Stochastic and deterministic approaches to generalised random Lotka-Volterra communities

In this project we studied the well-known generalised Lotka-Volterra model, usually employed in population dynamics to describe biodiversity and ecological communities. Considering the interactions between species to be random in nature, we sought to study through numerical simulations the dynamics and properties of these communities for different choices of model parameters such as the strength, variance and correlation of the interaction coefficients. We studied the model from a deterministic and stochastic approach, and we attempted to identify and explain the differences that both approaches may present.

We showed that by increasing the strength and the variance of the interactions, the species abun- dances generally grow and the system moves from a stable phase with a unique fixed point to an unstable phase with divergences, volatile trajectories or multiple fixed points. However, by increasing the competitive interactions in the community, the stability of the system is promoted.

We observe that some species go extinct throughout the dynamics and that the size of the surviving community decreases with the diversity/complexity in the interactions. Despite more extinctions, the average abundance per species increases with the diversity because the surviving species have generally higher abundances, except for ecosystems with the interactions of predator-prey nature, which do not allow for overly large abundances. These are known results which we confirmed and evaluated with the existing theoretical predictions.

When demographic stochasticity is introduced into the system, the model we study is individual- based, which is reduced to the Lotka-Volterra equations for infinite populations. It causes species abundances to fluctuate and the impact is evident: besides competitive extinctions, more species die out than in the deterministic model as a result of the fluctuations. Such differences between surviving communities in both models are more evident for ecosystems with a higher degree of competitive interactions, since species abundances are generally lower and species are thus more prone to go extinct due to fluctuations.

On the other hand, as the complexity of interactions increases, abundances grow and the number of extinctions due to fluctuations is reduced, so the surviving communities in both models become more comparable. Similar behavior is found in the relative fluctuations of the ecosystem, which are reduced by the variance in interactions. Nevertheless, for ecosystems with only predator-prey interactions, both the difference between surviving communities and the relative fluctuations increase with the diversity of interactions.

Although when increasing diversity/complexity in the interactions extinctions due to fluctuations are reduced, the extinction probability in the stochastic model for species that survive in the deterministic model generally rises, especially in the species with higher abundances where before it was essentially zero.

Finally, we also found that when approaching the unstable phase by increasing the variance in interactions, the abundances of stochastic species move increasingly away from the deterministic ones in the stationary state.