Dynamics of a system of three interacting populations with Allee effects and stocking

Abstract

A model of three interacting populations where two populations engage in competition and two populations are in predator–prey type interaction is proposed and analysed. One of the two competing populations is subject to Allee effects and is also a pest population. The other competing population is regarded as a control agent and is the host for the predator population. There is a constant level of the external control agents released into the interaction at each generation after parasitism. We provide asymptotic dynamics of the competition subsystem and prove that a Neimark–Sacker bifurcation occurs for the host–parasitoid subsystem when the unique interior steady state loses its stability. The three interacting populations are impossible to persist for all positive initial conditions. Sufficient conditions based on the initial population size of the population with Allee effects are derived for persistence of the three populations.

Keywords::

• Allee effects
• competitive exclusion
• Neimark–Sacker bifurcation
• persistence

AMS Subject Classification::

• 92D25
• 39A30

Acknowledgements

We thank one of the referees and the editor for helpful comments on the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.