CONSEQUENCES OF POPULATION-MODELS FOR THE DYNAMICS OF FOOD-CHAINS

A class of bioenergetic ecological models is studied for the dynamics of food chains with a nutrient at the base. A constant influx rate of the nutrient and a constant efflux rate for all trophic levels is assu med. Starting point is a simple model where prey is converted into pre dator with a fixed efficiency. This model is extended by the introduct ion of maintenance and energy reserves at all trophic levels, with two state variables for each trophic level, biomass and reserve energy. T hen the dynamics of each population are described by two ordinary diff erential equations. For all models the bifurcation diagram for the bi- trophic food chain is simple. There are three important regions; a reg ion where the predator goes to extinction, a region where there is a s table equilibrium and a region where a stable limit cycle exists. Bifu rcation diagrams for tri-trophic food chains are more complicated. Fli p bifurcation curves mark regions where complex dynamic behaviour (hig her periodic limit cycles as well as chaotic attractors) can occur. We show numerically that Shil'nikov homoclinic orbits to saddle-focus eq uilibria exists. The codimension 1 continuations of these orbits form a 'skeleton' for a cascade of flip and tangent bifurcations. The bifur cation analysis facilitates the study of the consequences of the popul ation model for the dynamic behaviour of a food chain. Although the pr edicted transient dynamics of a food chain may depend sensitively on t he underlying model for the populations, the global picture of the bif urcation diagram for the different models is about the same. (C) 1998 Elsevier Science Inc. All rights reserved.