Bifurcation analysis of an ecosystem model for spruce budworm outbreaks

Abstract This study develops and analyzes a mathematical model for outbreaks of the spruce budworm population. The system possesses a single boundary equilibrium and admits at most two positive equilibria. Under certain constraints, an interior equilibrium becomes a codimension-two Bogdanov–Takens singularity. By treating δ and h as bifurcation parameters, we obtain the curves for saddle-node, subcritical Hopf, and homoclinic bifurcations via reduction on the center manifold. These curves partition the parameter plane into four regions, corresponding to the absence of a positive equilibrium, coexistence of a focus and a saddle, a stable limit cycle, and a homoclinic loop, respectively. The first Lyapunov coefficient determines the transition between subcritical and supercritical Hopf bifurcations. Our results capture key dynamical transitions from low-density steady states to periodic outbreaks and sudden collapse, offering theoretical insights for pest management.