A central question in the study of ecology and evolution is: "Why are there so many species?" It has been shown that certain forms of the Lotka-Volterra (L-V) competition equations lead to an unlimited number of species. Furthermore, these authors note how any change in the nature of competition (the competition kernel) leads to a finite or small number of coexisting species. Here we build upon these works by further investigating the L-V model of unlimited niche packing as a reference model and evolutionary game for understanding the environmental factors restricting biodiversity. We also examine the combined eco-evolutionary dynamics leading up to the species diversity and traits of the ESS community in both unlimited and finite niche-packing versions of the model. As an L-V game with symmetric competition, we let the strategies of individuals determine the strength of the competitive interaction (like competes most with like) and also the carrying capacity of the population. We use a mixture of analytic proofs (for one and two species systems) and numerical simulations. For the model of unlimited niche packing, we show that a finite number of species will evolve to specific convergent stable minima of the adaptive landscape (also known as species archetypes). Starting with a single species, faunal buildup can proceed either through species doubling as each diversity-specific set of minima are reached, or through the addition of species one-by-one by randomly assigning a speciation event to one of the species. Either way it is possible for an unlimited number or species to evolve and coexist. We examine two simple and biologically likely ways for breaking the unlimited niche-packing: (1) some minimum level of competition among species, and (2) constrain the fundamental niche of the trait space to a finite interval. When examined under both ecological and evolutionary dynamics, both modifications result in convergent stable ESSs with a finite number of species. When the number of species is held below the number of species in an ESS coalition, we see a diverse array of convergent stable niche archetypes that consist of some species at maxima and some at minima of the adaptive landscape. Our results support those of others and suggest that instead of focusing on why there are so many species we might just as usefully ask, why are there so few species?

In inverted biomass pyramids (IBPs) prey are outnumbered by their predators when measured by biomass. We investigate how prey should behave in the face of danger from higher predator biomass, and how anti‐predator behavior (in the form of vigilance) can, in turn, affect the predator–prey system. In this study, we incorporate anti‐predator behaviors into a Lotka–Volterra predator–prey model in the form of fixed and facultative vigilance. Facultative vigilance models behavior as a dynamic foraging game, allowing us to assess optimal behavioral responses in the context of IBPs using a dynamical fitness optimization approach. We model vigilance as a tradeoff between safety and either the prey's maximum growth rate or its carrying capacity. We assess the population dynamics of predators and prey with fear responses, and investigate the role fear plays on trophic structure. We found that the ecology of fear plays an important role in predator–prey systems, impacting trophic structure and the occurrence of IBPs. Fixed vigilance works against IBP structure by always reducing the predator–prey biomass ratio at equilibrium with increasing levels of vigilance. Facultative vigilance can actually promote IBPs, as prey can now adjust their vigilance levels to cope with increased predation and the costs associated with vigilance. This is especially true when the effectiveness of vigilance is low and predators are very lethal. In general, these trends are true whether the costs of vigilance are felt on the prey's maximum growth rate or its carrying capacity. Just as the ecology of fear, when first introduced, was used to explain why top carnivores are rare in terrestrial systems, it can also be used to understand how big fierce predators can be common in IBPs.

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A. AbstractThe concept of the evolutionary stable strategy (ESS) has been fundamental to the development of evolutionary game theory. It represents an equilibrial evolutionary state in which no rare invader can grow in population size. With additional work, the ESS concept has been formalized and united with other stability concepts such as convergent stability, neighborhood invasion stability, and mutual invisibility. Other work on evolutionary models, however, shows the possibility of unstable and/or non-equilibrial dynamics such as limit cycles and evolutionary suicide. Such "pathologies" remain outside of a well-defined context, especially the currently defined stability concepts of evolutionary games. Ripa et al. (2009) offer a possible reconciliation between work on non-equilibrial dynamics and the ESS concept. They noticed that the systems they analyzed show non-equilibrial dynamics when under-saturated and "far" from the ESS and that getting "closer" to the ESS through the addition of more species stabilized their systems. To that end, we analyzed three models of evolution, two predator-prey models and one competition model of evolutionary suicide, to see how the degree of saturation affects the stability of the system. In the predator-prey models, stability is linked to the degree of saturation.Specifically, a fully saturated community will only show stable dynamics, and unstable dynamics occur only when the community is under-saturated. With the competition model, we demonstrate it to be permanently under-saturated, likely showing such extreme dynamics for this reason.Though not a general proof, our analysis of the models provide evidence of the link between community saturation and evolutionary dynamics. Our results offer a possible placement of these evolutionary "pathologies" into a wider framework. In addition, the results concur with previous results showing greater evolutionary response to less biodiversity and clarifies the effect of extrinsic vs. intrinsic non-equilibrial evolutionary dynamics on a community.