Feeding Threshold for Predators Stabilizes Predator-Prey Systems

D. Bontje; B. W. Kooi; G. A.K. van Voorn; S.A.L.M Kooijman

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 6, page 91-108
  • ISSN: 0973-5348

Abstract

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Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, several mechanisms 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 minimum threshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models are analysed, 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 of the 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.

How to cite

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Bontje, D., et al. "Feeding Threshold for Predators Stabilizes Predator-Prey Systems." Mathematical Modelling of Natural Phenomena 4.6 (2009): 91-108. <http://eudml.org/doc/222394>.

@article{Bontje2009,
abstract = { Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, several mechanisms 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 minimum threshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models are analysed, 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 of the 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. },
author = {Bontje, D., Kooi, B. W., van Voorn, G. A.K., Kooijman, S.A.L.M},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {algal-ciliate experiments; feeding threshold; mass-balance; paradox of enrichment; strong stability},
language = {eng},
month = {11},
number = {6},
pages = {91-108},
publisher = {EDP Sciences},
title = {Feeding Threshold for Predators Stabilizes Predator-Prey Systems},
url = {http://eudml.org/doc/222394},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Bontje, D.
AU - Kooi, B. W.
AU - van Voorn, G. A.K.
AU - Kooijman, S.A.L.M
TI - Feeding Threshold for Predators Stabilizes Predator-Prey Systems
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/11//
PB - EDP Sciences
VL - 4
IS - 6
SP - 91
EP - 108
AB - Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, several mechanisms 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 minimum threshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models are analysed, 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 of the 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.
LA - eng
KW - algal-ciliate experiments; feeding threshold; mass-balance; paradox of enrichment; strong stability
UR - http://eudml.org/doc/222394
ER -

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