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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13235_2025_646_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$n \cdot \left( {s + 1} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>·</mo> <mfenced close=")" open="("> <mrow> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> species can form a hypersaturated community, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13235_2025_646_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> is the number of ESS species at the strategy’s dimension <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13235_2025_646_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$s$</EquationSource> <EquationSource Format="MATHML"><math> <mi>s</mi> </math></EquationSource> </InlineEquation>. 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.

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

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2025年5月30日

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Kailas Shankar Honasoge,
Tania L. S. Vincent,
Gordon G. McNickle,
Roel Dobbe,
Kateřina Staňková,
Joel S. Brown,
Joseph Apaloo

摘要

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 $n \cdot \left( {s + 1} \right)$ $n \cdot (s + 1)$ species can form a hypersaturated community, where $n$ $n$ is the number of ESS species at the strategy’s dimension $s$ $s$ . 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.