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A Leslie–Gower type predator-prey model considering herd behavior

Abstract

In 1959 Crawford S. Holling formulated a classification to model the action of the predators over their prey, doing empirical works. In this taxonomy, he introduced only three types of functional responses dependent only on the prey population, which are described by saturated functions. Later, various other types have been proposed, including the functional responses dependent on both populations. This work concerns the study of the Leslie–Gower type predator-prey model, incorporating the Rosenzweig functional response described by a power law. The elected function does not conform to the types proposed by Holling since it is unbounded, being, besides, non-differentiable for \(x = 0\); nonetheless, the obtained system is Lipschitzian. Moreover, the existence of a separatrix curve \(\Sigma \) in the phase plane is proven, which is divided into two complementary sectors. According to the position of the initial conditions with respect to the curve, the trajectories can have different \(\omega \)-limit sets, which can be the equilibrium \(\left( 0,0\right) \), or a positive equilibrium, or a heteroclinic curve, or a stable limit cycle. These properties show the great difference of this model with the original Leslie–Gower model, in which a unique positive equilibrium exists, which is globally asymptotically stable, when it exists. Then, the analyzed system has a richer dynamic than the original system in which a linear functional response is considered, also unbounded. Numerical simulations and bifurcation diagrams are given to endorse our analytical results.

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Acknowledgements

The authors would like to thank professor professor Claudio Arancibia-Ibarra, of the Universidad de las Américas, Viña del Mar, Chile, for the elaboration of the Diagram of Bifurcations. The fourth author (KVP) was partially financed by Conicyt PAI/Academia 79150021.

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Correspondence to Eduardo González-Olivares.

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González-Olivares, E., Rivera-Estay, V., Rojas-Palma, A. et al. A Leslie–Gower type predator-prey model considering herd behavior. Ricerche mat (2022). https://doi.org/10.1007/s11587-022-00694-5

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  • DOI: https://doi.org/10.1007/s11587-022-00694-5

Keywords

  • Bifurcation
  • Limit cycle
  • Separatrix curve
  • Predator-prey model
  • Functional response

Mathematics Subject Classification

  • 92D25
  • 34C23
  • 58F14
  • 58F21