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All organisms are composed of multiple chemical elements such as carbon, nitrogen and phosphorus. While energy flow and element cycling are two fundamental and unifying principles in ecosystem theory, population models usually ignore the latter. Such models implicitly assume chemical homogeneity of all trophic levels by concentrating on a single constituent, generally an equivalent of energy. In this paper, we examine ramifications of an explicit assumption that both producer and grazer are composed of two essential elements: carbon and phosphorous. Using stoichiometric principles, we construct a two-dimensional Lotka-Volterra type model that incorporates chemical heterogeneity of the first two trophic levels of a food chain. The analysis shows that indirect competition between two populations for phosphorus can shift predator-prey interactions from a (+, -) type to an unusual (-, -) class. This leads to complex dynamics with multiple positive equilibria, where bistability and deterministic extinction of the grazer are possible. We derive simple graphical tests for the local stability of all equilibria and show that system dynamics are confined to a bounded region. Numerical simulations supported by qualitative analysis reveal that Rosenzweig's paradox of enrichment holds only in the part of the phase plane where the grazer is energy limited; a new phenomenon, the paradox of energy enrichment, arises in the other part, where the grazer is phosphorus limited. A bifurcation diagram shows that energy enrichment of producer-grazer systems differs radically from nutrient enrichment. Hence, expressing producer-grazer interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior. c # 2000 Society for Mathematical Biology 0092-8240/00/061137 + 26 $35.00/0 c # 2000 Society for Mathematical Biology 1138 I. Loladze et al

. This paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat. The model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations. In the case where only two species are engaged in competition, it is shown that competitive exclusion holds for any monotone growth response functions. Sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n -species case. In regard to the delay effects on the qualitative outcome of competition, it is demonstrated that when the delays are relatively small, the predictions of the model are identical with the predictions given by corresponding models without time delays. However, including large delays in the model may alter the predicted outcome of competition. The techniques used also work in the model when there...

Matrix game theory and optimisation models offer two radically different perspectives on the outcome of evolution. Optimisation models consider frequency-independent selection and envisage evolution as a hill-climbing process on a constant fitness landscape, with the optimal strategy corresponding to the fitness maximum. By contrast, in evolutionary matrix games selection is frequency-dependent and leads to fitness equality among alternative strategies once an evolutionarily stable strategy has been established. In this review we demonstrate that both optimisation models and matrix games represent limiting cases of the general framework of nonlinear frequency-dependent selection. Adaptive dynamics theory considers arbitrary nonlinear frequency and density dependence and envisages evolution as proceeding on an adaptive landscape that changes its shape according to which strategies are present in the population. In adaptive dynamics, evolutionarily stable strategies correspond to conditional fitness maxima: the ESS is characterised by the fact that it has the highest fitness if it is the established strategy. In this framework it can also be shown that dynamical attainability, evolutionary stability, and invading potential of strategies are pairwise independent properties. In optimisation models, on the other hand, these properties become linked such that the optimal strategy is always attracting, evolutionarily stable and can invade any other strategy. In matrix games fitness is a linear function of the potentially invading strategy and can thus never exhibit an interior maximum: Instead, the fitness landscape is a plane that becomes horizontal once the ESS is established. Due to

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The competitive exclusion principle (CEP) states that no equilibrium is possible if n species exploit fewer than n resources. This principle does not appear to hold in nature, where high biodiversity is commonly observed, even in seemingly homogenous habitats. Although various mechanisms, such as spatial heterogeneity or chaotic fluctuations, have been proposed to explain this coexistence, none of them invalidates this principle. Here we evaluate whether principles of ecological stoichiometry can contribute to the stable maintenance of biodiverse communities. Stoichiometric analysis recognizes that each organism is a mixture of multiple chemical elements such as carbon (C), nitrogen (N), and phosphorus (P) that are present in various proportions in organisms. We incorporate these principles into a standard predator–prey model to analyze competition between two predators on one autotrophic prey. The model tracks two essential elements, C and P, in each species. We show that a stable equilibrium is possible with two predators on this single prey. At this equilibrium both predators can be limited by the P content of the prey. The analysis suggests that chemical heterogeneity within and among species provides new mechanisms that can support species coexistence and that may be important in maintaining biodiversity.

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This paper is dedicated to Paul Waltman on the occasion of his retirement

The word ecology comes from the Greek