A geometric analysis of the Bazykin-Berezovskaya predator-prey model with Allee effect in an economic framework

TL;DR

This work analyzes a fast–slow Bazykin–Berezovskaya predator–prey model with Allee effect under an economic interpretation, using Geometric Singular Perturbation Theory to derive explicit thresholds that classify long-term behaviors. The authors construct fast and slow subsystems, establish a unified framework with exit–entry maps, and identify regimes leading to a stable equilibrium or a sustained production cycle, including a singular Hopf bifurcation. They corroborate the theory with numerical simulations, showing how the asymptotic predictions hold even for moderate values of the small parameter $\varepsilon$ and detailing how the limit $\varepsilon\to0$ can alter topological outcomes. The results provide insights into maximizing long-term production while avoiding resource depletion and offer a foundation for extensions to multiple resources or production variables.

Abstract

We study a fast-slow version of the Bazykin-Berezovskaya predator-prey model with Allee effect evolving on two timescales, through the lenses of Geometric Singular Perturbation Theory (GSPT). The system we consider is in non-standard form. We completely characterize its dynamics, providing explicit threshold quantities to distinguish between a rich variety of possible asymptotic behaviors. Moreover, we propose numerical results to illustrate our findings. Lastly, we comment on the real-world interpretation of these results, in an economic framework and in the context of predator-prey models.

Figures (6)

• Figure 1: Visualization of the dynamics as $m$ varies. Slow parts of the orbits are depicted in blue and with a single arrow along them, fast parts in red and with a double arrow. (a) $m<l$, the dynamic is attracted to $\mathbf{x_1}$, either immediately or after a slow permanence close to the $u$-axis, followed by a fast excursion away from it; (b) $l<m<\tilde{m}(l,\gamma,\varepsilon)$, same as (a), but the unstable equilibrium $\mathbf{x_4}$ appears in the economically relevant region \ref{['eq3']}; (c) $m=\tilde{m}(l,\gamma.\varepsilon)$, the system exhibits a heteroclinic cycle connecting $\mathbf{x_2}$ and $\mathbf{x_3}$ in the slow flow and $\mathbf{x_3}$ and $\mathbf{x_2}$ in the fast flow (this latter heteroclinic orbit is approximated by the curve $v=\alpha(u)$, recall \ref{['future']}); orbits starting inside this cycle are attracted to it, orbits starting outside are attracted to $\mathbf{x_1}$; (d) $\tilde{m}(l,\gamma,\varepsilon)<m<(l+1)/2$, orbits starting below the curve $v=\alpha(u)$ are attracted to the unique stable limit cycle (purple) around the unstable equilibrium $\mathbf{x_1}$, orbits starting above such curve are attracted to $\mathbf{x_1}$ (actually, we will show that there exists a value $\Bar{m}(l,\gamma,\varepsilon)$, such that $\Bar{m}(l,\gamma,\varepsilon) \to (l+1)/2$ as $\varepsilon \to 0$, which distinguishes between when the stable cycle enters the slow flow and when it does not, hence this case corresponds more precisely to $\tilde{m}<m<\Bar{m} \approx (l+1)/2$); (e) $(l+1)/2\leq m<1$, orbits starting below the curve $v=\alpha(u)$ are attracted to $\mathbf{x_4}$ (on which the limit cycle collapsed as $m\to {\frac{l+1}{2}}^-$), orbits starting above such curve are attracted to $\mathbf{x_1}$; (f) $m>1$, the dynamic is attracted either to $\mathbf{x_1}$ or to $\mathbf{x_3}$ (the curve $v=\alpha(u)$ again approximates the border between the two basins of attraction).
• Figure 2: Visualization of the entry–exit map on the line $\{x=x_0\}$.
• Figure 3: Blue dots represent initial conditions whose corresponding orbits tend to $\mathbf{x_1}$ as $t\to +\infty$, red dots represent initial conditions whose corresponding orbits tend to the stable limit cycle contained between the $u$-axis and the curve $v=\alpha(u)$ (black curve). Values of the parameters: $l=0.4$, $m=0.67$, $\gamma=1$, and (a) $\varepsilon=0.02$ or (b) $\varepsilon=0.01$. Notice that, as $\varepsilon$ decreases, the curve $\alpha$ approximates better the border between the corresponding basins of attraction.
• Figure 4: Plot of the maximum and minimum limit cycle values of $u$ and $w$ of system \ref{['eq23']} as $m$ varies. Values of the parameters: $l=0.4$, $\gamma=1$, and $\varepsilon=0.02$. For $m\in [0.7,0.71]$ the limit cycle does not exist, and the equilibrium $(\Bar{u}, \log \Bar{v})$, which corresponds to $\mathbf{x_4}$, is locally asymptotically stable (solid black line). For $m\in (\tilde{m}(l,\gamma,\varepsilon),0.7)$ the system exhibits a locally stable limit cycle around such equilibrium point (now unstable; dashed black line), which collapses on a heteroclinic cycle between $\mathbf{x_2}$ and $\mathbf{x_3}$ for $m\approx 0.6556$.
• Figure 5: Values of the parameters: $l=0.4$ and $\gamma=1$. (a) Fixed point $u_\infty$ of the map $\Pi_1 \circ \Pi_2$ in the limit $\varepsilon \to 0$ as $m$ varies between $(l-1)/\log l \approx 0.6548$ and $(l+1)/2=0.7$. (b) $|u_*-u_\infty|$, where $u_*$ is derived according to Section \ref{['section4_3']}, for different values of $m$ and as $\varepsilon$ decreases. Recall that $u_*$ is at worst an $\mathcal{O}(\varepsilon |\log \varepsilon|)$-approximation (dashed black line) of $u_\infty^\varepsilon$.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Theorem 1: Fenichel
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 1