Skip to main content
SpringerLink
Log in

Numerical Bifurcation Analysis of Ecosystems in a Spatially Homogeneous Environment

  • Published:
Acta Biotheoretica Aims and scope Submit manuscript

Abstract

The dynamics of single populations up to ecosystems, are often described by one or a set of non-linear ordinary differential equations. In this paper we review the use of bifurcation theory to analyse these non-linear dynamical systems. Bifurcation analysis gives regimes in the parameter space with quantitatively different asymptotic dynamic behaviour of the system. In small-scale systems the underlying models for the populations and their interaction are simple Lotka-Volterra models or more elaborated models with more biological detail. The latter ones are more difficult to analyse, especially when the number of populations is large. Therefore for large-scale systems the Lotka-Volterra equations are still popular despite the limited realism. Various approaches are discussed in which the different time-scale of ecological and evolutionary biological processes are considered together.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  • Abrams, P. A. (1996). Dynamics and interactions in food webs with adaptive foragers. In Polis, G. A. and K. O. Winemiller (Eds). Food webs, pages 113–121. Chapman & Hall, New York.

    Google Scholar 

  • Abrams, P. A. (1999). The adaptive dynamics of consumer choice. The American Naturalist 153: 83–97.

    Google Scholar 

  • Agrawal, A. A. (2003). Community genetics: New insights into community ecology by integrating population genetics. Ecology 84: 543–544.

    Google Scholar 

  • Arditi, R. and L. R. Ginzburg (1989). Coupling in predator-prey dynamics: ratiodependence. Journal of Theoretical Biology 139: 311–326.

    Google Scholar 

  • Aziz-Alaoui, M. A. (2002). Study of a Leslie-Gower-type tritrophic population model. Chaos, Solitons and Fractals 14: 1275–1293.

    Google Scholar 

  • Back, A., J. Guckenheimer, M. R. Myers, F. J. Wicklin, and P. A. Worfolk (1992). Dstool:computer assisted exploration of dynamical systems. Notices of the American Mathematical Society 39: 303–309.

    Google Scholar 

  • Bazykin, A. D. (1998). Nonlinear dynamics of interacting populations. World Scientific, Singapore.

    Google Scholar 

  • Begon, M., J. L., Harper and C. R. Townsend (1996). Ecology. Blackwell Science, Oxford.

    Google Scholar 

  • Belyea, L. R. and J. Lancaster (1999). Assembly rules within a contingent ecology. Oikos 86: 402–416.

    Google Scholar 

  • Boer, M. P., B. W. Kooi and S. A. L. M. Kooijman (1998). Food chain dynamics in the chemostat. Mathematical Biosciences 150: 4342.

    Google Scholar 

  • Boer, M. P., B. W. Kooi and S. A. L. M. Kooijman (1999). Homoclinic and heteroclinic orbits in a tri-trophic food chain. Journal of Mathematical Biology 39: 19–38.

    Google Scholar 

  • Boer, M. P., B. W. Kooi and S. A. L. M. Kooijman (2001). Multiple attractors and boundary crises in a tri-trophic food chain. Mathematical Biosciences 169: 109–128.

    Google Scholar 

  • Chen, X. and J. E. Cohen (2001). Global stability, local stability and permanence in model food webs. Journal of Theoretical Biology 212: 223–234.

    Google Scholar 

  • Cohen, J. E., F. Briand and C. M. Newman (1990). Community Food Webs. Biomathematics Volume 20. Springer-Verlag, New York.

    Google Scholar 

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

    Google Scholar 

  • de Ruiter, P. C., A.-M. Neutel and J. C. Moore (1995). Energetics, patterns of interaction strengths, and stability in real ecosystems. Science 269: 1257–1260.

    Google Scholar 

  • DeAngelis, D. L. (1992). Dynamics of Nutrient Cycling and Food Webs. Number 9 in Population and Community Biology series. Chapman & Hall, London.

    Google Scholar 

  • Diekmann, O. (2002). A beginners guide to adaptive dynamics. Monographs and proceedings. pp 63–100, Coimbra, Portugal Centro Internacional de Matematica.

    Google Scholar 

  • Dieckmann, U. and R. Law (1996). The dynamical theory of coevolution: a derivation from stochastic ecological processes. Journal of Mathematical Biology 34: 579–612.

    Google Scholar 

  • Doedel, E. J., A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang (1997). Auto 97: Continuation and bifurcation software for ordinary differential equations. Technical report, Concordia University, Montreal, Canada.

    Google Scholar 

  • Drake, J. A. (1990). The mechanism of community assembly and succession. Journal of Theoretical Biology 147: 213–233.

    Google Scholar 

  • Drossel, B., P. G. Higgs and A. J. McKane (2001). The influence of predator-prey population dynamics on the long-term evolution of food web structure. Journal of Theoretical Biology 208: 91–107.

