Uniqueness of limit cycles and multiple attractors in a gause-type predator-prey model with nonmonotonic functional response and allee effect on prey

Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

The main purpose of this work is to analyze a Gause type predatorprey model in which two ecological phenomena are considered: the Allee effect affecting the prey growth function and the formation of group defence by prey in order to avoid the predation. We prove the existence of a separatrix curves in the phase plane, determined by the stable manifold of the equilibrium point associated to the Allee effect, implying that the solutions are highly sensitive to the initial conditions. Trajectories starting at one side of this separatrix curve have the equilibrium point (0; 0) as their ω-limit, while trajectories starting at the other side will approach to one of the following three attractors: a stable limit cycle, a stable coexistence point or the stable equilibrium point (K; 0) in which the predators disappear and prey attains their carrying capacity. We obtain conditions on the parameter values for the existence of one or two positive hyperbolic equilibrium points and the existence of a limit cycle surrounding one of them. Both ecological processes under study, namely the nonmonotonic functional response and the Allee effect on prey, exert a strong inuence on the system dynamics, resulting in multiple domains of attraction. Using Liapunov quantities we demonstrate the uniqueness of limit cycle, which constitutes one of the main differences with the model where the Allee effect is not considered. Computer simulations are also given in support of the conclusions.

Original languageEnglish
Pages (from-to)345-367
Number of pages23
JournalMathematical Biosciences and Engineering
Volume10
Issue number2
DOIs
StatePublished - Apr 2013
Externally publishedYes

Keywords

  • Allee effect
  • Bifurcation
  • Limit cycle
  • Non-monotonic functional response
  • Predator-prey model
  • Stability

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