Mathematical analysis of trophic interactions: From bacteria competition to lemming cycles
Mechanistic and phenomenological models and careful parameter estimations are presented through both aquatic and terrestrial ecosystems. The stoichiometric modeling of the bacteria-algae lake system is relatively new, while the lemming population cycle has attracted the attention of several generations of theoretical and experimental biologists and continues to be an issue of controversy. Bacteria-algae interaction in epilimnion is modeled with explicit consideration of carbon (energy) and phosphorus (nutrient). Global qualitative analysis and bifurcation diagrams of this model are presented. Competition of bacterial strains are modeled to examine Nishimura's hypothesis that in severely P-limited environments, such as Lake Biwa, P limitation exerts more severe constraints on the growth of bacterial groups with higher nucleic acid contents, which allows low nucleic acid bacteria to be competitive. Through a series of carefully derived models of the well documented high-amplitude, large-period fluctuations of lemming populations at Point Barrow, one can argue that, when appropriately formulated, autonomous differential equations may capture much of the desirable rich dynamics such as the existence of a periodic solution with period and amplitude close to that of approximately periodic solutions produced by the more natural but mathematically daunting nonautonomous models. This, together with the bifurcation analysis, indicates that neither seasonal factors, nor the moss growth rate and lemming death rate, are the main determinants of the observed multi-year lemming cycles. What ecological factors control population cycles? For some species - collared lemmings and snowshoe hares in particular-maturation delay of predators and the functional response of predation appear to be the primary determinants. Maturation delay almost completely determines the cycle period, whereas the functional response greatly affects its amplitude and even its existence. This result is obtained from sensitivity analysis of all the parameters and comparison of the lemming-stoat and hare-lynx systems.