Does deterministic coexistence theory matter in a finite world?

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 counterpart 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.

Rainfall variability maintains grass‐forb species coexistence

“…While both species exhibited positive growth rates when rare under average historical conditions, variation in rainfall increased GRWR (comparing r i À r r to D 0 i ). This increase in GRWR from variable environmental conditions in turn increased the probability of long-term coexistence, even with natural fluctuations in realised growth rates due to stochasticity (Schreiber et al 2018).…”

Technical Comment on Pande et al. (2020): Why invasion analysis is important for understanding coexistence

“…1c,d) with 500 randomly chosen parameter sets: means and standard deviations of log per capita fecundity were Uniform [0,1], and δ was Uniform [0.2,0.5]. Mean persistence times were estimated by the Aldous algorithm (Schreiber et al 2018) with simulation lengths 10 5 and 10 6 for N = 200 and 1000, respectively, 25 replicates for each parameter set. Circle size is proportional to fecundity variability.…”

Antibiotics in microbial communities: an ecological frame of resistance

“…Populations at low abundance also become increasingly vulnerable to demographic stochasticity, and several researchers have argued that the immune system can clear the residual infection once the population is sufficiently small 48 , indicating that the kinds of qualitative indicators derived from dynamical analyses might be less informative in practical terms. Fortunately, efforts are already underway within the theoretical literature to more explicitly incorporate demographic stochasticity into coexistence analysis 67,68 , and there is no constraint on extending the types of models regularly adopted by community ecologists to incorporate the immune response.…”