This paper investigates the number of limit cycles in a predator-prey system with group defense, intially introduced by Wolkowicz and later examined by Rothe and Shafer in the 1980's. Under the assumption of large prey growth, the system reduces to a perturbed singular system, whose limit cycles can be analyzed using geometric singular perturbation methods—primarily through

the study of a slow-divergence integral. Our work completes partially the results previously obtained by Li and Zhu and by Hsu. We provide a comprehensive classification of all possible singular cycles capable of generating limit cycles and analyze the slow-divergence integral for the nine distinct types of cycle families that arise in a canard explosion. Based on these findings, we demonstrate that the maximum number of limit cycles emerging from the singular cycles is two in all cases, thereby confirming conjectures posed by Rothe-Shafer and Xiao-Ruan.

The Pfaffian property of graphs is of fundamental importance in graph theory, as it precisely characterizes those graphs for which the number of perfect matchings can be computed in polynomial time with respect to the number of edges. The study of Pfaffian graphs originated from the enumeration of perfect matching in planar graphs. References \cite{4,5,7} demonstrated that every planar graph is Pfaffian. Therefore, the Pfaffian property and planarity of graphs play a vital role in modern matching theory.

This paper contributes a complete characterization of the Pfaffian property and planarity of connected Cayley graphs over the dicyclic group $T_{4n}$ of order $4n$ $(n\geq 3)$, shows that the Cayley graph $Cay(T_{4n}, S)$ is Pfaffian if and only if $n$ is odd and $S=\{a^{k_1},a^{2n-k_1},ba^{k_2},ba^{n+k_2}\}$, where $1\leq k_1\leq n-1$, $0\leq k_2\leq n-1$ and $(k_1,n)=1$, and furthermore, shows that $Cay(T_{4n}, S)$ is never planar.

In this paper, we consider the following logarithmic Schrödinger equation:

\begin{equation*}

-\Delta u+\omega u =u\log|u|^2,~~ u\in H^1(\mathbb{R}^N),

\end{equation*}

where $N\geq3$, and $\omega>0$ is a constant. With an auxiliary equation, we obtain the existence of normalized solutions by using the constrained variational method.