Bulletin of Mathematical Biology

, Volume 58, Issue 5, pp 835–859 | Cite as

Weakly dissipative predator-prey systems

  • A. A. King
  • W. M. Schaffer
  • C. Gordon
  • J. Treat
  • M. Kot
Article

Abstract

In the presence of seasonal forcing, predator-prey models with quadratic interaction terms and weak dissipation can exhibit infinite numbers of coexisting periodic attractors corresponding to cycles of different magnitude and frequency. These motions are best understood with reference to the conservative case, for which the degree of dissipation is, by definition, zero. Here one observes the familiar mix of “regular” (neutrally stable orbits and tori) and chaotic motion typical of non-integrable Hamiltonian systems. Perturbing away from the conservative limit, the chaos becomes transitory. In addition, the invariant tori are destroyed and the neutrally stable periodic orbits becomes stable limit cycles, the basins of attraction of which are intertwined in a complicated fashion. As a result, stochastic perturbations can bounce the system from one basin to another with consequent changes in system behavior. Biologically, weak dissipation corresponds to the case in which predators are able to regulate the density of their prey well below carrying capacity.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Altieri, M. A. 1991. How best can we use biodiversity in agroecosystems?Outlook on Agriculture 20, 15–23.Google Scholar
  2. Altieri, M. A. and L. L. Schmidt. 1986. Cover crops affect insect and spider populations in apple orchards.California Agriculture 40, 15–17.Google Scholar
  3. Altieri, M. A. and W. H. Whitcomb. 1979. The potential use of weeds in the manipulation of beneficial insects.Hort. Sci. 14, 12.Google Scholar
  4. Ayres, M. 1993. Plant defense, herbivory and climate change. InBiotic Interactions and Global Change, P. M. Kareiva, J. G. Kingsolver and R. B. Huey (Eds.) Sunderland, MA: Sinauer Associates.Google Scholar
  5. Beddington, J. R., C. A. Free and J. H. Lawton. 1978. Characteristics of successuful natural enemies in models of biological control of insect pests.Nature 273, 513–519.CrossRefGoogle Scholar
  6. Brust, G. E. and L. R. King. 1994. Effects of crop rotation and reduced chemical inputs on pests and predators in maize agroecosystems.Agriculture, Ecosystems, and Environment 48, 77–89.CrossRefGoogle Scholar
  7. Crawley, M. J. 1992.Natural Enemies: The Population Biology of Predators, Parasites, and Diseases. Boston, MA: Blackwell Scientific.Google Scholar
  8. Debach, P. and D. Rosen. 1991.Biological Control by Natural Enemies, 2nd ed. Cambridge: Cambridge University Press.Google Scholar
  9. Doutt, R. L. and J. Nakata. 1973. Therubus grasshopper and its egg parasitoid: an endemic biotic system useful in grape-pest management.Environmental Entomology,2, 381–386.Google Scholar
  10. Ehrlich, P. R. and L. C. Birch. 1967. The “balance of nature” and “population control”.Amer. Natur. 101, 97–107.CrossRefGoogle Scholar
  11. Gleissman, S. R. and M. A. Altieri. 1982. Polyculture cropping has advantages.California Agriculture 36, 14–16.Google Scholar
  12. Godfrey, L. D. and T. F. Leigh. 1994. Alfalfa harvest strategy on lygus bug (Hemiptera miridae) and insect predator population density: implications for use as a trap crop in cotton.Environ. Entomol. 23, 1106–1118.Google Scholar
  13. Goel, N. S., S. C. Maitra and E. W. Montroll. 1971. On the Volterra and other nonlinear models of interacting populations.Rev. Mod. Phys. 43, 231–276.MathSciNetCrossRefGoogle Scholar
