Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T13:06:18.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Feeding Thresholdfor Predators Stabilizes Predator-Prey Systems

Published online by Cambridge University Press:  27 November 2009

D. Bontje ,
B. W. Kooi ,
G. A.K. van Voorn  and
S.A.L.M Kooijman
D. Bontje*
Affiliation:
Department of Theoretical Biology, Vrije Universiteit, De Boelelaan 1085, 1081HV Amsterdam, The Netherlands
B. W. Kooi
Affiliation:
Department of Theoretical Biology, Vrije Universiteit, De Boelelaan 1085, 1081HV Amsterdam, The Netherlands
G. A.K. van Voorn
Affiliation:
Biometris, Wageningen Universiteit & Research, 6708PB Wageningen, The Netherlands
S.A.L.M Kooijman
Affiliation:
Department of Theoretical Biology, Vrije Universiteit, De Boelelaan 1085, 1081HV Amsterdam, The Netherlands
*
daniel.bontje@falw.vu.nl
Get access

Abstract

Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, severalmechanisms have been proposed to resolve this paradox. In this paper we will show that a feeding threshold in the functional response for predators feeding on a prey population stabilizes the system and that there exists a minimumthreshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models areanalysed, the first being the classical Rosenzweig-MacArthur (RM) model with an adapted Holling type-II functional response to include a feeding threshold. This mathematical model can be studied using analytical tools, which gives insight into the mathematical properties of the two dimensional ode system and reveals underlying stabilisation mechanisms. The second model is a mass-balance (MB) model for a predator-prey-nutrient system with complete recycling ofthe nutrient in a closed environment. In this model a feeding threshold is also taken into account for the predator-prey trophic interaction. Numerical bifurcation analysis is performed on both models. Analysis results are compared between models and are discussed in relation to the analytical analysis of the classical RM model. Experimental data from the literature of a closed system with ciliates, algae and a limiting nutrient are used to estimate parameters for the MB model. This microbial system forms the bottom trophic levels of aquatic ecosystems and therefore a complete overview of its dynamics is essential for understanding aquatic ecosystem dynamics.

Keywords

algal-ciliate experimentsfeeding thresholdmass-balanceparadox of enrichmentstrong stability
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 6: Ecology (Part 1) , 2009 , pp. 91 - 108
Copyright
© EDP Sciences, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

