The Dynamics of the Spruce Budworm-Bird System on Time Scale

Abstract

We perform a geometric analysis focusing on relaxation oscillations and canard cycles within a singularly perturbed predator-prey system involving budworm and birds. The system undergoes a comprehensive stability analysis, leading to the identification of canard cycles in proximity to the Hopf bifurcation points. The study particularly highlights the transition from smaller Hopf-type cycles to larger relaxation cycles. And the expression of transition threshold $\mu_c(\sqrt{ε})$ of the spruce budworm-bird system is obtained innovatively. Furthermore, numerical simulations are carried out to validate the theoretical findings.