Environmental Variability Promotes Coexistence in Lottery Competitive Systems

In deterministic approaches to modeling, two species are generally regarded as capable of coexistence if the model has a stable equilibrium with both species in positive numbers. Temporal environmental variability is assumed to reduce the likelihood of coexistence by keeping species abundances away from equilibrium. Here we present a contrasting view based on a model of competition for space among coral reef fishes, or any similarly territorial animals. The model has no stable equilibrium point with both species in positive abundance, yet both species persist in the system provided environmental variability in birth rates is sufficiently high. In general the higher the environmental variability the more likely it is that coexistence will occur. This conclusion is not affected by one species having a mean advantage over the other. Not all kinds of environmental variability necessarily lead to coexistence, however, for when the death rates of the two species are highly variable and negatively correlated, the extinction of one species, determined by chance, is likely to occur. The results in this paper are shown to depend on the nonlinearity of the dynamics of the system. This nonlinearity arises from the simple fact that the animals have overlapping generations. When applied to the coral reef fish setting, our analysis confirms the view that coexistence can occur in a system where space is allocated largely at random, provided environmental variability is sufficiently great (Sale 1977); but our explanations and predictions differ in detail with those of Sale.