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International Conference on Stochastic Processes and Algebraic Structures

SPAS 2017: Stochastic Processes and Applications pp 437–465Cite as

Nonlinear Dynamics Simulations of Microbial Ecological Processes: Model, Diagnostic Parameters of Deterministic Chaos, and Sensitivity Analysis

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 271))

Abstract

Modeling of ecological processes is demonstrated using a newly developed nonlinear dynamics model of microbial populations, consisting of a 4-variable system of coupled ordinary differential equations. The system also includes a modified version of the Monod kinetics equation. The model is designed to simulate the temporal behavior of a microbiological system containing a nutrient, two feeding microbes and a microbe predator. Three types of modeling scenarios were numerically simulated to assess the instability caused by (a) variations of the nutrient flux into the system, with fixed initial microbial concentrations and parameters, (b) variations in initial conditions, with fixed other parameters, and (c) variations in selected parameters. A modeling framework, using the high-level statistical computing languages MATLAB and R, was developed to conduct the time series analysis in the time domain and phase space. In the time domain, the Hurst exponent, the information measure–Shannon’s entropy, and the time delay of temporal oscillations of nutrient and microbe concentrations were calculated. In the phase domain, we calculated a set of diagnostic criteria of deterministic chaos: global and local embedding dimensions, correlation dimension, information dimension, and a spectrum of Lyapunov exponents. The time series data are used to plot the phase space attractors to express the dependence between the system’s state parameters, i.e., microbe concentrations, and pseudo-phase space attractors, in which the attractor axes are used to compare the observations from a single time series, which are separated by the time delay. Like classical Lorenz or Rossler systems of equations, which generate a deterministic chaotic behavior for a certain range of input parameters, the developed mathematical model generates a deterministic chaotic behavior for a particular set of input parameters. Even a slight variation of the system’s input data might result in vastly different predictions of the temporal oscillations of the system. As the nutrient influx increases, the system exhibits a sharp transition from a steady state to deterministic chaotic to quasi-periodic and again to steady state behavior. For small changes in initial conditions, resulting attractors are bounded (contrary to that of a random system), i.e., may represent a ‘sustainable state’ (i.e., resilience) of the ecological system.

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Notes

  1. 1.

    Supplementary Information containing 4 Appendices can be found at URL https://goo.gl/xdCtHK

References

  1. Abarbanel, H.D.I.: Analysis of Observed Chaotic Data. Springer, New York (1996)

    Book  Google Scholar 

  2. Abarbanel, H.D.I., Tsimring, L.S.: Reader’s Guide to Toolkit for Nonlinear Signal Analysis. UCSD, San Diego, California (1998)

    Google Scholar 

  3. Abrams, P.A., Ginzburg, L.R.: Coupling in predator-prey dynamics: ratio dependence. Trends Ecol. Evol. 15(8), 337–341 (1973). https://doi.org/10.1016/s0169-5347(00)01908-x

    Article  Google Scholar 

  4. Arditi, R., Ginzburg, L.R.: Coupling in predator-prey dynamics: ratio dependence. J. Theor. Biol. 139, 311–326 (1989). https://doi.org/10.1016/s0022-5193(89)80211-5

    Article  Google Scholar 

  5. Becks, L., Hilker, F., Malchow, H., Jürgens, K., Arndt, H.: Experimental demonstration of chaos in a microbial food web. Nature 435, 1226–1229 (2005). https://doi.org/10.1038/nature03627

    Article  Google Scholar 

  6. Becks, L., Arndt, H.: Different types of synchrony in chaotic and cyclic communities. Nat. Commun. 4 1359, 9 (2013). https://www.nature.com/articles/ncomms2355

  7. Becks, L., Arndt, H.: Transitions from stable equilibria and back in an experimental food web. Ecology 89, 3222–3226 (2008)

    Article  Google Scholar 

  8. Beninca, E., Huisman, J., Heerkloss, R., Johnk, K., Branko, P., Van Nes, E., Cheffer, M., Ellner, S.: Chaos in a long-term experiment with a plankton community. Nature 451, 822–825 (2008). https://doi.org/10.1038/NATURE06512

    Article  Google Scholar 

  9. Curriero, F., Patz, J.A., Rose, J.B., Lele, S.: The association between extreme precipitation and waterborne disease outbreaks in the United States. Am. J. Public Health 91, 1194–1199 (2001)

