Parametric resonance, chaos and spatial structure in the Lotka-Volterra model

Abstract: We investigate the Lotka-Volterra model for predator-prey competition with a finite carrying capacity that varies periodically in time, modeling seasonal variations in nutrients or food resources. In the absence of time variability, the ordinary differential equations have an equilibrium point that represents coexisting predators and prey. The time dependence removes this equilibrium solution, but the equilibrium point is restored by allowing the predation rate also to vary in time. This equilibrium can undergo a parametric resonance instability, leading to subharmonic and harmonic time-periodic behavior, which persists even when the predation rate is constant. We also find period-doubling bifurcations and chaotic dynamics. If we allow the population densities to vary in space as well as time, introducing diffusion into the model, we find that variations in space diffuse away when the underlying dynamics is periodic in time, but spatiotemporal structure persists when the underlying dynamics is chaotic. We interpret this as a competition between diffusion, which makes the population densities homogeneous in space, and chaos, where sensitive dependence on initial conditions leads to different locations in space following different trajectories in time. Patterns and spatial structure are known to enhance resilience in ecosystems, suggesting that chaotic time-dependent dynamics arising from seasonal variations in carrying capacity and leading to spatial structure, might also enhance resilience.