Trends and oscillations in tick population dynamics

The life cycle of ticks is complex, proceeding through four developmental stages: egg, larva, nymph and adult. In temperate zones it is punctuated by seasonality, as in the case of the deer tick, which is the vector of the agent of Lyme disease in the northeastern parts of the United States. A model of 12 monthly stage-structured matrices were constructed and their product was used to follow the annual changes in the tick population. A judicious choice of the cut-off date for the yearly cycle enabled us to gain insight into the algebraic structure of the 11 x 11 annual transition matrix, and discover that it possesses only three non-zero eigenvalues: a dominant one that determines the main trend and two minor ones that add a riding oscillatory behavior, demonstrating the inherent nature of population swings even under constant external conditions. In addition, we find oscillations due to fluctuating environments that are strongly affected by the magnitude of the eigenvalues. Thus, both inherent and environmental oscillations determine patterns of tick population growth. In a non-fluctuating environment, tick population structure ultimately converges to a stable stage distribution, regardless of whether the population thrives or declines.