    Google Scholar 

  • Elton, C. (1927). Animal Ecology. Macmillan, New York.

    Google Scholar 

  • Ermentrout, B. (2002). Simulating, Analyzing, and Animating Dynamical Systems: A Guide to Xppaut for Researchers and Students. Society for Industrial and Applied Mathematics, Philadelphia.

    Google Scholar 

  • Fussmann, G. F., S. P. Ellner, K. W. Shertzer and N. G. Hairston Jr, (2000). Crossing the Hopf bifurcation in a live predator-prey system. Science 290: 1358–1360.

    Google Scholar 

  • Geritz, S. A. H., M. Gyllenberg, F. J. A. Jacobs and K. Parvinen (2002). Invasion dynamics and attractor inheritance. Journal of Mathematical Biology 44: 548–560.

    Google Scholar 

  • Geritz, S. A. H., E. Kisdi, G. Meszéna and J. A. J. Metz (1998). Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolutionary Ecology 12: 35–57.

    Google Scholar 

  • Geritz, S. A. H., J. A. J. Metz, E. Kisdi and G. Meszéna (1997). Dynamics of adaptation and evolutionary branching. Physical Review Letters 78: 2024–2027.

    Google Scholar 

  • Gilpin, M. E. (1979). Spiral chaos in a predator prey model. The American Naturalist 107: 306–308.

    Google Scholar 

  • Gragnani, A., O. De Feo and S. Rinaldi (1998). Food chains in the chemostat: relationships between mean yield and complex dynamics. Bulletin of Mathematical Biology 60: 703–719.

    Google Scholar 

  • Grebogi, C., E. Ott and J. A. Yorke (1987a). Basin boundary metamorphoses: Changes in accessible boundary orbits. Physica D 24: 243–262.

    Google Scholar 

  • Grebogi, C., E. Ott and J. A. Yorke (1987b). Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamical systems. Science 238: 632–638.

    Google Scholar 

  • Grimm, V. and C. Wissel (1997). Babel, or the ecological stability discussion: an inventory and analysis of terminology and a guide for avoiding confusion. Oecologia 109: 323–334.

    Google Scholar 

  • Grover, J. P. (1997). Resource Competition. Population and Community Biology Series. Chapman & Hall, London.

    Google Scholar 

  • Guckenheimer, J. and P. Holmes (1985). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Volume 42 of Applied Mathematical Sciences. Springer-Verlag, New York, 2nd edition.

    Google Scholar 

  • Gurney, W. S. C. and R. M. Nisbet (1998). Ecological Dynamics. Oxford University Press, Oxford.

    Google Scholar 

  • Gyllenberg, M., I. Hanski and T. Lindstrom (1997). Continuous versus discrete single species population models with adjustable reproductive strategies. Bulletin of Mathematical Biology 59: 679–705.

    Google Scholar 

  • Hairston, N. G., F. E. Smith and L. B. Slobodkin (1960). Community structure, population control, and competition. The American Naturalist 94: 421–425.

    Google Scholar 

  • Hanegraaf, P. P. F. and B. W. Kooi (2002). The dynamics of a tri-trophic food chain with two-component populations from a biochemical perspective. Ecological Modelling 152: 47–64.

    Google Scholar 

  • Hastings, A. (1996). What equilibrium behaviour of Lotka-Volterra models does not tell us about food webs. In: Polis, G. A. and K. O. Winemiller (Eds). Food Webs. pp. 211–217. Chapman & Hall, New York.

    Google Scholar 

  • Hastings, A. and T. Powell (1991). Chaos in a three-species food chain. Ecology 72: 896–903.

    Google Scholar 

  • Hofbauer, J. and K. Sigmund (1998). Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge.

    Google Scholar 

  • Hogeweg, P. and B. Hesper (1978). Interactive instruction population interactions. Computers in Biology and Medicine 8: 319–327.

    Google Scholar 

  • Huisman, J. and F. J. Weissing (2002). Oscillations and chaos generated by competition for interactively essential resources. Ecological Research 17: 175–181.

    Google Scholar 

  • Hunt, H. W. and D. H. Wall (2002). Modelling the effects of loss of soil biodiversity on ecosystem function. Global Change Biology 8: 33–50.

    Google Scholar 

  • Hutchinson, G. E. (1959). Homage to Santa Rosalia or why are there so many kinds of animals? The American Naturalist 93: 145–159.

    Google Scholar 

  • Jenkins, B., R. L. Kitching and S. L. Pimm (1992). Productivity, disturbance, and food web structure at a local spatial scale in experimental container habitats. Oikos 65: 249–255.

