A safe harbor can protect an endangered species from its predators

  • Malay BanerjeeEmail author
  • Bob W. Kooi
  • Ezio Venturino


The objective is the study of the dynamics of a prey–predator model where the prey species can migrate between two patches. The specialist predator is confined to the first patch, where it consumes the prey following the simple law of mass action. The prey is further “endangered” in that it suffers from the strong Allee effect, assumed to occur due to the lowering of successful matings. In the second patch the prey grows logistically. The model is formulated in a comprehensive way so as to include specialist as well as generalist predators, as a continuum of possible behaviors. This model described by a set of three ordinary differential equation is an extension of some previous models proposed and analysed in the literature on metapopulation models. The following analysis issues will be addressed: boundedness of the solution, equilibrium feasibility and stability, and dynamic behaviour dependency of the population and environmental parameters. Three types for both equilibria and limit cycles are possible: trivial, predator-free and coexistence. Classical analysis techniques are used and also theoretical and numerical bifurcation analysis. Besides the well-known local bifurcations, also a homoclinic connection as a global bifurcation is calculated. In view of the difficulty in the analysis, only the specialist case will be analysed. The obtained results indicate that the safe harbor can protect the endangered species under certain parametric restrictions.


Metapopulations Specialist predator Allee effect Local bifurcation Global bifurcation 

Mathematics Subject Classification

34C23 35B32 92D25 



EV has been partially supported by the project “Metodi numerici e computazionali per le scienze applicate” of the Dipartimento di Matematica “Giuseppe Peano”. Part of this work was carried out during a stay of BK and EV at the Banff International Research Station, whose the support is kindly acknowledged, in the framework of the program “New Mathematical Methods for Complex Systems in Ecology”.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Università degli Studi di Napoli "Federico II" 2020

Authors and Affiliations

  1. 1.Indian Institute of Technology KanpurKanpurIndia
  2. 2.Faculty of ScienceVU UniversityAmsterdamThe Netherlands
  3. 3.Dipartimento di Matematica “Giuseppe Peano”Università di TorinoTorinoItaly

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