Host–virus–predator coexistence in a grey-box model with dynamic optimization of host fitness


Lytic viruses are believed to affect both flow patterns and host diversity in microbial food webs. Models resolving host and virus communities into subgroups can represent both aspects. However, when flow pattern is the prime interest, such models may seem unnecessary complex. This has led to proposals of black-box models using only total community sizes as state variables. This simplification creates a coexistence problem, however, since predator and virus communities then compete for the same, shared, prey = host community. Mathematically, this problem can be solved by introducing feedbacks allowing community-level properties to adapt. The different mathematical alternatives for such feedback represent different ecological assumptions and thus different hypotheses for how the balance between predators and viruses is controlled in nature. We here explore a model where the feedback works through an increase in host community resistance in response to high virus abundances, thereby reducing virus production. We use a dynamic “strategy” index S to describe the balance between defensive and competitive abilities in the host community, and assume the rate of change in S to be proportional to the local slope of the per capita fitness gradient for the host. We explore how such a “grey-box” model can allow stable coexistence of viruses and predators, and how equilibrium food web structure, virus-to-host ratio, and partitioning of host production varies; both as functions of host community traits, and as functions of external bottom-up and top-down drivers.

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  1. 1.

    Riemann L, Middelboe M. Viral lysis of marine bacterioplankton: Implications for organic matter cycling and bacterial clonal composition. Ophelia. 2002;56:57–68.

    Article  Google Scholar 

  2. 2.

    Bratbak G, Jacobsen A, Heldal M. Viral lysis of Phaeocystis pouchetii and bacterial secondary production. Aquat Micro Ecol. 1998;16:11–6.

    Article  Google Scholar 

  3. 3.

    Fuhrman JA, Noble RT. Viruses and protists cause similar bacterial mortality in coastal seawater. Limnol Oceanogr. 1995;40:1236–42.

    Article  Google Scholar 

  4. 4.

    Thingstad TF, Lignell R. Theoretical models for the control of bacterial growth rate, abundance, diversity and carbon demand. Aquat Micro Ecol. 1997;13:19–27.

    Article  Google Scholar 

  5. 5.

    Thingstad TF. Elements of a theory for the mechanisms controlling abundance, diversity, and biogeochemical role of lytic bacterial viruses in aquatic systems. Limnol Oceanogr. 2000;45:1320–8.

    Article  Google Scholar 

  6. 6.

    Thingstad TF, Vage S, Storesund JE, Sandaa R-A, Giske J. A theoretical analysis of how strain-specific viruses can control microbial species diversity. Proc Natl Acad Sci USA. 2014;111:7813–8.

    CAS  Article  Google Scholar 

  7. 7.

    Våge S, Pree B, Thingstad TF. Linking internal and external bacterial community control gives mechanistic framework for pelagic virus-to-bacteria ratios. Environ Microbiol. 2016;18:3932–48.

    Article  Google Scholar 

  8. 8.

    Weitz JS, Stock CA, Wilhelm SW, Bourouiba L, Coleman ML, Buchan A, et al. A multitrophic model to quantify the effects of marine viruses on microbial food webs and ecosystem processes. ISME J. 2015;9:1352–64.

    Article  Google Scholar 

  9. 9.

    Gause GF. Experimental analysis of Vito Volterra's mathematical theory of the struggle for existence. Science. 1934;79:16–7.

    CAS  Article  Google Scholar 

  10. 10.

    Talmy D, Beckett SJ, Zhang AB, Taniguch DAA, Weitz JS, Follows MJ. (2019). Contrasting controls on microzooplankton grazing and viral Infection of microbial prey. Front Mar Sci. 2019;6.

  11. 11.

    Murray AG, Jackson GA. Viral dynamics: a model of the effects of size, shape motion and abundance of single celled planktonic organisms and other particles. Mar Ecol Prog Ser. 1992;89:103–16.

    Article  Google Scholar 

  12. 12.

    Blackburn N, Zweifel UL, Hagstrom A. Cycling of marine dissolved organic matter .2. A model analysis. Aquat Micro Ecol. 1996;11:79–90.

    Article  Google Scholar 

  13. 13.

    Våge S, Storesund JE, Giske J, Thingstad TF. Optimal defense strategies in an Idealized microbial food web under trade-off between competition and defense. PloS ONE. 2014;9:e101415.

    Article  Google Scholar 

  14. 14.

    Abrams PA. Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods. Ecol Lett. 2001;4:166–75.

    Article  Google Scholar 

  15. 15.

    Edwards KF, Kremer CT, Miller ET, Osmond MM, Litchman E, Klausmeier CA. Evolutionarily stable communities: a framework for understanding the role of trait evolution in the maintenance of diversity. Ecol Lett. 2018;21:1853–68.

    Article  Google Scholar 

  16. 16.

    Sandaa R-A, Pree B, Larsen A, Vage S, Topper B, Topper JP, et al. The response of heterotrophic prokaryote and viral communities to labile organic carbon inputs is controlled by the predator food chain structure. Viruses. 2017;9:E238.

    Article  Google Scholar 

  17. 17.

