A Holling-Tanner predator-prey model with strong Allee effect

TL;DR

This paper analyzes a modified Holling-Tanner predator-prey model incorporating a strong Allee effect, revealing complex dynamics including bifurcations, separatrices, and coexistence-extinction basins.

Contribution

It extends previous models by including a strong Allee effect, proving the existence of separatrices, homoclinic curves, and various bifurcations in the predator-prey dynamics.

Findings

Existence of separatrices dividing basins of attraction.

Presence of homoclinic curves leading to limit cycles.

Identification of saddle-node, Hopf, and Bogdanov-Takens bifurcations.

Abstract

We analyse a modified Holling-Tanner predator-prey model where the predation functional response is of Holling type II and we incorporate a strong Allee effect associated with the prey species production. The analysis complements results of previous articles by Saez and Gonzalez-Olivares (SIAM J. Appl. Math. 59 1867-1878, 1999) and Arancibia-Ibarra and Gonzalez-Olivares (Proc. CMMSE 2015 130-141, 2015)discussing Holling-Tanner models which incorporate a weak Allee effect. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to co-existence and extinction of the species. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle-node, Hopf, and Bogadonov-Takens bifurcations.