  We use dynamical generating functionals to study the stability and size of
communities evolving in Lotka-Volterra systems with random interaction
coefficients. The size of the eco-system is not set from the beginning.
Instead, we start from a set of possible species, which may undergo extinction.
How many species survive depends on the properties of the interaction matrix;
the size of the resulting food web at stationarity is a property of the system
itself in our model, and not a control parameter as in most studies based on
random matrix theory. We find that prey-predator relations enhance stability,
and that variability of species interactions promotes instability. Complexity
of inter-species couplings leads to reduced sizes of ecological communities.
Dynamically evolved community size and stability are hence positively
correlated.
