Bifurcation Analysis of a Predator-Prey Model with Allee Effect and Cooperative Hunting

TL;DR

The paper develops and analyzes a predator–prey model that incorporates prey Allee effects and predator cooperative hunting, augmenting the Rosenzweig–MacArthur framework with a nonlinear, density‑dependent functional response. It proves global existence and boundedness of solutions, characterizes interior and boundary equilibria, and provides comprehensive stability results. A detailed bifurcation analysis reveals transcritical, saddle‑node, Hopf, and heteroclinic phenomena, including scenarios where interior equilibria exchange stability with boundary states and where limit cycles arise. The work elucidates how Allee strength and cooperative hunting shape long‑term dynamics and global behavior, with implications for ecological management and potential extensions to stochastic perturbations.

Abstract

We propose a novel predator-prey model that integrate two ecologically significant mechanisms: the Allee effect in the prey population and cooperative hunting behavior among predators. Building upon the Rosenzweig-MacArthur framework, our model modifies the prey growth term to incorporate the Allee effect and introduces a nonlinear functional response reflecting predator cooperation. We establish the existence and boundedness of global solutions for the system and analyze the local and global stability of its equilibria. In addition, we perform a comprehensive bifurcation analysis, including transcritical, saddle-node, Hopf, and heteroclinic bifurcations, to explore how system dynamics change with key parameters. These results reveal rich and biologically relevant behaviors, such as multiple equilibria, transitions in stability, and the emergence of complex dynamical patterns.

Figures (10)

• Figure 1: Comparison of prey growth functions: strong Allee effect (blue), weak Allee effect (red), and logistic growth (yellow). The strong Allee effect results in negative growth in region $A_1$. For the same intrinsic growth rate $r_1$, the weak Allee effect consistently shows higher growth than the strong Allee effect. In regions $A_2$ and $A_3$, the weak Allee effect surpasses logistic growth, and in region $A_3$, the strong Allee effect also exceeds logistic growth.
• Figure 2: Graphs of $f^{(1)}(x,y)=0$ (blue curve, independent of $s$) and $g^{(2)}(x)$, together with their intersection points. Left panel (weak Allee effect): $r_1=0.45$, $k_1=3$, $k_0=-1$, $\lambda=0.2$, $A=0.8$, $b=0.5$, $h=0.7$, with $s = 0.3, 0.73, 0.78$. Right panel (strong Allee effect): $r_1=1.2$, $k_1=5$, $k_0=1$, $\lambda=1.2$, $A=0.2$, $b=0.5$, $h=0.9$, with $s = 0.38, 0.6, 0.8$. Here, $g_1^{(2)}, g_2^{(2)}, g_3^{(2)}$ denote the graphs of $g^{(2)}(x)$ corresponding to the three values of $s$ in each case. The intersections illustrate that the system admits 0, 1, or 2 interior equilibria, respectively.
• Figure 3: Transcritical bifurcation at the equilibrium point $E_1(k_1, 0)$ under the strong Allee effect. The curves represent the predator population ($P$) as a function of the prey population ($N$). Parameter values are set as follows: $r_1 = 1.5$, $k_1 = 3$, $k_0 = 1$, $A = 0.8$, $b = 0.5$, $h = 0.7$, and $s = 0.75$. The estimated critical value of $\lambda$ at the bifurcation point is $\lambda_{TB} =0.263$. The left and right panels correspond to $\lambda = 0.2$ and $\lambda = 0.3$, respectively.
• Figure 4: Nullclines of system \ref{['equ08']}: $f^{(1)}(x, y) = 0$ and $f^{(2)}(x, y) = 0$ (also expressed as $y = g^{(2)}(x)$). The left panel shows the curves for $\lambda = 0.2$, and the right panel for $\lambda = 0.23$. The remaining parameters are fixed at $r_1 = 1.5$, $k_1 = 3$, $k_0 = 1$, $A = 0.8$, $b = 0.5$, $h = 0.7$, and $s = 0.8$.
• Figure 5: Saddle-node bifurcation at an interior equilibrium under the strong Allee effect. The curves illustrate the predator population ($y$) versus the prey population ($x$). The left panel corresponds to $\lambda = 0.23$, showing no interior equilibria, while the right panel corresponds to $\lambda = 0.2$, demonstrating the emergence of two interior equilibria.

Theorems & Definitions (32)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 3.1
• proof