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Complex dynamics in a model microbial system

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Abstract

The forced double-Monod model (for a chemostat with a predator, a prey and periodically forced inflowing substrate) displays quasiperiodicity, phase locking, period doubling and chaotic dynamics. Stroboscopic sections reveal circle maps for the quasiperiodic regimes and noninvertible maps of the interval for the chaotic regimes. Criticality in the circle maps sets the stage for chaos in the model. This criticality may arise with an increase in the period or amplitude of forcing.

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Literature

  • Alexander, M. 1981. Why microbial predators and parasites do not eliminate their prey and hosts.Ann. Rev. Microbiol. 35, 113–133.

    Article  Google Scholar 

  • Allen, J. 1990. Chaos and phase-locking in predator-prey models in relation to the functional response.Florida Ent. 73, 100–110.

    Google Scholar 

  • Arnol’d, V. I. 1965. Small denominators, I. Mappings of the circumference onto itself.Am. math. Soc. Transl. 46, 213–284.

    Google Scholar 

  • Andronov, A. A., E. A. Vitt and S. E. Khaiken. 1966.Theory of Oscillators. Oxford: Pergamon Press.

    MATH  Google Scholar 

  • Bak, P. 1986. The devil’s staircase.Phys. Today 39, 38–45.

    Google Scholar 

  • Bazin, M. J., C. Curds, A. Dauppe, B. A. Owen and P. T. Saunders. 1983. Microbial predation dynamics.ACS Symposium Series 207, 253–264.

    Google Scholar 

  • Bellows, T. S. 1981. The descriptive properties of some models for density dependence.J. anim. Ecol. 50, 139–156.

    Article  MathSciNet  Google Scholar 

  • Berryman, A. A. and J. A. Millstein. 1989. Are ecological systems chaotic—and if not, why not?Trends Ecol. Evol. 4, 26–28.

    Article  Google Scholar 

  • Butler, G. J., S. B. Hsu and P. Waltman. 1983. Coexistence of competing predators in a chemostat.J. math. Biol. 17, 133–151.

    Article  MATH  MathSciNet  Google Scholar 

  • Canale, R. P. 1970. An analysis of models describing predator-prey interaction.Biotech. Bioengng 12, 353–378.

    Article  Google Scholar 

  • Canale, R. P., T. D. Lustig, P. M. Kehrberger and J. E. Salo. 1973. Experimental and mathematical modeling studies of protozoan predation on bacteria.biotech. Bioengng 15, 707–728.

    Article  Google Scholar 

  • Caswell, H. and D. E. Weeks. 1986. Two-sex models: chaos, extinction, and other dynamic consequences of sex.Am. Nat. 128, 707–735.

    Article  Google Scholar 

  • Chang, S.-J., M. Wortis and J. A. Wright. 1981. Iterative properties of a one-dimensional quartic map: Critical lines and tricritical behavior.Phys. Rev. A. 24, 2669–2694.

    Article  MathSciNet  Google Scholar 

  • Chang, S.-J., M. Wortis and J. A. Wright. 1982. Tricritical points and bifurcations in the quartic map. InNonlinear Problems: Present and Future, A. R. Bishop and D. K. Campbell (Eds), pp. 395–402. Amsterdam: North-Holland.

    Google Scholar 

  • Collet, P. and J. P. Eckmann. 1980.Iterated Maps on the Interval as Dynamical Systems. Basel: Birkhauser.

    MATH  Google Scholar 

  • Contois, D. E. 1959. Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures.J. gen. Microbiol. 33, 40–50.

    Google Scholar 

  • Cunningham, A. and R. M. Nisbet. 1983. Transients and oscillations in continuous culture. InMathematics in Microbiology, M. Bazin (Ed.), pp. 77–103. London: Academic Press.

    Google Scholar 

  • Curds, C. R. and A. Cockburn. 1968. Studies on the growth and feeding ofTetrahymena pyriformis in axenic and monoxenic culture.J. gen. Microbiol. 54, 343–358.

    Google Scholar 

  • Curds, C. R. and M. J. Bazin. 1977. Protozoan predation in batch and continuous culture.Adv. aquat. Microbiol. 1, 115–176.

