Probabilities of extinction, weak extinction permanence, and mutual exclusion in discrete, competitive, Lotka-Volterra systems

The probabilities of extinction, weak extinction, permanence, and mutual exclusion are calculated for models with up to five species by examining one million randomly chosen discrete, competitive, LotkarVolterra systems. The probability of permanence drops off very rapidly with the increase in the number of species. It drops to less than 1% with five species. The probability that at least one species will die out increases with the number of species. It reaches 95% with five species. When a group of species weakly dominates another species, the dominated species goes extinct. The probability that at least one species is weakly dominated is close to 50%. Mutual exclusion happens between 10% and 20% of the time when there are at least three species.