan, N. S. Wingreen, and Y. Meir, The value of information gathering in phage–bacteria warfare, PNAS Nexus3, pgad431 (2024)

    2024

  21. [30]

    Sinha, A

    V. Sinha, A. Goyal, S. L. Svenningsen, S. Semsey, and S. Krishna, In silico evolution of lysis-lysogeny strategies reproduces observed lysogeny propensities in temperate bacteriophages, Frontiers in Microbiology8, 10.3389/fmicb.2017.01386 (2017)

    2017

  22. [31]

    Casters, L

    Y. Casters, L. E. B¨ acker, K. Broux, and A. Aert- sen, Phage transmission strategies: are phages farming their host?, Current Opinion in Microbiology79, 102481 (2024)

    2024

  23. [32]

    Nguyen, V

    J. Nguyen, V. Fernandez, S. Pontrelli, U. Sauer, M. Ack- ermann, and R. Stocker, A distinct growth physiology enhances bacterial growth under rapid nutrient fluctua- tions, Nature Communications12, 3662 (2021)

    2021

  24. [33]

    D. A. Schwartz, W. R. Shoemaker, A. M˘ ag˘ alie, J. S. Weitz, and J. T. Lennon, Bacteria-phage coevolution with a seed bank, The ISME Journal17, 1315 (2023)

    2023

  25. [34]

    T. Goel, S. J. Beckett, and J. S. Weitz, Eco-evolutionary dynamics of temperate phages in periodic environments, Virus Evolution11, veaf019 (2025)

    2025

  26. [35]

    Luo, Z.-X

    H.-N. Luo, Z.-X. Wu, and J.-Y. Guan, Matlab Codes for the Phage-Bacteria Dynamics with Fluctuating Environ- ments, Github, (2025)

    2025

  27. [36]

    Glass and M

    L. Glass and M. C. Mackey,From Clocks to Chaos: The Rhythms of Life(Princeton University Press, 1988)

    1988

  28. [37]

    J. H. Peniston, M. Barfield, A. Gonzalez, and R. D. Holt, Environmental fluctuations can promote evolution- ary rescue in high-extinction-risk scenarios, Proceedings of the Royal Society B: Biological Sciences287, 20201144 (2020)

    2020

  29. [38]

    S. M. Carlson, C. J. Cunningham, and P. A. Westley, Evolutionary rescue in a changing world, Trends in Ecol- ogy & Evolution29, 521 (2014)

    2014

  30. [39]

    Glass, Synchronization and rhythmic processes in physiology, Nature410, 277 (2001)

    L. Glass, Synchronization and rhythmic processes in physiology, Nature410, 277 (2001)

    2001

  31. [40]

    Post and M

    E. Post and M. C. Forchhammer, Synchronization of an- imal population dynamics by large-scale climate, Nature 420, 168 (2002)

    2002

  32. [41]

    Shostak, Circadian clock, cell division, and cancer: From molecules to organism, International Journal of Molecular Sciences18, 10.3390/ijms18040873 (2017)

    A. Shostak, Circadian clock, cell division, and cancer: From molecules to organism, International Journal of Molecular Sciences18, 10.3390/ijms18040873 (2017)

    2017 doi

  33. [42]

    A. J. McKane and T. J. Newman, Predator-prey cycles from resonant amplification of demographic stochasticity, Phys. Rev. Lett.94, 218102 (2005)

    2005

  34. [43]

    R. B. Kaul, A. M. Kramer, F. C. Dobbs, and J. M. Drake, Experimental demonstration of an allee effect in micro- bial populations, Biology Letters12, 20160070 (2016)

    2016

  35. [44]

    Goswami, P

    M. Goswami, P. Bhattacharyya, and P. Tribedi, Allee effect: the story behind the stabilization or extinction of microbial ecosystem, Archives of Microbiology199, 185 (2017)

    2017

  36. [45]

    A. V. Letarov and M. A. Letarova, The burden of sur- vivors: How can phage infection impact non-infected bac- teria?, International Journal of Molecular Sciences24, 10.3390/ijms24032733 (2023)

    2023 doi

  37. [46]

    M. G. Weinbauer, Ecology of prokaryotic viruses, FEMS Microbiol Rev28, 127 (2004)

    2004

  38. [47]

    F. M. Stewart and B. R. Levin, The population biology of bacterial viruses: Why be temperate, Theoretical Pop- ulation Biology26, 93 (1984)

    1984

  39. [48]

    Sneppen, Models of life: epigenetics, diversity and cycles, Reports on Progress in Physics80, 042601 (2017)

    K. Sneppen, Models of life: epigenetics, diversity and cycles, Reports on Progress in Physics80, 042601 (2017)

    2017

  40. [49]

    O. S. Lund and K. Sneppen, Optimizing phage strategies in competitive boom-bust environments, Phys. Rev. E 111, 064417 (2025)

    2025

  41. [50]

    C. C. de Souza Silva, D. Cirne, O. Freitas, and P. R. A. Campos, Phenotypic evolution as an ornstein-uhlenbeck process: The effect of environmental variation and phe- notypic plasticity, Phys. Rev. E107, 024417 (2023)

    2023

  42. [51]

    K. Pal, S. Deb, and P. S. Dutta, Tipping points in spatial ecosystems driven by short-range correlated noise, Phys. Rev. E106, 054412 (2022)

    2022

  43. [52]

    Taitelbaum, R

    A. Taitelbaum, R. West, M. Assaf, and M. Mobilia, Pop- ulation dynamics in a changing environment: Random versus periodic switching, Phys. Rev. Lett.125, 048105 (2020)

    2020

  44. [53]

    C. P. Mancuso, H. Lee, C. I. Abreu, J. Gore, and A. S. Khalil, Environmental fluctuations reshape an un- expected diversity-disturbance relationship in a microbial community, eLife10, e67175 (2021)

    2021

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.