This paper describes the coevolution of phenotypes in a community comprising a population of predators and of prey. It is shown that evolutionary cycling is a likely outcome of the process. The dynamical systems on which this description is based are constructed from microscopic stochastic birth and death events, together with a process of random mutation. Births and deaths are caused in part by phenotype-dependent interactions between predator and prey individuals and therefore generate natural selection. Three outcomes of evolution are demonstrated. A community may evolve to a state at which the predator becomes extinct, or to one at which the species coexist with constant phenotypic values, or the species may coexist with cyclic changes in phenotypic values. The last outcome corresponds to a Red Queen dynamic, in which the selection pressures arising from the predator-prey interaction cause the species to evolve without ever reaching an equilibrium phenotypic state. The Red Queen dynamic requires an intermediate harvesting efficiency of the prey by the predator and sufficiently high evolutionary rate constant of the prey, and is robust when the model is made stochastic and phenotypically polymorphic. A cyclic outcome lies outside the contemporary focus on evolutionary equilibria, and argues for an extension to a dynamical framework for describing the asymptotic states of evolution.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. British Ecological Society is collaborating with JSTOR to digitize, preserve and extend access to Journal of Animal Ecology. Summary 1. Though models of life-history decisions are traditionally based on age-related changes in the costs and benefits of reproduction, in nature both costs and benefits vary with individual differences in phenotype as well as with environmental changes. Using long-term records of individual reproduction and survival in the Soay sheep of St Kilda, we show that the costs and benefits of breeding to animals of different weight categories vary with population density.3. Subsequently, we use stochastic dynamic programming to predict the optimal fecundity of animals belonging to each category at high and low population density. Optimal strategies of fecundity vary with population density as well as between different weight categories of sheep. However, there is no evidence that the sheep track density-related changes in optimal fecundity. Instead, their behaviour approximates to an average, weight-related optimum that is well adapted to the range of conditions that they encounter.

Abstract. Evolution takes place in an ecological setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals -the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the context of predator-prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator-prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamic; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species.

The evolutionary consequences of asymmetric competition between species are poorly understood in comparison with symmetric competition. A model for evolution of body size under asymmetric competition within and between species is described. The model links processes operating at the scale of the individual to that of macroscopic evolution through a stochastic mutation-selection process. Phase portraits of evolution in a phenotype space characteristically show character convergence and parallel character shifts, with character divergence being relatively uncommon. The asymptotic states of evolution depend very much on the properties of asymmetric competition. Given relatively weak asymmetries between species, a single equilibrium point exists; this is a local attractor, and its position is determined by the intra-and interspecific asymmetries. When the asymmetries are made stronger, several fixed points may come about, creating further equilibrium points which are local attractors. It is also possible for periodic attractors to occur; such attractors comprise Red Queen dynamics with phenotype values that continue to change without ever settling down to constant values. From certain initial conditions, evolution leading to extinction of one of the species is also a likely outcome.

We present a novel market-based method, inspired by retail markets, for resource allocation in fully decentralised systems where agents are self-interested. Our market mechanism requires no coordinating node or complex negotiation. The stability of outcome allocations, those at equilibrium, is analysed and compared for three buyer behaviour models. In order to capture the interaction between self-interested agents, we propose the use of competitive coevolution. Our approach is both highly scalable and may be tuned to achieve specified outcome resource allocations. We demonstrate the behaviour of our approach in simulation, where evolutionary market agents act on behalf of service providing nodes to adaptively price their resources over time, in response to market conditions. We show that this leads the system to the predicted outcome resource allocation. Furthermore, the system remains stable in the presence of small changes in price, when buyers' decision functions degrade gracefully.