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Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses.

ABSTRACT: In this paper we analyze the dynamics of a tritrophic food chain, with functional responses of Holling type III and II for the mesopredator and superpredator respectively and logistic growth rate for the prey. We show that there are parameter conditions for which the system has up to three equilibrium points. When three equilibrium points appear we show that each one of them exhibits a supercritical Hopf bifurcation. Moreover we can also show that there is a set of parameters for which the system exhibits a simultaneous Hopf bifurcation.

SUBMITTER: Ble G

PROVIDER: S-EPMC7902555 | biostudies-literature | 2021 Feb

REPOSITORIES: biostudies-literature

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Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses.

Blé Gamaliel G Castellanos Víctor V Castillo-Santos Francisco Eduardo FE

Heliyon 20210217 2

In this paper we analyze the dynamics of a tritrophic food chain, with functional responses of Holling type III and II for the mesopredator and superpredator respectively and logistic growth rate for the prey. We show that there are parameter conditions for which the system has up to three equilibrium points. When three equilibrium points appear we show that each one of them exhibits a supercritical Hopf bifurcation. Moreover we can also show that there is a set of parameters for which the syste ...[more]

PMID: 33665414

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Project description:BackgroundHolling (Can Entomol 91(5):293-320, 1959) was the first to describe a functional response between a predator's consumption-rate and the density of its prey. The same concept can be applied to the habitat selection of herbivores, specifically, the change in relative habitat use with the change in habitat availability. Functional responses in habitat selection at a home-range scale have been reported for several large herbivores. However, a link to Holling's original functional response types has never been drawn, although it could replace the current phenomenological view with a more mechanistically based understanding of functional responses.MethodsIn this study, discrete choice models were implemented as mixed-effects baseline-category logit models to analyze the variation in habitat selection of a large herbivore at seasonal and diurnal scales. Thus, changes in the use of land cover types with respect to their availability were investigated by monitoring 11 land cover types commonly used by roe deer (Capreolus capreolus) in the Bavarian Forest National Park, Germany. Functional response curves were then fitted using Holling's formulas.ResultsStrong evidence of non-linear functional responses was obtained for almost all of the examined land cover types. The shape of the functional response curves varied depending on the season, the time of day, and in some cases between sexes. These responses could be referenced to Holling's types, with a predominance of type II.ConclusionsOur results indicate that Holling's types can be applied to describe general patterns of the habitat selection behavior of herbivores. Functional responses in habitat selection may occur in situations requiring a trade-off in the selection of land cover types offering different resources, such as due to the temporally varying physiological needs of herbivores. Moreover, two associated parameters defining the curves (prey density and predation rate) can aid in the identification of temporal variations and in determinations of the strength of the cost-benefit ratio for a specific land cover type. Application of our novel approach, using Holling's equations to describe functional responses in the habitat selection of herbivores, will allow the assignment of general land cover attraction values, independent of availability, thus facilitating the identification of suitable habitats.

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Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses.

Publications

Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses.

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