Ecological communities from random generalised Lotka-Volterra dynamics with non-linear feedback

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.