Species assembly in model ecosystems, I: Analysis of the population model and the invasion dynamics

Recently we have introduced a simplified model of ecosystem assembly (Capitán et al., 2009) for which we are able to map out all assembly pathways generated by external invasions in an exact manner. In this paper we provide a deeper analysis of the model, obtaining analytical results and introducing some approximations which allow us to reconstruct the results of our previous work. In particular, we show that the population dynamics equations of a very general class of trophic-level structured food-web have an unique interior equilibrium point which is globally stable. We show analytically that communities found as end states of the assembly process are pyramidal and we find that the equilibrium abundance of any species at any trophic level is approximately inversely proportional to the number of species in that level. We also find that the per capita growth rate of a top predator invading a resident community is key to understand the appearance of complex end states reported in our previous work. The sign of these rates allows us to separate regions in the space of parameters where the end state is either a single community or a complex set containing more than one community. We have also built up analytical approximations to the time evolution of species abundances that allow us to determine, with high accuracy, the sequence of extinctions that an invasion may cause. Finally we apply this analysis to obtain the communities in the end states. To test the accuracy of the transition probability matrix generated by this analytical procedure for the end states, we have compared averages over those sets with those obtained from the graph derived by numerical integration of the Lotka-Volterra equations. The agreement is excellent.