Loose Coupling of Predator-Prey Cycles: Entrainment, Chaos, and Intermittency in the Classic Macarthur Consumer-Resource Equations

The consumer-resource equations have classically been formulated such that damping is inevitable, an unfortunate restriction since actual systems are known to exhibit permanent oscillations. Modern formulations of a single consumer-resource pair have included conditions that permit limit cycles, but rarely in the connected form originally proposed by MacArthur for the basic consumer-resource system. The purpose of this study was to explore the qualitative behavior of such a connected system. At low levels of connection between the consumer and its nonpreferred resource, the two subsystems rapidly come into phase with one another, thus exhibiting the well-known phenomenon of entrainment. As the level of connectedness increases, the sequence of period doubling is observed that leads to a chaotic attractor. Finally, when the symmetry assumption is relaxed, the system exhibits intermittent chaos, with populations occasionally flushing between long periods of rarity. These general patterns appear to be insensitive to the particular consumer-resource model, depending only on the assumption that both consumer-resource subsystems exhibit limit-cycle-type behavior.