Abstract
Much of the recent work on species coexistence is based on studying per-capita growth rates of species when rare (invasion growth rates) in deterministic models where populations have continuous densities and extinction only occurs as densities approach zero over an infinite time horizon. In nature, extinctions occur in finite time and rarity corresponds to small, discrete populations whose dynamics are not well approximated by deterministic models. To understand whether the biological significance of these discrepancies, we parameterized a stochastic counter-part of a classical deterministic model of competition using data from annual plants competing on serpentine soils. While the minimum of the invasion growth rates explained up to 60% of the variation in the predicted coexistence times, species pairs with similar invasion growth rates had coexistence times that differed by several orders of magnitude. By integrating the deterministic invasion growth rates and coexistence equilibrium population sizes, a simplified stochastic model explained over 99% of the variation in the coexistence times. This simplified model corresponds to uncoupled single species models whose parameters are determined from the two species model. This simplified model shows that coexistence times are approximately one-half of the harmonic mean of these single species’ persistence times. Furthermore, coexistence times increase and saturate with invasion growth rates, but increase exponentially with equilibrium population sizes. When the minimum of the invasion growth rate is sufficiently greater than one, coexistence times of 1, 000 years occur even when the inferior species has < 50 individuals at the deterministic coexistence equilibrium. When the fitness inferior has the lower equilibrium population size (which occurs for 6 out of 8 of the deterministically coexisting pairs), niche overlap and fitness differences negatively impact coexistence times, which is consistent with the deterministic theory. However, when the fitness inferior has the higher equilibrium population size (2 species pairs), coexistence times can exhibit a humped shaped relationship with fitness differences–increasing and then decreasing with fitness differences. Collectively our results support the use of deterministic theory to infer the controls over coexistence in finite systems, while also highlighting when ecologists must look beyond invasion growth rates and consider species equilibrium population sizes.
