Further stability criteria for singular Hopf bifurcation in planar predator–prey systems

In this paper, we investigate the singular Hopf bifurcation in predator-prey systems, where bifurcation occurs as the eigenvalues become singular when the singular perturbation parameter ε→0. In Krupa and Szmolyan [SIAM J. Math. Anal. (2001)], the first Lyapunov coefficient for singular Hopf bifurcation is given as L1(ε)=ε8(A+O(ε)), with the bifurcation being supercritical for A<0 and subcritical for A>0. As far as we know, there are no general results regarding the stability of singular Hopf bifurcation when A=0. This paper aims to address this gap for planar predator-prey systems. We present further stability criteria for singular Hopf bifurcation in planar predator-prey systems of Leslie and Gause types. Additionally, numerical simulations are conducted to support and validate our analytical findings.