    Article  Google Scholar 

  10. Codeco, C.T., Lele, S., Pascual, M., Bouma, M., Ko, A.I.: A stochastic model for ecological systems with strong nonlinear response to environmental drivers: application to two water-borne diseases. J. R. Soc. Interface 5, 247–252 (2008)

    Article  Google Scholar 

  11. Dormand, J., Prince, P.: A family of embedded Runke-Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)

    Article  MathSciNet  Google Scholar 

  12. Faybishenko, B.: Chaotic dynamics in flow through unsaturated fractured media. Adv. Water Resour. 25(7), 793–816 (2002)

    Article  Google Scholar 

  13. Faybishenko, B., Molz, F.: Nonlinear rhizosphere dynamics yields synchronized oscillations of microbial populations, carbon and oxygen concentrations induced by root exudation. Proc. Environ. Sci. 19, 369–378 (2013)

    Article  Google Scholar 

  14. Faybishenko, B., Murdoch, L., Molz, F.: The Emergence of Chaotic Dynamics in Complex Microbial Systems: Fully Mixed versus 1-D Models. Paper Presented at the Goldschmidt Conference, June 8-13 2014, Sacramento (2014)

    Google Scholar 

  15. Hanemaaijer, M., Röling, W., Olivier, B., Khandelwal, R., Teusink, B., Bruggeman, F.: Systems modeling approaches for microbial community studies: from metagenomics to inference of the community structure. Front. Microbiol. 6 (2015). https://doi.org/10.3389/fmicb.2015.00213.

  16. Gillman, M., Hails, R.: An Introduction to Ecological Modelling: Putting Practice into Theory. Blackwell Scientific Publications, Oxford (1997)

    Google Scholar 

  17. Graham, D., Knapp, C., Van Vleck, E., Bloor, K., Lane, T., Graham, C.: Experimental demonstration of chaotic instability in biological nitrification. ISME J. 1, 385–393 (2007)

    Article  Google Scholar 

  18. Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Physica D9, 189–208 (1983)

    Article  MathSciNet  Google Scholar 

  19. Groffman, P.M., Law, N.L., Belt, K.T., Band, L.E., Fisher, G.T.: Nitrogen fluxes and retention in urban watershed ecosystems. Ecosystems 7, 393–403 (2004)

    Google Scholar 

  20. Grover, J.P.: Resource Competition. Springer, New York (1997)

    Chapter  Google Scholar 

  21. Gunderson, L.: Ecological resilience-theory to practice. Annu. Rev. Ecol. Syst. 31, 421–439 (2000)

    Article  Google Scholar 

  22. Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series methods: the TISEAN package. Chaos 10, 413–421 (1999)

    Article  Google Scholar 

  23. Hernandez, M.-J.: Dynamics of transitions between population interactions: a nonlinear interaction-function defined. Proc. R. Soc. Lond. B 265, 1433–1440 (1998)

    Article  Google Scholar 

  24. Hirsch, M.: Systems of differential equations which are competitive or cooperative: i. Limit sets. SIAM J. Math. Anal. 13, 167–179 (1982)

    Article  MathSciNet  Google Scholar 

  25. Holling, C.S.: Surprise for science, resilience for ecosystems and incentives for people. Ecol. Appl. 6, 733–735 (1996)

    Article  Google Scholar 

  26. Kot, M., Sayler, G., Schultz, T.: Complex dynamics in a model microbial system. Bull. Math. Biol. 54, 619–648 (1992)

    Article  Google Scholar 

  27. Ludwig, D., Walker, B., Holling, C.: Sustainability, stability, and resilience. Conserv. Ecol. 1, 7, (1997). http://www.consecol.org/vol1/iss1/art7

  28. May, R.M.: Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269, 471–477 (1977)

    Article  Google Scholar 

  29. May, R.M.: Models for two interacting populations. In: May, R.M. (ed.) Theoretical Ecology: Principles and Applications, 2nd edn., pp. 78–104. Sunderland, Sinauer (1981)

    Google Scholar 

  30. Molz, F., Faybishenko, B., Agarwal, D.: A Broad Exploration of Nonlinear Dynamics in Microbial Systems Motivated by Chemostat Experiments Producing Deterministic Chaos. LBNL-2001172, Berkeley (2018)