    Google Scholar 

  • Dent, V. E., M. J. Bazin and P. T. Saunders. 1976. Behaviour ofDictyostelium discoideum amoebae andEscherichia coli grown together in a chemostat culture.Arch. Microbiol. 109, 187–194.

    Article  Google Scholar 

  • Devaney, R. L. 1986.An Introduction to Chaotic Dynamical Systems. Menlo Park: benjamin/Cummings.

    MATH  Google Scholar 

  • Drake, J. F. and H. M. Tsuchiya. 1976. Predation, onEscherichia coli byColpoda steinii.Appl. environ. Microbiol. 331, 870–874.

    Google Scholar 

  • Drake, J. F. and H. M. Tsuchiya. 1977. Growth kinetics ofColpoda steinii onEscherichia coli.Appl. environ. Microbiol. 34, 18–22.

    Google Scholar 

  • Farmer, J. D. 1985. Sensitive dependence on parameters in nonlinear dynamics.Phys. Rev. Lett. 55, 351–354.

    Article  MathSciNet  Google Scholar 

  • Feigenbaum, M. 1978. Quantitative universality for a class of nonlinear transformations.J. stat. Phys. 19, 25–52.

    Article  MATH  MathSciNet  Google Scholar 

  • Feigenbaum, M. 1979. Universal metric properties of nonlinear transformations.J. stat. Phys. 21, 669–706.

    Article  MATH  MathSciNet  Google Scholar 

  • Feigenbaum, M. 1983. Universal behavior in nonlinear systems.Physica D 7, 16–39.

    Article  MathSciNet  Google Scholar 

  • Fredrickson, A. G. 1983. Interactions of microbial populations in mixed culture situations.ACS Symposium Series 207, 201–227.

    Article  Google Scholar 

  • Gilpin, M. E. 1979. Spiral chaos in a predator-prey model.Am. Nat. 113, 306–308.

    Article  MathSciNet  Google Scholar 

  • Glazier, J. A. and A. Libchaber. 1988. Quasi-periodicity and dynamical systems: an experimentalist’s view.IEEE Trans. Circ. Syst. 35, 790–809.

    Article  MathSciNet  Google Scholar 

  • Graham, J. M. and R. P. Canale. 1982. Experimental and modeling studies of a four-trophic level predator-prey system.Microb. Ecol. 8, 217–232.

    Article  Google Scholar 

  • Grebogi, C., E. Ott and J. A. Yorke. 1983. Crises, sudden changes in chaotic attractors, and transient chaos.Physica D 7, 181–200.

    Article  MathSciNet  Google Scholar 

  • Hassell, M. P., J. H. Lawton and R. M. May. 1976. Patterns of dynamical behavior in single-species populations.J. anim. Ecol. 45, 471–486.

    Article  Google Scholar 

  • Hastings, A. and T. Powell. 1992. Chaos in a three species food chain.Ecology, in press.

  • Holling, C. S. 1959. Some characteristics of simple types of predation and parasitism.Can. Ent. 91, 385–395.

    Article  Google Scholar 

  • Inoue, M. and H. Kamifukumoto. 1984. Scenarios leading to chaos in a forced Lotka-Volterra model.Prog. theor. Phys. 71, 930–937.

    Article  MATH  MathSciNet  Google Scholar 

  • Jacobson, M. V. 1981. Absolutely continuous invariant measure for one-parameter families of one-dimensional maps.Communs Math. Phys. 81, 39–88.

    Article  Google Scholar 

  • Jensen, M. H., P. Bak and T. Bohr. 1983. Complete devil’s staircase, fractal dimension, and universality of mode-locking structure in the circle map.Phys. Rev. Lett. 50, 1637–1639.

    Article  Google Scholar 

  • Jensen, M. H., P. Bak and T. Bohr. 1984. Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps.Phys. Rev. A 30, 1960–1969.

    Article  MathSciNet  Google Scholar 

  • Jost, J. L., J. F. Drake, A. G. Fredrickson and J. M. Tsuchiya. 1973. Interactions ofTetrahymena pyriformis, Escherichia coli, Azotobacter vinelandii, and glucose in a minimal medium.J. Bacteriol. 113, 834–840.

