Abstract
We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the ``inside'' one is an unstable separatrix and the ``outside'' one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armstrong, R.A., McGehee, R.: Competitive exclusion, Amer. Natur. 115, 151–170 (1980)
Butler, G.J., Wolkowicz, G.S.K.: A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math. 45, 138–151 (1985)
Butler, G.J., Wolkowicz, G.S.K.: Exploitative competition in a chemostat for two complementary, and possible inhibitory, resources, Math. Biosci. 83, 1–48 (1987)
Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X. Auto 97: Continuation and bifurcation software for ordinary differential equations, Technical report, Concordia University, Montreal, Canada, 1997
Chi C-W, Hsu S.B., Wu L-L.: On the asymmetric May-Leonard model of three competing species, SIAM J. Appl. Math. 58, 211–226 (1998)
Gause, G.F.: The Struggle for Existence, Williams and Wilkins, Baltimore, Maryland, 1934
Grover, J.P.: Resource Competition, Population and Community Biology Series, 19, Chapman & Hall, New York, 1997
Guckenheimer, J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983
Gyllenberg, M., Yan, P., Wang, Y.: A 3D competitive Lotka-Volterra system with three limit cycles: a falsification of a conjecture of Hofbauer and So, preprint
Hardin, G.: The competitive exclusion principle, Science 131, 1292–1298 (1960)
Hirsch, M.W.: Systems of differential equations which are competitive or cooperative. III: Competing species. Nonlinearity 1, 51–71 (1988)
Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems, London Math. Soc. Student Texts 7, Cambridge University Press, Cambridge, 1988
Hofbauer, J., So, J.W.-H.: Multiple limit cycles for three dimesnional Lotka-Volterra Equations, Appl. Math. Lett. 7, 65–70 (1994)
Holm, N.P., Armstrong, D.E.: Role of nutrients limitation and competition in controlling the populations of Asterionella formosa and Microcystis aeruginosa in semicontinuous culture, Limonl. Oceanogr. 26, 622–634 (1981)
Hsu, S.B., Hubbell, S., Waltman, P.: A mathematical theory of single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math. 32, 366–383 (1977)
Hsu, S.B.: Limiting behavior for competing species, SIAM J. Appl. Math. 34, 760–763 (1978)
Hsu, S.B., Cheng, K.S., Hubbell, S.P.: Exploitative competition of microorganism for two complementary nutrients in continuous culture, SIAM J. Appl. Math. 41, 422–444 (1981)
Huisman, J., Weissing, F.J.: Biodiversity of plankton by species oscillations and chaos, Nature 402, 407–410 (1999)
Huisman, J., Weissing, F.J.: Fundamental unpredictability in multispecies competition, Amer. Naturalist 157, 488–494 (2001)
Leon, J.A., Tumpson, D.B.: Competition between two species for two complementary or substitutable resources, J. Theor. Biol. 50, 185–201 (1975)
Li, B.: Analysis of Chemostat-Related Models With Distinct Removal Rates, Ph.D thesis, Arizona State University, 1998
Li, B.: Global asymptotic behaviour of the chemostat: general response functions and different removal rates, SIAM J. Appl. Math. 59, 411–422 (1999)
Li, B., Smith, H.L.: How many species can two essential resources support? SIAM J. Appl. Math. 62, 336–66 (2001)
Li, B.: Periodic coexistence in the chemostat with three species competing for three essential resources, Math. Biosciences 174, 27–40 (2001)
Lu, Z., Luo, Y.: Two limit cycles in three-dimensional Lotka-Volterra systems, Computer and Mathematics with Applications, 44, 51–66 (2002)
Lu, Z., Luo, Y.: Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle, Computer and Mathematics with Applications, 46, 51–66 (2003)
May, R.M., Leonard, W.J.: Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29, 243–253 (1975)
Novick, A., Sziliard, L.: Description of the chemostat. Science. 112, 715–716 (1950)
Phillips, O.M.: The equilibrium and stability of simple marine biological systems, 1. Primary nutrient consumers, Amer. Natur. 107, 73–93 (1973)
K. O. Rothhaup, Laboratory experiments with a mixotrophic chrysophyte and obligately phagotrophic and phototrophic competitors. Ecology 77, 716–724 (1996)
Schuster, P., Sigmund, K., Wolf, R.: On ω-limit for competition between three species, SIAM J. Appl. Math. 37, 49–54 (1979)
Smith, H.L.: Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs 41, Amer. Math. Soc., Providence, RI, 1995
Smith, H.L., Li, B.: Competition for essential resources: A brief review, Fields Institute Communications, 36, 213–227 (2003)
Smith, H.L., Waltman, P.: The Theory of the Chemostat, Cambridge University Press, 1995
Sommer, U.: Comparison between steady states and non-steady competition: experiments with natural phytoplankton. Limnol. Oceanogr. 30, 335–346 (1985)
Sommer, U.: Nitrate-and silicate-competition among Antarctic phytoplankton, Mar. Biol. 91, 345–351 (1986)
Tilman, D.: Tests of resources competition theory using four species of Lake Michigan algae, Ecology 62, 802–815 (1981)
Tilman, D.: Plant Strategies and the Dynamics and Structure of Plant Communities, Princeton University Press, Princeton, N.J., 1981
Tilman, D.: Resource competition and Community Structure, Princeton University Press, Princeton, N.J., 1982
Van Donk, E., Kilham, S.S.: Temperature effects on silicon-and phosphorus-limited growth and competitive interactions among three diatoms, J. Phycol. 26, 40–50 (1990)
Von Liebig, J.: Die organische Chemie in ihrer Anwendung auf Agrikultur und Physiologie. Friedrich Vieweg, Braunschweig, 1840
Wolkowicz, G.S.K., Lu, Z.: Global dynamics of a mathematical model of competition in the chemostat: general response function and differential death rates, SIAM J. Appl. Math. 52, 222–233 (1992)
Zeeman, M.L.: Hopf Bifurcation in competitive three-dimensional Lotka-Volterra systems, Dynamics and Stability of Systems, 8, 189–217 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by NSF grant DMS 0211614. KY 40292, USA.
This author's research was supported in part by NSF grant DMS 0107160
Rights and permissions
About this article
Cite this article
Baer, S., Li, B. & Smith, H. Multiple limit cycles in the standard model of three species competition for three essential resources. J. Math. Biol. 52, 745–760 (2006). https://doi.org/10.1007/s00285-005-0367-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-005-0367-x