  14. Gumowski, I. and C. Mira. 1980.Recurrences and Discrete Dynamical Systems. Berlin: Springer-Verlag.Google Scholar
  15. Hairston, N. G. 1989.Ecological Experiment. Cambridge: Cambridge University Press.Google Scholar
  16. Hairston, N. G., F. E. Smith and L. B. Slobodkin. 1960. Community structure, population control and competition.Amer. Natur. 94, 421–425.CrossRefGoogle Scholar
  17. Hénon, M. 1983. Numerical explorations of Hamiltonian systems. InChaotic Behavior of Deterministic Systems, G. Iooss, R. H. Helleman and R. Stora (Eds). Amsterdam: North-Holland.Google Scholar
  18. Huffaker, C. B. and P. S. Messenger. 1976.Theory and Practice of Biological Control. New York: Academic Press.Google Scholar
  19. Inoue, M. and H. Kamifukumoto. 1984. Scenarios leading to chaos is forced Lotka-Volterra model.Prog. Theor. Phys. 71, 930–937.MATHMathSciNetCrossRefGoogle Scholar
  20. Kerner, E. H. 1957. A statistical mechanics of interacting biological species.Bull. Math. Biophys. 19, 121–146.MathSciNetGoogle Scholar
  21. Kerner, E. H. 1959. Further considerations on the statistical mechanics of biological associations.Bull. Math. Biophys. 23, 141–157.MathSciNetGoogle Scholar
  22. Kot, M., G. S. Sayler and T. W. Schulz. 1992. Complex dynamics in a model microbial system.Bull. Math. Biol. 54, 619–648.MATHGoogle Scholar
  23. Lagerlöf, J. and H. Wallin. 1993. The abundance of arthropods along two field margins with different types of vegetation composition: an experimental study.Agriculture, Ecosystems, and Environment 43, 141–154.CrossRefGoogle Scholar
  24. Leigh, E. G. 1965. On the relation between the productivity, biomass, diversity, and stability of a community.Proc. Natl. Acad. Sci. U.S.A. 53, 777–783.MATHCrossRefGoogle Scholar
  25. Leigh, E. G. 1968. The ecological role of Volterra's equations. InSome Mathematical Problems in Biology, M. Gerstenhaber (Ed), Providence, RI: American Mathematical Society.Google Scholar
  26. Leigh, E. G. 1975. Population fluctuations, community stability, and environmental variability. InEvolution of Species Communities, M. Cody and J. Diamond (Eds) Cambridge, MA: Belknap Press.Google Scholar
  27. Levins, R. 1968.Evolution in Changing Environments. Princeton, NJ: Princeton University Press.Google Scholar
  28. Lichtenberg, A. J. and M. A. Lieberman. 1992.Regular and Chaotic Dynamics, 3rd ed. New York: Springer-Verlag.Google Scholar
  29. Lin, C. C. and L. A. Segel. 1974.Mathematics Applied to Deterministic Problems in the Natural Sciences. New York: MacMillan.Google Scholar
  30. MacArthur, R. H. 1969. Species packing and what competition minimizes.Proc. Natl. Acad. Sci. U.S.A. 64, 1369–1371.CrossRefGoogle Scholar
  31. MacArthur, R. H. 1970. Species packing and competitive equilibrium for many species.Theoret. Population Biol. 1, 1–11.CrossRefGoogle Scholar
  32. MacDonald, S. W.,et al. 1985. Fractal basin boundaries.Physica D 17, 125–153.MathSciNetCrossRefGoogle Scholar
  33. Marion, J. B. 1970.Classical Dynamics of Particles and Systems. New York: Academic Press.Google Scholar
  34. May, R. M. 1973.Complexity and Stability in Model Ecosystems. Princeton, NJ: Princeton University Press.Google Scholar
  35. Maynard Smith, J. 1974.Models in Ecology. London: Cambridge University Press.Google Scholar
  36. Mira, C. 1987.Chaotic Dynamics. Singapore: World Scientific.Google Scholar
  37. Morris, R. F. 1963. The dynamics of endemic spruce budworm populations.Mem. Entomol. Soc. Canada 21, 1–322.Google Scholar