    Google Scholar 

  • Kaneko, K. 1986.Collapse of Tori and Genesis of Chaos in Dissipative Systems. Singapore: World Scientific.

    MATH  Google Scholar 

  • Kot, M. and W. M. Schaffer. 1984. The effects of seasonality on discrete models of population growth.Theoret. popul. Biol. 26, 340–360.

    Article  MATH  MathSciNet  Google Scholar 

  • Kot, M. and W. M. Schaffer. 1986. Discrete-time growth-dispersal models.Math. Biosci. 80, 109–136.

    Article  MATH  MathSciNet  Google Scholar 

  • Kot, M., W. M. Schaffer, G. L. Truty, D. J. Graser and L. F. Olsen. 1988. Changing criteria for imposing order.Ecol. Mod. 43, 75–110.

    Article  Google Scholar 

  • Kuang, Y. 1989. Limit cycles in a chemostat-related model.SIAM J. appl. Math. 49, 1759–1767.

    Article  MATH  MathSciNet  Google Scholar 

  • Leven, R. W., B. P. Kock and G. S. Markman. 1987. Periodic, quasiperiodic, and chaotic motion in a forced predator-prey ecosystem. InDynamical Systems and Environmental Models, H. G. Bothe, W. Ebeling, A. B. Kurzhanski and M. Peschel (Eds), pp. 95–104. Berlin: Akademie-Verlag.

    Google Scholar 

  • Li, T. Y. and J. A. Yorke. 1975. Period three implies chaos.Am. math. Mon. 82, 985–992.

    Article  MATH  MathSciNet  Google Scholar 

  • May, R. M. 1972. On relationships among various types of population models.Am. Nat. 107, 46–57.

    Article  Google Scholar 

  • May, R. M. 1974. Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos.Science 186, 645–647.

    Google Scholar 

  • May, R. M. 1976. Simple mathematical models with very complicated dynamics.Nature 261, 459–467.

    Article  Google Scholar 

  • May, R. M. and G. F. Oster. 1976. Bifurcations and dynamic complexity in simple ecological models.Am. Nat. 110, 573–599.

    Article  Google Scholar 

  • May, R. M. 1979. Bifurcations and dynamic complexity in ecological systems.Annls. N.Y. Acad. Sci. 316, 517–529.

    MATH  MathSciNet  Google Scholar 

  • May, R. M. 1980a. Mathematical models in whaling and fisheries management. In:Some Mathematical Questions in Biology, Vol. 13, G. F. Oster (Ed.), pp. 1–64. Providence: Mathematical Society.

    Google Scholar 

  • May, R. M. 1980b. Nonlinear phenomena in ecology and epidemiology.Annls., N.Y. Acad. Sci.,357, 282–291.

    MathSciNet  Google Scholar 

  • May, R. M. 1985. Regulation of populations with non-overlapping generations by microparasites: a purely chaotic system.Am. Nat. 125, 573–584.

    Article  Google Scholar 

  • Maynard Smith, J. 1968.Mathematical Ideas in Biology. Cambridge, U.K.: Cambridge University Press.

    Google Scholar 

  • MacKay, R. S. and C. Tresser. 1984. Transition to chaos for two-frequency systems.J. Phys. Lett. 44, L741-L746.

    Google Scholar 

  • Metropolis, N., M. L. Stein and P. R. Stein. 1973. On finite limit sets of transformations on the unit interval.J. Comb. Theor. 15, 25–44.

    Article  MATH  MathSciNet  Google Scholar 

  • Monod, J. 1942.Recherches sur la Croissance des Cultures Bacteriennes Paris: Hermann.

    Google Scholar 

  • Monod, J. 1950. La technique de culture continue; theorie et application.Annls. Inst. Pasteur 79, 390–401.

    Google Scholar 

  • Nisbet, R. M., A. Cunningham and W. S. C. Gurney. 1983. Endogenous metabolism and the stability of microbial predator-prey systems.Biotech. Bioengng 25, 301–306.

