LOW-FREQUENCY AND HIGH-FREQUENCY OSCILLATIONS IN 3-DIMENSIONAL FOOD-CHAIN SYSTEMS

The dynamic behavior of a three-trophic-level food chain (prey-predator-superpredator) is analyzed in this paper. Prey is logistic, while predator and superpredator have functional response of the Holling type. Moreover, the trophic levels are assumed to be characterized by increasing and quite diversified time responses. Using a singular perturbation approach, explicit conditions for the persistence of the three populations are derived, and the structure of the corresponding attractors is noted, as well as the nature of transients. The analysis shows that the system can have low-frequency cycles due to the interactions between predator and superpredator. Nevertheless, the interactions between prey and predator can give rise to high-frequency oscillations, which can arise during the transients toward the attractor or during the low-frequency cycle. This periodic burst of high-frequency oscillations develops, in particular, when the two predators are fairly efficient. These results note in a very concise way some of the possible consequences of the interactions between very fast and very slow components of a dynamical system.