    Hessen D, VanDonk E. Morphological changes in Scenedesmus induced by substances released from Daphnia. ArcHydrobiol. 1993;127:129–40.

    Google Scholar 

  18. 18.

    Matz C, Jürgens K. Interaction of nutrient limitation and protozoan grazing determines the phenotypic structure of a bacterial community. Microb Ecol. 2003;45:384–98.

    CAS  Article  Google Scholar 

  19. 19.

    Salcher MM, Pernthaler J, Psenner R, Posch T. Succession of bacterial grazing defense mechanisms against protistan predators in an experimental microbial community. Aquat Micro Ecol. 2005;38:215–29.

    Article  Google Scholar 

  20. 20.

    Franze G, Pierson JJ, Stoecker DK, Lavrentyev PJ. Diatom-produced allelochemicals trigger trophic cascades in the planktonic food web. Limnol Oceanogr. 2018;63:1093–108.

    CAS  Article  Google Scholar 

  21. 21.

    Thingstad TF, Øvreås L, Egge JK, Løvdal T, Heldal M. Use of non-limiting substrates to increase size; a generic strategy to simultaneously optimize uptake and minimize predation in pelagic osmotrophs? Ecol Lett. 2005;8:675–82.

    Article  Google Scholar 

  22. 22.

    Tsagaraki TM, Pree B, Leiknes O, Larsen A, Bratbak G, Ovreas L, et al. Bacterial community composition responds to changes in copepod abundance and alters ecosystem function in an Arctic mesocosm study. ISME J. 2018;12:2694–705.

    CAS  Article  Google Scholar 

  23. 23.

    Winter C, Bouvier T, Weinbauer MG, Thingstad TF. Trade-offs between competition and defense specialists among unicellular planktonic organisms: The “Killing the Winner” hypothesis revisited. Microbiol Mol Biol Rev. 2010;74:42–57.

    CAS  Article  Google Scholar 

  24. 24.

    Weitz JS, Hartman H, Levin SA. Coevolutionary arms races between bacteria and bacteriophage. Proc Natl Acad Sci USA. 2005;102:9535–40.

    CAS  Article  Google Scholar 

  25. 25.

    Menge DNL, Weitz JS. Dangerous nutrients: evolution of phytoplankton resource uptake subject to virus attack. J Theoret Biol. 2009;257:104–15.

    Article  Google Scholar 

  26. 26.

    Litchman E, Klausmeier CA, Schofield OM, Falkowski PG. The role of functional traits and trade-offs in structuring phytoplankton communities: scaling from cellular to ecosystem level. Ecol Lett. 2007;10:1170–81.

    Article  Google Scholar 

  27. 27.

    Ferenci T. Trade-off mechanisms shaping the diversity of bacteria. Trends Microbiol. 2016;24:209–23.

    CAS  Article  Google Scholar 

  28. 28.

    Abedon ST, Hyman P, Thomas C. Experimental examination of bacteriophage latent-period evolution as a response to bacterial availability. Appl Environ Microbiol. 2003;69:7499–506.

    CAS  Article  Google Scholar 

  29. 29.

    Goldhill DH, Turner PE. The evolution of life history trade-offs in viruses. Curr Opin Virol. 2014;8:79–84.

    Article  Google Scholar 

  30. 30.

    Parikka KJ, Le Romancer M, Wauters N, Jacquet S. Deciphering the virus-to-prokaryote ratio (VPR): insights into virus-host relationships in a variety of ecosystems. Biol Rev. 2017;92:1081–1100.

    Article  Google Scholar 

  31. 31.

    Wigington CH, Sonderegger D, Brussaard CPD, Buchan A, Finke JF, Fuhrman JA, et al. Re-examination of the relationship between marine virus and microbial cell abundances (vol 1, pg 15024, 2016). Nat Microbiol. 2017;2:1571–1571.

    CAS  Article  Google Scholar 

  32. 32.

    Parvinen K, Heino M, Dieckmann U. Function-valued adaptive dynamics and optimal control theory. J Math Biol. 2013;67:509–33.

    Article  Google Scholar 

  33. 33.

    Våge S, Bratbak G, Egge J, Heldal M, Larsen A, Norland S, et al. Simple models combining competition, defence and resource availability have broad implications in pelagic microbial food webs. Ecol Lett. 2018;21:1440–52.

    Article  Google Scholar 

  34. 34.

    Dugdale RC, Goering JJ. Uptake of new and regenerated forms of nitrogen in primary productivity. Limnol Oceanogr. 1967;12:196–206.

    CAS  Article  Google Scholar 

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This work was supported by EU H2020-INFRAIA project Aquacosm (No 731065), Trond Mohn Stiftelse starting grant SIMPLEX (TMS2019REK02) and RCN project VirVar (No 294363). The authors are grateful for constructive comments by the editor and three anonymous reviewers.

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Correspondence to Tron Frede Thingstad.

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Thingstad, T.F., Våge, S. Host–virus–predator coexistence in a grey-box model with dynamic optimization of host fitness. ISME J 13, 3102–3111 (2019).

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