    Article  Google Scholar 

  • Olsen, L. F. 1987. Low dimensional strange attractors in epidemics of childhood diseases in Copenhagen, Denmark, InChaos in Biological Systems, H. Degn, A. V. Holden and L. F. Olsen (Eds), NATO ASI Series, Series A, Vol. 138. New York: Plenum.

    Google Scholar 

  • Olsen, L. F., W. M. Schaffer and G. L. Truty. 1988. Oscillations and chaos in epidemics: a nonlinear dynamics study of six childhood diseases in Copenhagen, Denmark.Theor. Popul. Biol. 33, 344–370.

    Article  MATH  MathSciNet  Google Scholar 

  • Olsen, L. F. and W. M. Schaffer. 1990. Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics.Science 249, 499–504.

    Google Scholar 

  • Omenn, G. S. 1988.Environmental Biotechnology—Reducing Risks from Environmental Chemicals Through Biotechnology. New York: Plenum Press.

    Google Scholar 

  • Poincaré, H. 1892.Les Méthodes Nouvelles de la Méchanique Celeste. Paris: Gauthier-Villars.

    Google Scholar 

  • Pool, R. 1989a. Is it chaos, or is it just noise?.Science 243, 25–28.

    Google Scholar 

  • Pool, R. 1989b. Ecologists flirt with chaos.Science 243, 310–313.

    Google Scholar 

  • Press, W. H., B. P. Flannery, S. A. Teukolsky and W. T. Vetterling. 1988.Numerical Recipes in C. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Proper, G. and J. C. Garver. 1966. Mass cultures of the protozoaColpoda steinii.Biotechnol. Bioengng 8, 287–296.

    Article  Google Scholar 

  • Ratnam, D. A., S. Pavlou and A. G. Fredrickson. 1982. Effects of attachment of bacteria to chemostat walls in a microbial predator-prey relationship.Biotech. Bioengng 24, 2675–2694.

    Article  Google Scholar 

  • Ricker, W. E. 1954. Stocks and recruitment.J. Fish. Res. Bd Can.,11, 559–623.

    Google Scholar 

  • Rodgers, T. D. 1981. Chaos in systems in population biology.Prog. theor. Biol. 6, 91–146.

    Google Scholar 

  • Rosenzweig, M. L. 1971. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time.Science 171, 385–387.

    Google Scholar 

  • Ruelle, D. and F. Takens. 1971. On the nature of turbulence.Communs math. Phys. 20, 167–192.

    Article  MATH  MathSciNet  Google Scholar 

  • Salt, G. W. 1967. Predation in an experimental protozoa population (Woodruffia-Paramecium).Ecol. Monogr. 37, 113–144.

    Article  Google Scholar 

  • Sarkovskii, A. N. 1964. Coexistence of cycles of a continuous map of a line into itself.Ukr. mat. Zh. 16, 61–71.

    MathSciNet  Google Scholar 

  • Schaffer, W. M. 1985. Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology?IMA J. Math. appl. med. Biol. 2, 221–252.

    MATH  MathSciNet  Google Scholar 

  • Schaffer, W. M. and M. Kot. 1985. Nearly one dimensional dynamics in a simple epidemic.J. theor. Biol. 112, 403–427.

    Article  MathSciNet  Google Scholar 

  • Schaffer, W. M. 1989. Perceiving order in the chaos of nature. InEvolution of Life Histories. M. Boyce (Ed.), pp. 313–350. New Haven: Yale University Press.

    Google Scholar 

  • Schaffer, W. M. and M. Kot. 1986a. Differential systems in ecology and epidemiology. InChaos, A. V. Holden (Ed.), pp. 158–178. Princeton: Princeton University Press.

    Google Scholar 

  • Schaffer, W. M. and M. Kot. 1986b. Chaos in ecological systems: the coals that Newcastle forgot.Trends Ecol. Evol. 1, 58–63.

    Article  Google Scholar 

  • Schaffer, W. M., L. F. Olsen, G. L. Truty, S. L. Fulmer and D. J. Graser. 1988. Periodic and chaotic dynamics in childhood infections. InFrom Chemical to Biological Organization. M. Markus, S. Muller and G. Nicolis (Eds), pp. 331–347. Berlin: Springer.

