Abstract
The non-linear behavior of a differential equations-based predator-prey model, incorporating a spatial refuge protecting a consant proportion of prey and with temperature-dependent parameters chosen appropriately for a mite interaction on fruit trees, is examined using the numerical bifurcation code AUTO 86. The most significant result of this analysis is the existence of a temperature interval in which increasing the amount of refuge dynamically destabilizes the system; and on part of this interval the interaction is less likely to persist in that predator and prey minimum population densities are lower than when no refuge is available. It is also shown that increasing the amount of refuge can lead to population outbreaks due to the presence of multiple stable states. The ecological implications of a refuge are discussed with respect to the biological control of mite pests.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature
Arrowsmith, D. K. and C. M. Place. 1982.Ordinary Differential Equations. London: Chapman & Hall.
DeBach, P. and D. Rosen. 1991.Biological Control By Natural Enemies. Cambridge: Cambridge University Press.
Doedel, E. J. 1984. The computer-aided bifurcation analysis of predator-prey models.J. Math. Biol. 20, 1–14.
Doedel, E. J. and J. P. Kernevez. 1986.AUTO: Software For Continuation and Bifurcation Problems in Ordinary Differential Equations. Pasadena, California: California Institute of Technology.
Fransz, H. G. 1974.The Functional Response to Prey Density in an Acarine System (Simulation Monographs). Wageningen: Pudoc.
Hassell, M. P. 1978.The Dynamics of Arthropod Predator-Prey Systems. Princeton: Princeton University Press.
Hoy, M. A. 1985. Almonds (California). InSpider Mites: Their Biology, Natural Enemies and Control, W. Helle and M. W. Sabelis (Eds), pp. 299–310. World Crop Pests Vol. 1B. Amsterdam: Elsevier.
Hoyt, S. C. 1969. Integrated chemical control of insects and biological control of mites on apple in Washington.J. Eco. Entomol. 62, 74–86.
Hoyt, S. C., L. K. Tanigoshi and R. W. Browne. 1979. Economic injury level studies in relation to mites on apple. InRecent Advances in Acarology, Vol. 1, J. G. Rodriguez (Ed.), pp. 3–12. New York: Academic Press.
Huffaker, C. B., 1958. Experimental studies on predation: dispersion factors and predator-prey oscillations.Hilgardia 27, 343–383.
Huffaker, C. B. and C. E. Kennett. 1956. Experimental studies on predation: predation and cyclamen-mite populations on strawberries in California.Hilgardia 26, 191–222.
Huffaker, C. B., K. P. Shea and S. G. Herman. 1963. Experimental studies on predation (III). Complex dispersion and levels of food in an acarine predator-prey interaction.Hilgardia 34, 305–330.
Liu, W., S. A. Levin and Y. Iwasa. 1986. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models.J. Math. Biol. 23, 187–204.
Logan, J. A. and D. W. Hilbert. 1983. Modeling the effect of temperature on arthropod population systems. InAnalysis of Ecological Systems: State of the Art in Ecological Modeling, W. K. Lauenroth, G. V. Skogerboe and M. Flug (Eds), pp. 113–122. Amsterdam: Elsevier.
Lotka, A. J. 1925.Elements of Physical Biology. Baltimore: williams and Wilkins.
May, R. M. 1973.Stability and Complexity in Model Ecosystems. Princeton: Princeton University Press.
Maynard Smith, J. 1974.Models in Ecology. Cambridge: Cambridge University Press.
Nicholson, A. J. 1933. The balance of animal populations.J. Anim. Ecol. 2, 132–178.
Nicholson, A. J. and V. A. Bailey. 1935. The balance of animal populations. Part I.Proc. Zool. Soc. Lond. 3, 551–598.
Taylor, R. J., 1984.Predation. New York: Chapman & Hall.
Van de Vrie, M. 1985. Apple. InSpider Mites: Their Biology, Natural Enemies and Control, W. Helle and M. W. Sabelis (Eds), pp. 311–325. World Crop Pests Vol. 1B. Amsterdam: Elsevier.
Volterra, V. 1926. Variazioni e Fluttuazioni del Numero d'Individui in Specie Animali Conviventi.Mem. Acad. Lincei. 2, 31–113.
White, E. G. and C. B. Huffaker. 1969. Regulatory processes and population cyclicity in laboratory populations ofAnagasta kühniella (Zeller) (Lepidoptera: Phycitidae) II. Parasitism, predation, competition, and protective cover.Res. Pop. Ecol. 11, 150–185.
Wollkind, D. J. and J. A. Logan. 1978. Temperature-dependent predator-prey mite ecosystem on apple tree foliage.J. Math. biol. 6, 265–283.
Wollkind, D. J., J. B. Collings and J. A. Logan. 1988. Metastability in a temperature-dependent model system for predator-mite outbreak interactions on fruit trees.Bull. math. Biol. 50, 379–409.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Collings, J.B. Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge. Bltn Mathcal Biology 57, 63–76 (1995). https://doi.org/10.1007/BF02458316
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02458316