Hamiltonian limits and subharmonic resonance in models of population fluctuations

Abstract

It is shown that the dynamics of models of predator-prey interactions in the presence of seasonality are profoundly structured by Hamiltonian limits, i.e., limiting cases where the flow satisfies Hamilton's canonical equations of motion. We discuss the dynamics at nonintegrable Hamiltonian limits, focusing on the existence of subharmonic periodic orbits, which correspond to multi-annual fluctuations. Perturbing away from a Hamiltonian limit, subharmonic periodic orbits are annihilated in tangent bifurcations, which compose the boundaries of resonance horns. All resonance horns emanate from the Hamiltonian limit and penetrate well into the realm of biologically-realistic parameter values. There, they indicate the "color" of the dynamics, i.e., the spectrum of dominant frequencies, whether the dynamics be regular or chaotic. Our observations provide both an account of the phase coherence often observed in population dynamics and a method for investigating more complex models of predator-prey dynamics, which may involve multiple Hamiltonian limits. This method is applied to the celebrated problem of the cyclic fluctuations of boreal hare populations. We present a model of the population dynamics of the boreal forest community based on known demographic mechanisms and parameterized entirely by measurements reported in the literature. The aforementioned method reveals the geometry potentially underlying the observed fluctuations. The model is quantitatively consistent with observed fluctuations. We derive specific, testable predictions of the model relating to the roles of herbivore functional response, browse abundance and regeneration, starvation mortality, and composition of the predator complex.