Abstract
Some mathematical techniques for the analysis of satiation-based predation models previously developed by the first author are applied in the present paper to a model by the second author for predation by the predatory miteMetaseiulus occidentalis (Nesbitt). It turns out that for this predator the predation rate should keep increasing at high prey densities as the square root of the prey density,x. This particular shape of the functional response is shown to occur if and only if the upper satiation threshold for prey capture coincides with the maximum gut capacity. The functional response predicted by the model, moreover, is in fair quantitative agreement with predation rates observed by the third author in artificial arenas.
A further analysis of the model shows that the variance of the catch should also increase as the square root ofx. This prediction is consistent in a qualitative manner with the continued increase in the variance of the catch. However, quantitatively, the observed variances are even too large to be compatible with any model in which the feeding rate is subject to regulation by a negative feedback. Therefore, the difference between predicted and observed variances is hypothesized to be due to nonhomogeneities in the experimental material. The inferred additional variance component proportional tox accords fairly well with the trend apparent in the data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cox, D.R., 1962. Renewal Theory. Methuen, London, 142 pp.
Diekmann, O., Metz, J.A.J. and Sabelis, M.W., 1988. Mathematical models of predator-prey-plant interactions in a patch environment. Exp. Appl. Acarol., 5: 319–342.
Doucet, P.G. and van Straalen, N.M., 1980. Analysis of hunger from feeding rate observation. Anim. Behav., 28: 913–921.
Fransz, H.G., 1974, The functional response to prey density in an acarine system. PUDOC, Wageningen, 143 pp.
Heijmans, H.J.A.M., 1984. Holling’s hungry mantid model of the invertebrate functional response: III. Stable satiation distribution. J. Math. Biol., 21: 115–143.
Holling, C.S., 1959. The components of predation as revealed by the study of small mammal predation of the European pine sawfly. Can. Entomol., 41: 293–320.
Holling, C.S., 1966. The functional response of invertebrate predators to prey density. Mem. Entomol. Soc. Can., 48: 86 pp.
Johnson, D.M., Akre, B.G. and Crowley, P.H., 1975. Modeling arthropod predation: wasteful killing by damselfy najads. Ecology, 56: 1081–1093.
Metz, J.A.J. and Diekmann, O. (Editors), 1986. The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, 68; Springer, Berlin, 511 pp.
Metz, J.A.J. and van Batenburg, 1985a. Holling’s “hungry mantid” model for the invertebrate functional response: I. The full model and some of its limits. J. Math. Biol., 22:209–238.
Metz, J.A.J. and van Batenburg, 1985b. Holling’s “hungry mantid” model for the invertebrate functional response: II. Negligible handling time. J. Math. Biol., 22: 239–257.
Sabelis, M.W., 1981. Biological control of two-spotted spider mites using phytoseiid predators. Agric. Res. Rep. 910, PUDOC, Wageningen, 242 pp.
Sabelis, M.W., 1986. The functional response of predatory mites to the density of two-spotted spider mites. In: J.A.J. Metz and O. Diekmann (Editors), The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, 68; Springer, Berlin, pp. 298–321.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Metz, J.A.J., Sabelis, M.W. & Kuchlein, J.H. Sources of variation in predation rates at high prey densities: an analytic model and a mite example. Exp Appl Acarol 5, 187–205 (1988). https://doi.org/10.1007/BF02366094
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02366094