Singular bifurcations in a modified Leslie-Gower model

TL;DR

This work analyzes a slow–fast predator–prey model with a Holling type II response and a weak Allee effect, focusing on a degenerate transcritical organizing singularity. Using blow-up desingularization and an intrinsic Hopf criticality criterion, the authors characterize singular Hopf behavior and establish the existence of transitory canards and relaxation oscillations that pass through the degenerate point, with a supercritical Hopf demonstrated in an open parameter region. They further identify a nearby Takens–Bogdanov point numerically, and construct contraction-based transition maps to describe the global flow near the degeneracy, validating the theory with MatCont bifurcation analysis. The results advance understanding of how degenerate slow–fast structures organize complex oscillatory dynamics in ecological models and illuminate pathways for further study of degenerate bifurcations such as TB points and cyclicity in coupled predator–prey systems.

Abstract

We study a predator-prey system with a generalist Leslie-Gower predator, a functional Holling type II response, and a weak Allee effect on the prey. The prey's population often grows much faster than its predator, allowing us to introduce a small time scale parameter $\varepsilon$ that relates the growth rates of both species, giving rise to a slow-fast system. Zhu and Liu (2022) show that, in the case of the weak Allee effect, Hopf singular bifurcation, slow-fast canard cycles, relaxation oscillations, etc., exist. Our main contribution lies in the rigorous analysis of a degenerate scenario organized by a (degenerate) transcritical bifurcation. The key tool employed is the blow-up method that desingularizes the degenerate singularity. In addition, we determine the criticality of the singular Hopf bifurcation using recent intrinsic techniques that do not require a local normal form. The theoretical analysis is complemented by a numerical bifurcation analysis, in which we numerically identify and analytically confirm the existence of a nearby Takens-Bogdanov point.

Figures (14)

• Figure 4: Dynamics of the slow-fast system \ref{['slowfast1']} for (a) $C< -AMQ$ and (b) $C= -AMQ$, where $\mathcal{M}_0^0$ and $\mathcal{M}_0^1$ are the critical submanifolds. The fold point $P=(u_p,v_p)$ and $T_\mathcal{C}=(0,-AM)$ are shown, as well as a schematic representation of the slow flow in the different components of the critical manifold. The green line $\ell (u)$ denotes the nullcline of variable the $v$. Hence, we are depicting the case in which an equilibrium of the slow flow coincides with the fold point, which may lead to canards, and corresponds to the parameter $Q=Q_H$. Indeed, in the figures, the magenta curve is a singular canard orbit.
• Figure 6: Left: schematic of the blow-up of a generic fold point. Right: schematic of the blow-up of a canard point. For these pictures we have used the local setup of the model under study. Hence, the right branch of the critical manifold is attracting, while the left branch is repelling. In the left picture, the slow flow (magenta) is directed toward the fold point on both branches of the critical manifold. On the right picture, the slow flow "passes through" the fold point, compare with Figure \ref{['dinamicasf']}.
• Figure 7: Singular cycles that we focus on. The solid red cycle in (a) represents a candidate orbit for a regular relaxation oscillation, which appear when $C<-AMQ$. Such a cycle has already been studied in Zhu2022, but here we include its analysis for completeness, see Section \ref{['sec:relaxationoscillations']}. The two on the bottom (b) and (c) are degenerate and, to the best of our knowledge, novel. At the singular level, they exist for $C=-AMQ$. As we show in Section \ref{['sec_desing']}, the one depicted in (b), called a transitory canard, is obtained by combining a maximal canard at $P$ and a (degenerate) saddle-like transition at $T_{\mathcal{C}}$. The one depicted in (c) combines a generic jump at $P$ and the same degenerate saddle-like transition at $T_{\mathcal{C}}$.
• Figure 8: We present slices of \ref{['eq:sigma']} for the indicated values of parameters. In this way, we verify the region of parameters for which the Hopf bifurcation is super- ($\sigma<0$) or sub- ($\sigma>0$) critical. The black curve indicates $\sigma=0$.
• Figure 9: Following the analysis presented in this section, we provide a sketch of the dynamics of \ref{['eq:K1e1']} near the origin. The limit dynamics are presented in green and blue (for $\varepsilon_1=0$ and $r_1=0$ respectively), while a sample orbit of \ref{['eq:K1e1']} is shown in red.

Theorems & Definitions (46)

• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• Theorem 2
• Remark 4