Multiple stability and uniqueness of the limit cycle in a Gause-type predatorprey model considering the Allee effect on prey

Eduardo González-Olivares, Héctor Meneses-Alcay, Betsabé González-Yaez, JAIME JUAN FERNANDO MENA LORCA, Alejandro Rojas-Palma, Rodrigo Ramos-Jiliberto

Research output: Contribution to journalArticlepeer-review

39 Scopus citations

Abstract

In this work, a bidimensional differential equation system obtained by modifying the well-known predatorprey RosenzweigMacArthur model is analyzed by considering prey growth influenced by the Allee effect. One of the main consequences of this modification is a separatrix curve that appears in the phase plane, dividing the behavior of the trajectories. The results show that the equilibrium in the origin is an attractor for any set of parameters. The unique positive equilibrium, when it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose uniqueness is established by calculating the Lyapunov quantities. Therefore, both populations could either reach deterministic extinction or long-term deterministic coexistence. The existence of a heteroclinic curve is also proved. When this curve is broken by changing parameter values, then the origin turns out to be an attractor for all orbits in the phase plane. This implies that there are plausible conditions where both populations can go to extinction. We conclude that strong and weak Allee effects on prey population exert similar influences on the predatorprey model, thereby increasing the risk of ecological extinction.

Original languageEnglish
Pages (from-to)2931-2942
Number of pages12
JournalNonlinear Analysis: Real World Applications
Volume12
Issue number6
DOIs
StatePublished - Dec 2011
Externally publishedYes

Keywords

  • Allee effect
  • Bifurcations
  • Ecological extinction
  • Heteroclinic curve
  • Limit cycle
  • Predatorprey model

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