Consequences of double Allee effect on the number of limit cycles in a predatorprey model

Eduardo González-Olivares, Betsabé González-Yaez, JAIME JUAN FERNANDO MENA LORCA, Alejandro Rojas-Palma, José D. Flores

Research output: Contribution to journalArticlepeer-review

28 Scopus citations

Abstract

The main goal of this work is to show a comparative analysis of simple continuous time predatorprey models considering the Allee effect affecting the prey population, also known as depensation in fisheries sciences. This phenomenon may be expressed by different mathematical forms, yielding a distinct number of limit cycles surrounding a positive equilibrium point, when two of these different formalizations are considered in the same system. It is known that the Volterra predation model, using the most usual form to express the Allee effect, has a unique limit cycle. In this work, considering a more complex mathematical expression, the existence of two limit cycles is proved, by means of the Lyapunov quantities. We argue that the second equation explains the existence of two Allee effects affecting the same population, which could justify the difference observed between the models. These results imply that the election of mathematical formulation can have consequences on the fit of the observed data, thus leading to mistakes for ecologists. We conclude that the oscillatory behaviors and overall dynamics depend strongly on the algebraic expression of the Allee effect, making difficult the proposition of general results. Nevertheless, the techniques reviewed in this paper emerge as key tools to analyze the existence of limit cycles in the presence of multiple Allee effects.

Original languageEnglish
Pages (from-to)3449-3463
Number of pages15
JournalComputers and Mathematics with Applications
Volume62
Issue number9
DOIs
StatePublished - Nov 2011
Externally publishedYes

Keywords

  • Allee effect
  • Bifurcation
  • Limit cycles
  • Predatorprey model
  • Separatrix curve
  • Stability

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