    Google Scholar 

  • Khibnik, A. I., Y. A. Kuznetsov, V. V. Levitin and E. V. Nikolaev (1993). Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. Physica D 62: 360–371.

    Google Scholar 

  • Kondoh, M. (2003). Foraging adaption and the relationship between food-web complexity and stability. Science 299: 1388–1391.

    Google Scholar 

  • Kooi, B. W. and M. P. Boer (2001). Bifurcations in ecosystem models and their biological interpretation. Applicable Analysis 77: 29–59.

    Google Scholar 

  • Kooi, B. W. and M. P. Boer (2003). Chaotic behaviour of a predator-prey system. Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms 10: 259–272.

    Google Scholar 

  • Kooi, B. W. and S. A. L. M. Kooijman (2000). Invading species can stabilize simple trophic systems. Ecological Modelling 133: 57–72.

    Google Scholar 

  • Kooi, B. W., M. P. Boer and S. A. L. M. Kooijman (1998). On the use of the logistic equation in food chains. Bulletin of Mathematical Biology 60: 231–246.

    Google Scholar 

  • Kooi, B. W., M. P. Boer and S. A. L. M. Kooijman (1999). Resistance of a food chain to invasion by a top predator. Mathematical Biosciences 157: 217–236.

    Google Scholar 

  • Kooi, B. W., L. D. J. Kuijper, M. P. Boer and S. A. L. M. Kooijman (2002a). Numerical bifurcation analysis of a tri-trophic food web with omnivory. Mathematical Biosciences 177: 201–228.

    Google Scholar 

  • Kooi, B. W., L. D. J. Kuijper, M. P. Boer and S. A. L. M. Kooijman (2002b). Aggregation methods in food chains with nutrient recycling. Ecological Modelling 157: 69–86.

    Google Scholar 

  • Kooijman, S. A. L. M. (2000). Dynamic Energy and Mass Budgets in Biological Systems. Cambridge University Press, Cambridge.

    Google Scholar 

  • Kooijman, S. A. L. M. and R. M. Nisbet (2000). How light and nutrients affect life in a closed bottle. In: Jorgenson, S. E., (Ed.), Thermodynamics and ecology, pages 19–60. CRC Publ., Boca Raton, FL.

    Google Scholar 

  • Kooijman, S. A. L. M., T. Andersen and B. W. Kooi (2002). Dynamic energy budget representations of stoichiometric constraints on population dynamics. Ecology. In press.

  • Kuijper, L. D. J., B. W. Kooi, C. Zonneveld and S. A. L. M. Kooijman (2003). Omnivory and food web dynamics. Ecological Modelling 163: 19–32.

    Google Scholar 

  • Kuznetsov, Y. A. (1998). Elements of Applied Bifurcation Theory, Volume 112 of Applied Mathematical Sciences. Springer-Verlag, New York, 2nd edition.

    Google Scholar 

  • Kuznetsov, Y. A. and V. V. Levitin (1997). CONTENT: Integrated environment for the analysis of dynamical systems. Centrum voor Wiskunde en Informatica (CWI), Kruislaan 413, 1098 SJ Amsterdam, The Netherlands, 1.5 edition.

    Google Scholar 

  • Kuznetsov, Y. A. and S. Rinaldi (1996). Remarks on food chain dynamics. Mathematical Biosciences 124: 1–33.

    Google Scholar 

  • Kuznetsov, Y. A., O. De Feo, and S. Rinaldi (2001). Belyakov homoclinic bifurcations in a tritrophic food chain model. SIAM Journal of Applied Mathematics 62: 462–487.

    Google Scholar 

  • Law, R. and R. D. Morton (1996). Permanence and the assembly of ecological communities. Ecology 77: 762–775.

    Google Scholar 

  • Letellier, C. and M. A. Aziz-Alaoui (2002). Analysis of the dynamics of a realistic ecological model. Chaos, Solitons and Fractals 13: 95–107.

    Google Scholar 

  • Lockwood, J. L., R. D. Powell, M. P. Nott and S. L. Pimm (1997). Assembling ecological communities in time and space. Oikos 80: 549–553.

    Google Scholar 

  • Luckinbill, L. S. (1974). The effects of space and enrichment on a predator-prey system. Ecology 55: 1142–1147.

    Google Scholar 

  • Marrow, P., U. Dieckmann and R. Law (1996). Evolutionary dynamics of predator-prey systems: An ecological perspective. Journal of Mathematical Biology 34: 556–578.

    Google Scholar 

  • Matsuda, H., M. Hori and P. A. Abrams (1996). Effects of predator-specific defence on biodiversity and community complexity in two-trophic-level communities. Evolutionary Ecology 10: 13–28.

    Google Scholar 

  • May, R. M. (1973). Stability and Complexity in Model Ecosystems. Princeton University Press, New Haven.

