The summary of the talk by Bob Kooi:
Food chains in the chemostat

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 assumed. Starting point is a simple model where prey is converted into predator with a fixed efficiency. This model is extended by the introduction of maintenance and energy reserves at all trophic levels, with two state variables for each trophic level, biomass and reserve energy. Then the dynamics of each population is described by two ordinary differential equations. For all models the bifurcation diagram for the bi-trophic food chain is simple. There are three important regions; a region where the predator goes to extinction, a region where there is a stable equilibrium and a region where a stable limit cycle exists. Bifurcation diagrams for tri-trophic food chains are more complicated. Flip bifurcation curves mark regions where complex dynamic behaviour (higher periodic limit cycles as well as chaotic attractors) can occur. We show numerically that Shil'nikov homoclinic orbits to saddle-focus equilibria exists. The codimension 1 continuations of these orbits form a 'skeleton' for a cascade of flip and tangent bifurcations. The bifurcation analysis facilitates the study of the consequences of the population model for the dynamic behaviour of a food chain and invasion of a top predator into a food chain in a chemostat. Although the predicted transient dynamics of a food chain may depend sensitively on the underlying model for the populations, the global picture of the bifurcation diagram for the different models is about the same.

We investigate invasion by studying the long-term behaviour after introduction of a small amount of top predator. To that end we look at the stability of the boundary attractors; equilibria, limit cycles as well as chaotic attractors using bifurcation analysis. It will be shown that the invasibility criterion is the positiveness of the Lyapunov exponent associated with the change of the biomass of the top predator. It appears that the region in the control parameter space where a predator can invade increases with its growth rate. The resulting system becomes more resistant to further invasion when the top predator grows faster. This implies that short food chains with moderate growth rate of the top predator are liable to be invaded by fast growing invaders which consume the top predator. There may be, however, biological constraints on the top predator's growth rate. Predators are generally larger than prey while larger organisms commonly grow slower. As a result, the growth rate generally decreases with the trophic level. This may enable short food chains to be resistant to invaders. We will relate these results to ecological community assembly and the debate on the length of food chains in nature.

Reference papers:
Kooi, B.W. and Boer, M.P. and Kooijman, S.A.L.M. "Consequences of population models on the dynamics of food chains", Mathematical Biosciences, 153(2), 99-124, 1998.
Kooi, B.W. and Boer, M.P. and Kooijman, S.A.L.M. "Resistance of a food chain to invasion by a top predator", Mathematical Biosciences, accepted 1998.

The first two lectures on food chains, dealing with relatively simple models and complex dynamic behaviour, were more or less theoretically orientated. On the other hand, the last three lectures on complex food webs with a very simple model structure dealt with more practically problems and relationships with experimental data. In these food web models a linear functional response is assumed and this yields simple criteria for existence and stability of equilibria. We can conclude that there is still a large gap between theory and practice. The theme of the mini-symposium was in accordance with the renewed spirit of our NLS-programme, that is, to broaden the group of biologists that could benefit from the activities of the programme. The lectures were appreciated and the discussions were very lively. In summary, this visit was worthwhile.