Some paradoxes in the Tilman model, how to avoid or accept them

In making models for biological systems one expects to grasp the biology and be able to use ones intuition to predict the outcome. This paper is about the discrepancy between what is expected and what is the outcome of the analysis of the model, a paradox. The Tilman model of consumers competing for resources has some aspects that may appear counterintuitive. Within the standard Tilman model for a single consumer and a single resource, when a very efficient consumer rapidly eats all available food, the resource density becomes zero, but if there is no food, how can the consumer survive? The paradox can be resolved by realising that in the short term indeed rapid consumption may lead to starvation and a decline in the consumer population, but in the long term a small resource and consumer density remain. A single consumer living on two essential nutrients leaves the density of the non-limiting nutrient above its critical level, so what is done with the extra food? We explain that the extra food is not consumed, just because it is available, but that in fact the additional food is not consumed at all, because the consumer has no use for it. For a model with two consumer species competing for a single nutrient one of the consumers will eventually disappear, even if the food economics of the surviving one is worse and there is still much food wasted. Both are inherent aspects of the model, and the paradox can be avoided if the difference between the consumer species is small, in which case it will take very long to reach equilibrium. We argue that in general an extended stability analysis, in which not only the asymptotically stable state is considered, but also the unstable steady states and all time scales involved in the transient dynamics, gives more insight in the dynamics involved and thus can help in avoiding apparent contradictions in ecological models, or accepting them.