    Google Scholar 

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

    Google Scholar 

  • McCann, K. and P. Yodzis (1994). Biological conditions for chaos in a tree-species food chain. Ecology 75: 561–564.

    Google Scholar 

  • McCann, K. and P. Yodzis (1995). Bifurcation structure of a tree-species food chain model. Theoretical Population Biology 48: 93–125.

    Google Scholar 

  • Meszéna, G., E., Kisdi, U. Dieckmann, S. A. H. Geritz and J. A. J. Metz (2001). Evolutionarily optimisation model and matrix games in a unified perspective of adaptive dynamics. Selection 2: 193–210.

    Google Scholar 

  • Metz, J. A. J., S. A. H. Geritz and R. M. Nisbet (1992). How should we define 'fitness' for general ecological scenarios? Trends in Ecology and Evolution 7: 198–202.

    Google Scholar 

  • Metz, J. A. J., S. A. H. Geritz, G., Meszéna, F. J. A. Jacobs and J. S. van Heerwaarden (1996). Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In: Strien, S. J. van and S. M. Verduyn Lunel (Eds). Stochastic and spatial structures of dynamical systems. pp 183–231. North-Holland, Amsterdam.

    Google Scholar 

  • Morin, P. J. (1999). Community Ecology. Blackwell Science, Malden, Massachusetts.

    Google Scholar 

  • Morton, R. D., R. Law, S. L. Pimm and J. A. Drake (1996). On models for assembling ecological communities. Oikos 75: 493–499.

    Google Scholar 

  • Neutel, A.-M., J. A. P. Heesterbeek and P. C. de Ruiter (2002). Stability in real food webs: Weak links in long loops. Science 296: 1120–1123.

    Google Scholar 

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

    Google Scholar 

  • Ott, E. (1993). Chaos in Dynamical Systems. Cambridge University Press, Cambridge.

    Google Scholar 

  • Pimm, S. L. and R. L. Kitching (1987). The determinants of food chain lengths. Oikos 50: 302–307.

    Google Scholar 

  • Pimm, S. L. and J. H. Lawton (1977). Number of trophic levels in ecological communities. Nature 268: 329–331.

    Google Scholar 

  • Post, W. M. and S. L. Pimm (1983). Community assembly and food web stability. Mathematical Biosciences 64: 169–192.

    Google Scholar 

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

    Google Scholar 

  • Scheffer, M., S. Carpenter, J. A. Foley, C. Folke and B. Walker (2001). Catastrophic shifts in ecosystems. Nature 413: 591–596.

    Google Scholar 

  • Shertzer, K. W., S. P. Ellner, G. F. Fussmann and N. G. Hairston Jr (2002). Predatorprey cycles in an aquatic microcosm: testing hypotheses of mechnism. Journal of Animal Ecology 71: 802–815.

    Google Scholar 

  • Smith, H. L. and P. Waltman (1994). The Theory of the Chemostat. Cambridge University Press, Cambridge.

    Google Scholar 

  • Stearns, S. C. and P. Magwene (2003). The naturalist in a world of genomics. The American Naturalist 161: 171–180.

    Google Scholar 

  • Sterner, R. W., A. Bajpai and T. Adams (1997). The enigma of food chain length: absence of theoretical evidence for dynamic constraints. Ecology 78: 2258–2262.

    Google Scholar 

  • Tilman, D. (1982). Resource competition and community structure. Princeton University Press, Princeton.

    Google Scholar 

  • Upadhyay, R. K. (2003). Multiple attractors and crisis route to chaos in a model food-chain. Chaos, Solitons and Fractals 16: 737–747.

    Google Scholar 

  • van den Berg, H. A. (1998). Propagation of permanent perturbations in food chains and food webs. Ecological Modelling 107: 225–235.

    Google Scholar 

  • Vos, M., B. W. Kooi, W. M., Mooij and D. L. DeAngelis (2003). Inducible defences and the paradox of enrichment. submitted.

  • Yodzis, P. (1988). The indeterminacy of ecological interactions as perceived through perturbation experiments. Ecology 69: 508–515.

    Google Scholar 

  • Yodzis, P. (1989). Introduction to Theoretical Ecology. Harper & Row Publishers Inc., New York.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Biology, Institute of Ecological Science, Faculty of Earth and Life Sciences, Vrije Universiteit, de Boelelaan 1087, 1081 HV, Amsterdam, The Netherlands

    B.W. Kooi

Authors
  1. B.W. Kooi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kooi, B. Numerical Bifurcation Analysis of Ecosystems in a Spatially Homogeneous Environment. Acta Biotheor 51, 189–222 (2003). https://doi.org/10.1023/A:1025146207201

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1025146207201

Keywords

Search

Navigation