Hyper Diversity, Species Richness, and Community Structure in ESS and Non-ESS Communities.

In mathematical models of eco-evolutionary dynamics with a quantitative trait, two species with different strategies can coexist only if they are separated by a valley or peak of the adaptive landscape. A community is ecologically and evolutionarily stable if each species' trait sits on global, equal fitness peaks, forming a saturated ESS community. However, the adaptive landscape may allow communities with fewer (undersaturated) or more (hypersaturated) species than the ESS. Non-ESS communities at ecological equilibrium exhibit invasion windows of strategies that can successfully invade. Hypersaturated communities can arise through mutual invasibility where each non-ESS species' strategy lies in another's invasion window. Hypersaturation in ESS communities with more than 1 species remains poorly understood. We use the G-function approach to model niche coevolution and Darwinian dynamics in a Lotka-Volterra competition model. We confirm that up to 2 species can coexist in a hypersaturated community with a single-species ESS if the strategy is scalar-valued, or 3 species if the strategy is bivariate. We conjecture that at most $n·\left(s+1\right)$ species can form a hypersaturated community, where $n$ is the number of ESS species at the strategy's dimension $s$ . For a scalar-valued 2-species ESS, 4 species coexist by "straddling" the would-be ESS traits. When our model has a 5-species ESS, we can get 7 or 8, but not 9 or 10, species coexisting in the hypersaturated community. In a bivariate model with a single-species ESS, an infinite number of 3-species hypersaturated communities can exist. We offer conjectures and discuss their relevance to ecosystems that may be non-ESS due to invasive species, climate change, and human-altered landscapes.

Supplementary information: The online version contains supplementary material available at 10.1007/s13235-025-00646-2.