    Google Scholar 

  31. Molz, F., Faybishenko, B.: Increasing evidence for chaotic dynamics in the soil-plant-atmosphere system: a motivation for future research. Proc. Environ. Sci. 19, 681–690 (2013)

    Article  Google Scholar 

  32. Rodriguez-Iturbe, I., Entekhabi, D., Bras, R.L.: Nonlinear dynamics of soil moisture at climate scales: 1. Stochastic analysis. Water Resour. Res. 27(8), 1899–1906 (1991)

    Article  Google Scholar 

  33. Rosenstein, M., Collins, J., De Luca, C.: A practical method for calculating largest Lyapunov exponent from small data sets. Physica D 65, 117–124 (1993)

    Article  MathSciNet  Google Scholar 

  34. Rosenzweig, M., MacArthur, R.: Graphical representation and stability conditions of predator-prey interactions. Am. Nat. 97(895), 209–223 (1963)

    Article  Google Scholar 

  35. Smale, S.: On the differential equations of species in competition. J. Math. Biol. 3, 5–7 (1976)

    Article  MathSciNet  Google Scholar 

  36. Tsonis, A.A.: Chaos: From Theory to Applications. Plenum Press, New York (1992)

    Google Scholar 

  37. Tsonis, A., Elsner, J.: Chaos, strange attractors and weather. Bull. Am. Meteorol. Soc. 7, 14–23 (1989)

    Article  Google Scholar 

  38. MATLAB, Release.: The MathWorks Inc. Natick, Massachusetts (2012)

    Google Scholar 

  39. R Core Team R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2016). https://www.R-project.org/

  40. Constantine, W., Percival, D.: Fractal: A Fractal Time Series Modeling and Analysis Package. R Package Version 2.0-4 (2017). https://CRAN.R-project.org/package=fractal

  41. Hausser, J., Strimmer, K.: Entropy: Estimation of Entropy, Mutual Information and Related Quantities. R package version 1.2.1 (2014). https://CRAN.R-project.org/package=entropy

  42. Garcia, C.A.: NonlinearTseries: Nonlinear Time Series Analysis. R package version 0.2.3 (2015). https://CRAN.R-project.org/package=nonlinearTseries

  43. Di Narzo, A.F.: TseriesChaos: Analysis of Nonlinear Time Series. R package version 0.1-13. https://CRAN.R-project.org/package=tseriesChaos

  44. Ligges, U., Machler, M.: Scatterplot3d - an R package for visualizing multivariate data. J. Stat. Softw. 8(11), 1–20 (2003)

    Article  Google Scholar 

  45. Zeileis, A., Grothendieck, G.: zoo: S3 infrastructure for regular and irregular time series. J. Stat. Softw. 14(6), 1–27 (2005). https://doi.org/10.18637/jss.v014.i06

    Article  Google Scholar 

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Acknowledgements

BF and DA research supported by the U.S. DOE, Office of Science, Office of Biological and Environmental Research, and Office of Science, Office of Advanced Scientific Computing under the DOE Contract No. DE-AC02-05CH11231. FM acknowledges the support of the Clemson University, Department of Environmental Engineering and Earth Sciences. The invitation by Sergei Silvestrov and Dmitrii Silvestrov to present the paper at the SPAS2017 International Conference on Stochastic Processes and Algebraic Structures, and their kind assistance with LaTeX typesetting of the manuscript for publication are gratefully appreciated.

Author information

Authors and Affiliations

  1. Lawrence Berkeley National Laboratory, Energy Geosciences Division, University of California, Berkeley, CA, USA

    Boris Faybishenko

  2. Environmental Engineering and Earth Sciences Dept, Clemson University, Anderson, SC, USA

    Fred Molz

  3. Lawrence Berkeley National Laboratory, Computer Research Division, University of California, Berkeley, CA, USA

    Deborah Agarwal

Authors
  1. Boris Faybishenko
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  2. Fred Molz
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  3. Deborah Agarwal
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Corresponding author

Correspondence to Boris Faybishenko .

Editor information

Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

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Faybishenko, B., Molz, F., Agarwal, D. (2018). Nonlinear Dynamics Simulations of Microbial Ecological Processes: Model, Diagnostic Parameters of Deterministic Chaos, and Sensitivity Analysis. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Stochastic Processes and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 271. Springer, Cham. https://doi.org/10.1007/978-3-030-02825-1_19

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