  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.
