  We study a model of competition for resource through a chemostat-type model
where species consume the common resource that is constantly supplied. We
assume that the species and resources are characterized by a continuous trait.
As already proved, this model, although more complicated than the usual
Lotka-Volterra direct competition model, describes competitive interactions
leading to concentrated distributions of species in continuous trait space.
Here we assume a very fast dynamics for the supply of the resource and a fast
dynamics for death and uptake rates. In this regime we show that factors that
are independent of the resource competition become as important as the
competition efficiency and that the direct competition model is a good
approximation of the chemostat. Assuming these two timescales allows us to
establish a mathematically rigorous proof showing that our resource-competition
model with continuous traits converges to a direct competition model. We also
show that the two timescales assumption is required to mathematically justify
the corresponding classic result on a model consisting of only finite number of
species and resources (MacArthur, R. Theor. Popul. Biol. 1970:1, 1-11). This is
performed through asymptotic analysis, introducing different scales for the
resource renewal rate and the uptake rate. The mathematical difficulty relies
in a possible initial layer for the resource dynamics. The chemostat model
comes with a global convex Lyapunov functional. We show that the particular
form of the competition kernel derived from the uptake kernel, satisfies a
positivity property which is known to be necessary for the direct competition
model to enjoy the related Lyapunov functional.
