  We investigate the outcome of generalised Lotka-Volterra dynamics of
ecological communities with random interaction coefficients and non-linear
feedback. We show in simulations that the saturation of non-linear feedback
stabilises the dynamics. This is confirmed in an analytical
generating-functional approach to generalised Lotka-Volterra equations with
piecewise linear saturating response. For such systems we are able to derive
self-consistent relations governing the stable fixed-point phase, and to carry
out a linear stability analysis to predict the onset of unstable behaviour. We
investigate in detail the combined effects of the mean, variance and
co-variance of the random interaction coefficients, and the saturation value of
the non-linear response. We find that stability and diversity increases with
the introduction of non-linear feedback, where decreasing the saturation value
has a similar effect to decreasing the co-variance. We also find co-operation
to no longer have a detrimental effect on stability with non-linear feedback,
and the order parameters mean abundance and diversity to be less dependent on
the symmetry of interactions with stronger saturation.
