[Paper Review] The Basins of Attraction in a Modified May-Holling-Tanner Predator-Prey Model with Allee Effect

Claudio Arancibia–Ibarra|arXiv (Cornell University)|Dec 29, 2018

Mathematical and Theoretical Epidemiology and Ecology Models25 references4 citations

TL;DR

This paper analyzes a modified May–Holling–Tanner predator-prey model with a linear functional response, Allee effect in prey, and alternative food for predators. Using dynamical systems theory, it identifies separatrices and basins of attraction that separate coexistence, oscillation, and extinction, and proves the existence of saddle-node, Hopf, and Bogdanov–Takens bifurcations, showing that strong Allee effects reduce coexistence regions and increase sensitivity to parameter changes.

ABSTRACT

We analyse a modified May-Holling-Tanner predator-prey model considering an Allee effect in the prey and alternative food sources for predator. Additionally, the predation functional response or predation consumption rate is linear. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation, co-existence and extinction of the predator-prey population. 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. We use simulations to illustrate the behaviour of the model.

Motivation & Objective

• To study the impact of a strong or weak Allee effect on predator-prey dynamics in a modified Holling–Tanner model.
• To analyze how alternative food sources for predators affect system stability and coexistence.
• To investigate the role of parameter variation in shaping basins of attraction and determining long-term population outcomes.
• To identify and characterize bifurcations such as saddle-node, Hopf, and Bogdanov–Takens in the system.
• To determine how the Allee threshold and predator growth rate influence the size and structure of coexistence basins.

Proposed method

• A topologically equivalent system is derived via a diffeomorphism to simplify analysis of the original model with Allee effect and alternative food.
• Equilibrium points are analyzed using Jacobian matrix traces to determine stability, with special attention to $P_3$, $P_4$, and $P_2$.
• Separatrices are identified as stable manifolds of saddle equilibria, particularly $P_1$, which partition basins of attraction.
• Numerical simulations are used to visualize basins of attraction and illustrate dynamic transitions under varying parameters.
• Bifurcation diagrams and parameter sweeps are employed to locate saddle-node, Hopf, and Bogdanov–Takens bifurcations.
• The model is analyzed under varying $S = s/(Kr)$, $M$, $Q$, and $C$ to assess sensitivity and coexistence regions.

Experimental results

Research questions

• RQ1 How does the inclusion of a strong Allee effect alter the basins of attraction in the modified May–Holling–Tanner model?
• RQ2 What role does the stable manifold of $P_1$ play in separating long-term outcomes such as coexistence and extinction?
• RQ3 Under what parameter conditions does the system undergo a saddle-node bifurcation, and how does this affect equilibrium structure?
• RQ4 How do changes in the predator’s intrinsic growth rate $s$ or prey carrying capacity $K$ affect the size of coexistence basins?
• RQ5 What is the impact of the Allee threshold $m$ on the stability and persistence of predator-prey coexistence?

Key findings

• The equilibrium point $(0, C)$ is globally attracting when $S < S_k$, corresponding to extinction of the prey and persistence of the predator via alternative food.
• For $S_k < S < S_k^*$, the equilibrium $P_3$ is surrounded by a stable limit cycle, indicating sustained population oscillations.
• When $S > S_k^*$, the equilibrium $P_3$ becomes stable, indicating coexistence without oscillations.
• The Allee effect reduces the basin of attraction for positive equilibria, particularly under strong Allee conditions, increasing the risk of extinction.
• The system exhibits a saddle-node bifurcation when the discriminant $\Delta = 0$, leading to the creation or annihilation of equilibrium points.
• A Bogdanov–Takens bifurcation occurs at a cusp point when $S = f(E)$, indicating complex dynamic transitions involving multiple bifurcations.