    Google Scholar 

  • Schell, M., S. Fraser and R. Kapral. 1983. Subharmonic bifurcation in the sine map: An infinite hierarchy of cusp bistabilites.Phys. Rev. A 28, 373–378.

    Article  MathSciNet  Google Scholar 

  • Schuster, H. G. 1988.Deterministic Chaos: An Introduction. Weinheim: VCH.

    Google Scholar 

  • Skjolding, H., B. Branner-Jorgensen, P. L. Christiansen and H. E. Jensen. 1983. Bifurcations in discrete dynamical systems with cubic maps.SIAM J. appl. Math. 43, 520–534.

    Article  MATH  MathSciNet  Google Scholar 

  • Stefan, P. 1977. A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line.Communs Math. Phys. 54, 237–248.

    Article  MATH  MathSciNet  Google Scholar 

  • Stubbs M. 1977. Density dependence in the life-cycles of animals and its importance inK andr-strategies.J anim. Ecol. 46, 677–688.

    Article  Google Scholar 

  • Sudo, R., K. Kobayashi and S. Aiba. 1975. Some experiments and analysis of a predator-prey model: Interaction betweenColpidium campylum andAlcaligenes faecalis in continuous and mixed culture.Biotech. Bioengng. 17, 167–184.

    Article  Google Scholar 

  • Swift, S. T., I. Y. Najita, K. Ohtaguchi and A. G. Fredrickson. 1982a. Continuous culture of the ciliateTetrahymena pyriformis onEscherichia coli.Biotech. Bioengng 24, 1953–1964.

    Article  Google Scholar 

  • Swift, S. T., I. Y. Najita., K. Ohtaguchi and A. G. Fredrickson. 1982b. Some physiological aspects of the autecology of the suspension-feeding protozoanTetrahymena pyriformis.Microb. Ecol. 8, 201–215.

    Article  Google Scholar 

  • Takens, F. 1981. Detecting strange attractors in turbulence.Lect. Notes Math. 898, 366–381.

    Article  MATH  MathSciNet  Google Scholar 

  • Taylor, W. D., M. D. Gates and J. Berger. 1976. Morphological changes during the growth cycle of axenic and monoxenicTetrahymena pyriformis.Can. J. Zool. 54, 2011–2018.

    Google Scholar 

  • Thomas, W. R., M. J. Pomerantz and M. E. Gilpin. 1980. Chaos, asymmetric growth, and group selection for dynamical stability.Ecology 61, 1312–1320.

    Article  Google Scholar 

  • Thompson, J. M. T. and Stewart, H. B. 1986.Nonlinear Dynamics and Chaos. Chichester: Wiley.

    MATH  Google Scholar 

  • Tsuchiya, H. M. J. F. Drake, J. L. Jost and A. G. Fredrickson. 1972. Predator-prey interactions ofDictyostelium discoideum andEscherichia coli in continuous culture.J. Bacteriol. 110, 1147–1153.

    Google Scholar 

  • van den Ende, P. 1973. Predator-prey interactions in continuous culture.Science 181, 562–564.

    Google Scholar 

  • Varon, M. and B. P. Zeigler, 1978. Bacterial predator-prey interaction at low prey density.Appl. environ. Microbiol. 36, 11–17.

    Google Scholar 

  • Waltman, P. 1983.Competition Models in Population Biology. Philadelphia: Society for Industrial and Applied Mathematics.

    Google Scholar 

  • Watson, P. J., O. Kazuhisa and A. G. Fredrickson. 1981. Kinetics of growth of the ciliateTetrahymena pyriformis onEscherichia coli.J. gen. Microbiol. 122, 323–333.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, 98195, Seattle, WA, USA

    Mark Kot

  2. Department of Animal Science-Veterinary Medicine, University of Tennessee, 37996, Knoxville, TN, USA

    Gary S. Sayler

  3. Department of Animal Science-Veterinary Medicine, University of Tennessee, 37996, Knoxville, TN, USA

    Terry W. Schultz

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Kot, M., Sayler, G.S. & Schultz, T.W. Complex dynamics in a model microbial system. Bltn Mathcal Biology 54, 619–648 (1992). https://doi.org/10.1007/BF02459637

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