One of the central goals of ecology is to predict the distribution and abundance of organisms. Here, we show that, in ecosystems of high biodiversity, the outcome of multispecies competition can be fundamentally unpredictable. We consider a competition model widely applied in phytoplankton ecology and plant ecology in which multiple species compete for three resources. We show that this competition model may have several alternative outcomes, that the dynamics leading to these alternative outcomes may exhibit transient chaos, and that the basins of attraction of these alternative outcomes may have an intermingled fractal geometry. As a consequence of this fractal geometry, it is impossible to predict the winners of multispecies competition in advance.

With the eutrophication of many freshwaters and coastal environments, phytoplankton blooms have become a common phenomenon. This article uses a reaction-diffusion model to investigate the implications of mixing processes for the dynamics and species composition of phytoplankton blooms. The model identifies four key parameters for bloom development: incident light intensity, background turbidity, water column depth, and turbulent mixing rates. The model predicts that the turbulent mixing rate is a major determinant of the species composition of phytoplankton blooms. In wellmixed environments, the species with lowest "critical light intensity" should become dominant. But at low mixing rates, the species with lowest critical light intensity is displaced if other species obtain a better position in the light gradient. Instead of a gradual change in species composition, the model predicts steep transitions between the dominance regions of the various species. The model predicts a low species diversity: phytoplankton blooms in eutrophic environments should be dominated by one or a few species only. The model predictions are consistent with laboratory competition experiments and many existing field data. We recommend examining competition in phytoplankton blooms under well-controlled laboratory conditions, and we derive scaling rules that facilitate translation from the laboratory to the field. Most phytoplankton competition studies published to date have utilized well-mixed laboratory systems (Tilman 1977;Sommer 1985;Grover 1991;Rothhaupt 1996;Ducobu et al. 1998;Huisman et al. 1999a). In contrast, the word "plankton" stems from the Greek neuter of plagkto, which means roaming or wandering (Hutchinson 1974). In many aquatic environments, phytoplankton species are not thoroughly mixed but slowly wander through the aquatic medium, often passively by turbulent diffusion and sinking and sometimes also actively by means of flagellae or buoyancy regulation (Reynolds 1984(Reynolds , 1997. The implications of slow mixing processes for phytoplankton competition are not well understood. However, field data and experiments clearly demonstrate that the intensity of mixing has a major impact on phytoplankton bloom development and on the species composition of phytoplankton blooms (Eppley et al. 1978;Reynolds et al. 1983;Viner and Kemp 1983;Steinberg and Zimmermann 1988;Jones and Gowen 1990;Cloern 1991;Visser et al. 1996;Berman and Shteinman 1998). KeywordsGeneral ecological theory predicts that incomplete mixing should promote species coexistence (Levin 1974;Atkinson and Shorrocks 1981;Powell and Richerson 1985;Hsu and Waltman 1993;Tilman 1994). Coexistence in spatially explicit competition models is usually brought about by two different mechanisms. First, environmental heterogeneity may favor one species at one location, another species at another location, and so on, so that, integrated over the entire habitat, species can coexist. Second, even in an initially homogeneous environment, incomplete mixing combined with in...

We present an evolutionarily stable strategy (ESS) model to analyze selection on seasonal variation in the brood sex ratio, as observed in several species of raptorial birds. The model is specifically tailored to the life history of the European kestrel, and it reflects the maturation time hypothesis, the idea that a seasonal sex ratio trend has evolved because of sex differences in the dependence of age of first breeding on date of birth. First we show how to derive a fitness function in the context of a seasonal environment. Model parameters are estimated from field data in order to derive quantitative predictions. Since little is known about constraints on sex ratio control in birds, we analyze three scenarios, each corresponding to a different strategy set. We consider a model without constraints on sex ratio control, a model where the sex ratio trend is constrained to be linear, and a mechanistic model incorporating a plausible mechanism of sex ratio control in birds. One of the models yields an ESS sex ratio trend that closely resembles the trend observed in the field. However, the predictions are very sensitive to the choice of strategy set. Moreover, the selective forces generated by sex differences in maturation are rather weak. In fact, the mechanistic model shows that seemingly negligible costs of sex ratio control may be sufficient to overcome the adaptive value of adjusting the sex ratio.

In a variety of species, females exhibit preferences for multiple male ornaments. Several hypotheses have been proposed to explain this phenomenon. Which, if any, of these hypotheses is the most plausible in general remains largely unresolved based on the available empirical data. Yet theoretical studies conclude that the evolution of preferences for multiple signals of male quality is unlikely, especially when the use of an additional cue in mate choice strongly increases the overall cost of choice. This would imply that most male courtship characters do not reflect the male's genetic quality but instead evolved through Fisherian sexual selection. However, the existing models focus on ornaments that signal overall genetic quality and do not address the possibility that different ornaments provide information about different aspects of quality. Therefore, we develop a model in which the ornaments act as signals for distinct quality components. When the ornaments provide overlapping information about these quality components, we retrieve the results of earlier models. However, when the ornaments provide independent information, preferences for multiple ornaments may evolve, even when exhibiting multiple preferences is costly. We discuss our results in relation to the multiple-message and redundant-signal hypotheses for ornament diversity and identify parallels between Fisherian and good-genes mechanisms for the evolution of multiple ornaments.

Males and females have opposing interests when it comes to the honesty of signals used in mate choice. The existence of this sexual conflict has long been acknowledged, but its consequences have not been fully investigated. By applying adaptive dynamics methods and individual-based computer simulations to a standard model for good-genes sexual selection, we show that sexual conflict over condition-dependent signaling can prevent the handicap process from ever attaining an evolutionary equilibrium. We outline the parameter conditions and properties of the underlying genetics conducive to nonequilibrium behavior and discuss the potential of such behavior to explain the elaboration and frequent phylogenetic loss of sexually selected traits. We also evaluate its consequences for wellestablished insights of sexual selection theory previously shown to apply when female mating preference and male ornament expression do converge on stable equilibrium levels. Contrary to equilibrium expectation, a continual change of condition-dependent signaling enables the evolution of a costly preference for a pure epistatic indicator and the evolution of preferences for redundant signals or a large number of independent ornaments. We thus conclude that seemingly general results of sexual selection theory, insofar as these are based on equilibrium considerations, do not extend to cases where nonequilibrium behavior occurs.