Does coexistence theory matter in a finite world?

Background/Question/Methods: Much of the recent empirical and theoretical work on species coexistence is based on studying per-capita growth rates of species when rare (invasion growth rates) in models that treat populations as continuous matter, and where extinction only occurs asymptotically as densities approach zero over an infinite time horizon. In nature, extinctions occur in finite time and rarity corresponds to small, discrete collections of individuals whose dynamics are not well approximated by continuous density models. To understand to what extent these discrepancies are biologically significant, we studied individual-based, stochastic counterparts of a classical competition model parameterized by competing annual plant species living on serpentine soils in California.

Results/Conclusions: While invasion growth rates of the inferior competitor 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 abundances, a simplified stochastic model explained over 99% of the variation in the coexistence times. We show that coexistence times are approximately the harmonic mean of the single species' persistence times in the simplified model. Hence, the species more susceptible to extinction has a disproportionately large influence on the coexistence time. Using this simplified model, we identify how invasion growth rates and equilibrium abundances contribute to coexistence times. Coexistence times increase in a saturated manner with invasion growth rates and increase exponentially with equilibrium abundance. Remarkably, when the invasion growth rate of the inferior competitor is sufficiently greater than one (which occurs for 75% of the deterministically coexisting pairs of serpentine annuals), coexistence times of 1,000 years occur even when the inferior species has <50 individuals at the deterministic coexistence equilibrium. When the inferior competitor has the lower equilibrium abundance (which occurs for 7 out of 8 of the deterministically coexisting pairs of serpentine annuals), niche overlap and fitness differences negatively impact coexistence times; a prediction consistent with the deterministic theory. However, when the inferior competitor has the higher equilibrium abundance (which occurs for 1 of the species pairs), coexistence times can exhibit a humped shaped relationship with fitness differences or niche overlap--increasing and then decreasing with these quantities. Collectively our results support making inferences about coexistence times from the deterministic coexistence theory, while highlighting the importance of looking beyond invasion growth rates.