Finding all multiple stable fixpoints of n-species Lotka-Volterra competition models

One way to explore assembly of extant and novel communities from species pools, and by that biodiversity and species ranges, is to study the equilibrium behavior of dynamic competition models such as the Lotka-Volterra competition (LVC) model. We present a novel method (COMMUSTIX) to determine all stable fixpoints of the general LVC model with abundances x from a given pool of n species. To that purpose, we split the species in potentially surviving species (x_i>0) and in others going extinct (x_i=0). We derived criteria for the stability of x_i=0 and for the equilibrium of x_i>0 to determine possible combinations of extinct and surviving species by iteratively applying a mixed binary linear optimization algorithm. We tested this new method against (a) the numerical solution at equilibrium of the LVC ordinary differential equations (ODEs) and (b) the fixpoints of all combinations of surviving and extinct species (possible only for small n), tested for stability and non-negativity. The tests revealed that COMMUSTIX is reliable, it detects all multiple stable fixpoints (SFPs), which is not guaranteed by solving the ODEs, and more efficient than the combinations method. With COMMUSTIX, we studied the dependence of the fixpoint behavior on the competition strengths relative to the intra-specific competition. If inter-specific competition was considerably lower than intra-specific competition, only globally SFPs occurred. In contrast, if all inter-specific was higher than intra-specific competition, multiple SFPs consisting of only one species occurred. If competition strengths in the species pool ranged from below to above the intra-specific competition, either global or multiple SFPs strongly differing in species composition occurred. The species richness over all SFPs was high for pools of species with similar, either weak or strong competition, and lower for species with dissimilar or close to intra-specific competition strengths. The new approach is a reliable and efficient tool for further extensive examinations of the dependence of community compositions on parameter settings of the LVC model.