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Study of a Tritrophic Food Chain Model with Non-differentiable Functional Response

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Abstract

We study a model of three interacting species in a food chain composed by a prey, an specific predator and a generalist predator. The capture of the prey by the specific predator is modelled as a modified Holling-type II non-differentiable functional response. The other predatory interactions are both modelled as Holling-type I. Moreover, our model follows a Leslie-Gower approach, in which the function that models the growth of each predator is of logistic type, and the corresponding carrying capacities depend on the sizes of their associated available preys. The resulting model has the form of a set of nonlinear ordinary differential equations which includes a non-differentiable term. By means of topological equivalences and suitable changes of parameters, we find that there exists an Allee threshold for the survival of the prey population in the food chain, given, effectively, as a critical level for the generalist predator. The dynamics of the model is studied with analytical and computational tools for bifurcation theory. We present two-parameter bifurcation diagrams that contain both local phenomena (Hopf, saddle-node transcritical, cusp, Bogdanov-Takens bifurcations) and global events (homoclinic and heteroclinic connections). In particular, we find that two types of heteroclinic cycles can be formed, both of them containing connections to the origin. One of these cycles is planar involving the absence of the specific predator. In turn, the other heteroclinic cycle is formed by connections in the full three-dimensional phase space.

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Acknowledgements

We thank professor Eduardo González-Olivares for suggesting the study of non-differentiable predation models and for his invaluable comments and advice.

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  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

    Viviana Rivera & Pablo Aguirre

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  1. Viviana Rivera
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  2. Pablo Aguirre
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Correspondence to Pablo Aguirre.

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V.R. was funded by PIIC DGIIP-UTFSM. P.A. was partially funded by CONICYT-FONDECYT Iniciación grant 11150306 and Proyecto Basal CMM-U. de Chile

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Rivera, V., Aguirre, P. Study of a Tritrophic Food Chain Model with Non-differentiable Functional Response. Acta Appl Math 165, 19–43 (2020). https://doi.org/10.1007/s10440-019-00239-3

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