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Dynamics of a pollinator-plant-herbivore mathematical model
  • Víctor Castellanos,
  • Miguel De la Rosa Castillo,
  • Faustino Sánchez-Garduño
Víctor Castellanos
Universidad Juárez Autónoma de Tabasco

Corresponding Author:[email protected]

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Miguel De la Rosa Castillo
Universidad Juarez Autonoma de Tabasco
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Faustino Sánchez-Garduño
Universidad Nacional Autónoma de México
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Abstract

In this paper we carried out the analysis of a three-dimensional ODE nonlinear autonomous system which is derived with the aim of describing the interaction between three populations. These take the form of two mutualistic (pollinators and plants) and a third population (herbivores) is introduced. This one is feeded by consuming plants which, in turn, damages the pollinators population too by reducing the rate of visits (to plants) behalf the pollinators. The specific type of interactions between the populations are described by two types of functional responses of type IV. One of these measures what we call “saciety rate” of consuming plants behalf the herbivores. The main result contained in this paper is the proof of the existence of an attracting limit cycle for the ODE system. This emerges from a supercritical Hopf bifurcation. Its existence is proved by using the Hopf-Andronov bifurcation theorem and its stability is proved by using the first Lyapunov coefficient. In addition of the analysis, a series of numerical simulations are carried out on the full ODE system. These show how the stability feature of an equilibrium point changes: from asymptotic locally stable to unstable hence the emergence of a stable limit cycle within the positive octant of the phase space.