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.