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Dynamics of a nonlinear mathematical model for three interacting populations

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Abstract

In this paper, we derive and analyze both analytically and numerically a mathematical model for three interacting populations. These take the form of a herbivore, a plant, and a pollinator. The full model is a nonlinear reaction–diffusion–advection system, which is derived on the basis of a series of plausible and widely supported ecological hypothesis. The study we present here deals with the conditions for the coexistence of the three interacting species. The analysis is carried out in two stages. For the homogeneous case, the mathematical model reduces to a nonlinear three-dimensional autonomous ODE system, which, as the ecological parameters change, exhibits different dynamical behaviors, namely a limit cycle and a positive attractor. The non-homogeneous case is studied mainly by means of numerical simulations of the full model defined on a rectangular region and considering appropriate initial and boundary conditions. Our results strongly suggest the stabilizing role played by the herbivore population which, in turn means that the introduction of this population into the mutualistic pollinator–plant interaction, favors the coexistence of the three interacting species.

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Notes

  1. The concepts Holling functional and numerical responses of the different types comes from a predator–prey interaction. Here, we are borrowing it such a concept. In our context, the first one captures the fact the pollinators have a limited capacity for visiting plants. The rate of pollinators visits (per pollinator) to plants, is a monotonic increasing function whose graph is asymptotic to a horizontal straight line.

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Acknowledgments

The authors thank the anonymous referees for their detailed revision of the manuscript they did. Their comments and suggestions strongly contributed to improve the contents of this paper.

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Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510 , Mexico, D.F., Mexico

    Faustino Sánchez-Garduño

  2. División Académica de Ciencias Básicas, UJAT, Km 1 Carretera Cunduacán–Jalpa de Méndez, c. p. 86690 , Cunduacán, Tabasco, México

    Víctor Castellanos & Ingrid Quilantán

Authors
  1. Faustino Sánchez-Garduño
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  2. Víctor Castellanos
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  3. Ingrid Quilantán
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Correspondence to Faustino Sánchez-Garduño.

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In celebration of “Mathematics of Planet Earth-2013”.

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Sánchez-Garduño, F., Castellanos, V. & Quilantán, I. Dynamics of a nonlinear mathematical model for three interacting populations. Bol. Soc. Mat. Mex. 20, 147–170 (2014). https://doi.org/10.1007/s40590-014-0010-1

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  • DOI: https://doi.org/10.1007/s40590-014-0010-1

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