Slow-fast dynamics of Hopfield spruce-budworm model with memory effects

In this paper we consider a kind of the spruce-budworm system with memory effects. On the basis of geometric singular perturbation theory, the transition of the solution trajectory is illustrated, and the existence of the relaxation oscillation with a rapid movement process alternating with a slow movement process is proved. The characteristic of the relaxation oscillation, it is indicated, is dependent on the structure of the slow manifold. Moreover, the approximate expression of the relaxation oscillation and its period are obtained analytically. Finally we present two simulations to demonstrate the validity of the analytical conclusion.