  38. Murdoch, W. W. 1966. Community structure, population control and competition—A critique.Amer. Natur. 100, 219–226.CrossRefGoogle Scholar
  39. Murdoch, W. W.. 1975. Diversity, complexity, stability and pest control.J. Appl. Ecol. 12, 795–807.CrossRefGoogle Scholar
  40. Murray, J. D. 1993.Mathematical Biology, 2nd ed, Berlin: Springer-Verlag.Google Scholar
  41. Paoletti, M. G.et al. 1989. Animal and plant interactions in agroecosystems: the case of the woodland remnants in northeastern Italy.Ecol. Int. Bull. 17, 79–91.Google Scholar
  42. Pimentel, D.et al. 1991. Environmental and economic impacts of reducting U.S. agricultural pesticide use. InCRC Handbook of Pest Management in Agriculture, D. Pimentel (Ed), Vol. II. Boca Raton, FL: CRC Press.Google Scholar
  43. Rand, D. A. and H. Wilson. 1991. Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics.Proc. Roy. Soc. Lond. Ser. B 246, 179–184.Google Scholar
  44. Rinaldi, S. and S. Muratori. 1993. Conditioned chaos in seasonally perturbed predator-prey models.Ecol. Model. 69, 79–97.CrossRefGoogle Scholar
  45. Rinaldi, S., S. Muratori and Y. A. Kuznetsov. 1993. Multiple attractors, catastrophes, and chaos in seasonally perturbed predator-prey communities.Bull. Math. Biol. 55, 15–35.MATHGoogle Scholar
  46. Rosen, R. 1970.Dynamical System Theory in Biology. 1. Stability Theory and its Applications. New York: Wiley Interscience.Google Scholar
  47. Rosenzweig, M. L. 1971. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time.Science 171, 385–387.Google Scholar
  48. Ruelle, D. 1979. Sensitive dependence on initial conditions.Ann. N.Y. Acad. Sci. 316, 408–416.MATHMathSciNetGoogle Scholar
  49. Schaffer, W. M. 1988. Perceiving order in the chaos of nature. InEvolution of Life Histories of Mammals, M. S. Boyce (Ed), pp. 313–350. New Haven, CT: Yale University Press.Google Scholar
  50. Schwerdtfäger, F. 1941. Über die ursachen des massenwechels des insekten.Z. Angew. Ent. 28, 254–303.Google Scholar
  51. Slobodkin, L. B., F. E. Smith and N. G. Hairston. 1967. Regulation in terrestrial ecosystems and the implied balance of nature.Am. Nat. 101, 109–124.CrossRefGoogle Scholar
  52. Stern, V. 1969. Implanting alfalfa in cotton to control lygus bugs and other insects.Proc. Tall Timbers Conf. Ecol. Anim. Control Habitat Mgmt. 1, 54.Google Scholar
  53. Tabor, M. 1989.Chaos and Integrability in Nonlinear Systems, New York: Wiley.Google Scholar
  54. Thompson, J. M. T. and H. B. Stewart. 1986.Nonlinear Dynamics and Chaos, Chichester: Wiley.Google Scholar
  55. Tsang, K. Y. and M. A. Lieberman. 1986. Transient chaotic distributions in dissipative systems.Physica D 21, 401–414.MATHMathSciNetCrossRefGoogle Scholar
  56. Varley, G. C. 1949. Special review: population changes in German forest pests.J. Anim. Ecol. 18, 117–122.CrossRefGoogle Scholar
  57. Whitcomb, W. H. and K. E. Godfrey. 1991. The use of predators in insect control. InCRC Handbook of Pest Management in Agriculture, D. Pimentel (Ed.), Vol. II. Boca Raton, FL: CRC Press.Google Scholar

Copyright information

© Society for Mathematical Biology 1996

Authors and Affiliations

  • A. A. King
    • 1
  • W. M. Schaffer
    • 1
    • 2
  • C. Gordon
    • 2
  • J. Treat
    • 2
  • M. Kot
    • 3
  1. 1.Program in Applied MathematicsThe University of ArizonaTucsonU.S.A.
  2. 2.Department of Ecology and Evolutionary BiologyThe University of ArizonaTucsonU.S.A.
  3. 3.Department of MathematicsThe University of TennesseeKnoxvilleU.S.A.

Personalised recommendations