A Holling–Tanner Predator–Prey Model with Strong Allee Effect

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][1868][1869][1870][1871][1872][1873][1874][1875][1876][1877][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.

A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

“…This manuscript also extends the properties of the weasel and field-vole interaction studied in [21] by showing the impact of these modifications in the original Holling-Tanner model. In addition, the analysis of the modified models (systems (4), (6) and (7)) shows a strong connection with the findings presented in [24,43]. Study results suggest that the system does not lose the coexistence and oscillation of the population when a constant alternative food source for weasels is included.…”

“…In particular, the Hopf curve of the model with strong Allee effect shows that system (4) can have up to two positive equilibrium points which can collapse at q =q 1 (respectively for m < 0 at q =q 2 ). Therefore, for q <q 1 system (4) presents two positive equilibrium points and thus the coexistence of both populations is observed [6]. In contrast, the Hopf curve of the model with week Allee effect (m < 0) shows that if q >q 2 then system (4) with m < 0 has always one positive equilibrium point which is stable or unstable surrounded by a stable limit cycle and thus the coexistence or oscillation of both population is observed.…”

“…In summary, the bifurcation diagrams of the models studied in [6,7,38] are often qualitatively similar but their solutions behave quantitatively differently. In other words, it is observed that there are two groups of modes which support equivalent ecological behaviour due to the addition of the modifications in the May-Holling-Tanner model.…”

“…The model with alternative food for predators corresponds to system (6). This model was studied in [5,9,18] where the authors presented the stability of the equilibrium points and conditions for periodic solutions.…”

“…Factoring out (u − G) from the equation resultant (14) of the intersection of the nullcline of system (10) [6], leaves as with second order polynomial equation…”