P. A. Abrams, H. Matsuda. Prey adaptation as a cause of predator-prey cycles. Evolution, 51 (1997),1742–1750.
Abrams, P. A., Walters, C. J.. Invulnerable prey and the paradox of enrichment. Ecology, 77 (1996), 1125–1133. CrossRef
Beddington, J. R.. Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol., 44 (1975), 331–340. CrossRef
Berryman, A. A., Hawkins, B. A.. The refuge as an integrating concept in ecology and evolution. Oikos, 115 (2006), No. 1, 192–196. CrossRef
Bontje, D., Kooi, B. W., Liebig, M., Kooijman, S. A. L. M.. Modelling long-term ecotoxicological effects on an algal population under dynamic nutrient stress. Water Research, 43 (2009), 3292–3300. CrossRef
Brack, W., Bakker, J., de Deckere, E., Deerenberg, C, van Gils, J., Hein, M, Jurajda, P, Kooijman, S., Lamoree, M., Lek, S., López de, M.-J. Alda, A. Marcomini, I. Muñoz, S. Rattei, H. Segner, K. Thomas, P. von der Ohe, B. Westrich, D. de Zwart, M. Schmitt-Jansen. Models for assessing and forecasting the impact of environmental key pollutants on freshwater and marine ecosystems and biodiversity. Environmental Science & Pollution Research, 12 (2005), No. 5, 252–256. CrossRef
D. L. DeAngelis. Dynamics of Nutrient Cycling and Food Webs. Number 9 in Population and Community Biology series. Chapman & Hall, London (1992).
DeAngelis, D. L., Goldstein, R. A., O'Neill, R. V.. A model for trophic interaction. Ecology, 56 (1975), 881–892. CrossRef
Grimm, V., Wissel, C.. Babel, or the ecological stability discussions: an inventory and analysis of terminology and a guide for avoiding confusion. Oecologia, 109 (1997), 323–334. CrossRef
Gross, T., Ebenhöh, W., Feudel, U.. Enrichment and foodchain stability: the impact of different forms of predator-prey interaction. J. Theor. Biol., 227 (2004), 349–358. CrossRef
Jansen, V. A. A.. Regulation of predator-prey systems through spatial interactions: a possible solution to the paradox of enrichment. Oikos, 74 (1995), 384–390. CrossRef
Kirk, K. L.. Enrichment can stabilize population dynamics: autotoxins and density depencence. Ecology, 79 (1998), 2456–2462. CrossRef
Kooi, B. W.. Numerical bifurcation analysis of ecosystems in a spatially homogeneous environment. Acta Biotheoretica, 51 (2003), No. 3, 189–222. CrossRef
Kooi, B. W., Poggiale, J. C., Auger, P., Kooijman, S. A. L. M.. Aggregation methods in food chains with nutrient recycling. Ecological Modelling, 157 (2002), No. 1, 69–86. CrossRef
Kretzschmar, M., Nisbet, R. M., McCauley, E. A predator–prey model for zooplankton grazing on competing algal populations. Theor. Popul. Biol., 44 (1993), 32–66. CrossRef
Kuwamura, M, Nakazawa, T, Ogawa, T. A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment. Journal of Mathematical Biology, 58 (2009), No. 3, 459–479. CrossRef
Liebig, M., Schmidt, G., Bontje, D., Kooi, B. W., Streck, G., Traunspurger, W., Knacker, T.. Direct and indirect effects of pollutants on algae and algivorous ciliates in an aquatic indoor microcosm. Aquatic Toxicology, 88 (2008), 102–110. CrossRef
Maplesoft. Maple. Maplesoft, Waterloo, Ontario, Canada (2003).
Mitra, A., Flynn, K. J.. Importance of interactions between food quality, quantity, and gut transit time on consumer feeding, growth, and trophic dynamics. The American Naturalist, 169 (2007), No. 5, 632–646. CrossRef
A. M. Mood, F. A. Graybill, D. C. Boes. Introduction to the Theory of Statistics. McGraw-Hill, Inc., New York, 3 th edition (1974).
Müller, H., Schlegel, A.. Responses of three freshwater planktonic ciliates with different feeding modes to cryptophyte and diatom prey. Aquat. Microb. Ecol., 17 (1999), 49–60. CrossRef
Mullin, M. M., Stewart, E. F., Fuglister, F. J.. Ingestion by planktonic grazers as a function of concentration of food. Limnol. Oceanogr., 20 (1975), 259–262. CrossRef
R. A. Park, J. S. Clough. Aquatox for windows: A modular fate and effects model for aquatic ecosystems. Technical Report 2, EPA (2004).
Park, R.A., Clough, J. S., Wellman, M. C.. Aquatox: Modeling environmental fate and ecological effects in aquatic ecosystems. Ecological Modelling, 213 (2008), 1–15. CrossRef
Petrovskii, S., B–L Li, H. Malchow. Transition to spatiotemporal chaos can resolve the paradox of enrichment. Ecol. Complex., 1 (2004), 37–47. CrossRef
H. W. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery. Numerical Recipes in C. Cambridge University Press, Cambridge, 2nd edition (1992).
Rosenzweig, M. L., MacArthur, R. H.. Graphical representation and stability conditions of predator-prey interactions. The American Naturalist, 97 (1963), 209–223. CrossRef
Saage, A., Vadstein, O., Sommer, U.. Feeding behaviour of adult Centropages hamatus (Copepoda, Calanoida): Functional response and selective feeding experiments. Journal of Sea Research, 62 (2009), 16–21. CrossRef
Scheffer, M., De Boer, R. J.. Implications of spatial heterogeneity for the paradox of enrichment. Ecology, 76 (1995), 2270–2277. CrossRef
Strom, S. L., Miller, C. B., Frost, B. W.. What sets lower limits to phytoplankton stocks in high-nitrate, low-chlorophyll regions of the open ocean? Marine Ecology Progress Series, 193 (2000), 19–31. CrossRef
G. A. K. Van Voorn, D. Stiefs, T. Gross, B. W. Kooi, U. Feudel, S. A. L. M. Kooijman. Stabilization due to predator interference: comparison of different analysis approaches. Mathematical Biosciences and Engineering, 5 (2008), No. 3,:567–583.
Vos, M., Kooi, B. W., DeAngelis, D. L., Mooij, W. M.. Inducible defences and the paradox of enrichment. Oikos, 105 (2004), 471–480. CrossRef
T. Weisse, N. Karstens, V.C.L. Meyer, L. Janke, S. Lettner, K. Teichgräber. Niche separation in common prostome freshwater ciliates: the effect of food and temperature. Aquatic Microbial Ecology, 26 (2001),167–179.
Weisse, T., Kirchhoff, B. Feeding of the heterotrophic freshwater dinoflagellate Peridiniopsis berolinense on cryptophytes: analysis by flow cytometry and electronic particle counting. Aquat. Microb. Ecol., 12 (1997), 153–164. CrossRef