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We present a general epidemiological model of host-parasite interactions that includes various forms of superinfection. We use this model to study the effects of different host life-history traits on the evolution of parasite virulence. In particular, we analyze the effects of natural host death rate on the evolutionarily stable parasite virulence. We show that, contrary to classical predictions, an increase in the natural host death rate may select for lower parasite virulence if some form of superinfection occurs. This result is in agreement with the experimental results and the verbal argument presented by Ebert and Mangin (1997). This experiment is discussed in the light of the present model. We also point out the importance of superinfections for the effect of nonspecific immunity on the evolution of virulence. In a broader perspective, this model demonstrates that the occurrence of multiple infections may qualitatively alter classical predictions concerning the effects of various host life-history traits on the evolution of parasite virulence.

Multi-patch systems, in which several species interact in patches connected by dispersal, offer a general framework for the description and analysis of spatial ecological systems. This paper describes how to analyse the local stability of spatially homogeneous solutions in such systems. The spatial arrangement of the patches and their coupling is described by a matrix. For a local stability analysis of spatially homogeneous solutions it turns out to be sufficient to know the eigenvalues of this matrix. This is shown for both continuous and discrete time systems. A bookkeeping scheme is presented that facilitates stability analyses by reducing the analysis of a k-species, n-patch system to that of n uncoupled k-dimensional single-patch systems. This is demonstrated in a worked example for a chain of patches. In two applications the method is then used to analyse the stability of the equilibrium of a predator-prey system with a pool of dispersers and of the periodic solutions of the spatial Lotka-Volterra model.

Many ecological interactions that are called mutualistic are in fact mixtures of antagonistic and mutualistic aspects. For example, plasmids exploit their bacterial hosts but also protect them against external threats. In this study, we analyse the conditions for the evolution of what we call 'dangerous liaisons': interactions combining mutualistic and antagonistic aspects. Starting point of our analysis is a model that was proposed as early as 1934. In this model, partners have to form a complex (either temporary or long lasting) in order to interact. Using this model framework we then set out to define and tease apart private interests of the interacting partners from their common good. This dichotomy provides a unifying perspective to classify ecological interactions. We discuss some examples to illustrate how the outcome of the interaction may depend on densities or on other contextual variables. Finally, we note that having a common good is not a necessary condition for partners to have aligned interests. In a dangerous liaison partners may have interest to cooperate even when this does not bolster the common good. M. 6an Baalen,

We present a general epidemiological model of host-parasite interactions that includes various forms of superinfection. We use this model to study the effects of different host life-history traits on the evolution of parasite virulence. In particular, we analyze the effects of natural host death rate on the evolutionarily stable parasite virulence. We show that, contrary to classical predictions, an increase in the natural host death rate may select for lower parasite virulence if some form of superinfection occurs. This result is in agreement with the experimental results and the verbal argument presented by Ebert and Mangin (1997). This experiment is discussed in the light of the present model. We also point out the importance of superinfections for the effect of nonspecific immunity on the evolution of virulence. In a broader perspective, this model demonstrates that the occurrence of multiple infections may qualitatively alter classical predictions concerning the effects of various host life-history traits on the evolution of parasite virulence.

Inter-generational temporal variability of the environment is important in the evolution and adaptation of phenotypic traits. We discuss a population-dynamic approach which plays a central role in the analysis of evolutionary processes. The basic principle is that the phenotypes with the greatest long-term average growth rate will dominate the entire population. The calculation of longterm average growth rates for populations under temporal stochasticity can be highly cumbersome. However, for a discrete non-overlapping population, it is identical to the geometric mean of the growth rates (geometric mean fitness), which is usually different from the simple arithmetic mean of growth rates. Evolutionary outcomes based on geometric mean fitness are often very different from the predictions based on the usual arithmetic mean fitness. In this paper we illustrate the concept of geometric mean fitness in a few simple models. We discuss its implications for the adaptive evolution of phenotypes, e.g. foraging under predation risks and clutch size. Next, we present an application: the risk-spreading egg-laying behaviour of the cabbage white butterfly, and develop a two-patch population dynamic model to show how the optimal solution diverges from the usual arithmetic mean approach. The dynamics of these stochastic models cannot be predicted from the dynamics of simple deterministic models. Thus the inclusion of stochastic factors in the analyses of populations is essential to the understanding of not only population dynamics, but also their evolutionary dynamics.

Many arthropod predator-prey systems on plants typically have a patchy structure in space and at least two essentially different phases at each of the trophic levels: a phase of within-patch population growth and a phase of between-patch dispersal. Coupling of the trophic levels takes place in the growth phase, but it is absent in the dispersal phase. By representing the growth phase as a simple presencelabsence state of a patch, metapopulation dynamics can be described by a system of ordinary differential equations with the classic Lotka-Volterra model as a limiting case (e.g. when the dispersal phases are of infinitely short duration).When timescale arguments justify ignoring plant dynamics, it is shown that the otherwise unsrdbk! Lotka-Volterra model becomes stable by any of the following extensions: ( I ) a dispersal phase of the prey, (2) variability in prey patches with respect to the risk of detection by predators,(3) (sufficiently high) interception of dispersing predators in predator-invaded prey patches, and (4) prey dispersal from predator-invaded prey patches. The parameter domain of stability shrinks when the duration of within-patch predator-prey interaction is fixed rather than variable, and when predators do not disperse from a patch until after prey extermination. A dispersal phase of the predator has a destabilizing effect in contrast to a dispersal phase of the prey. When the timescale of plant dynamics is not very different from predator-prey patch dynamics, the Lotka-Volterra predator-prey patch model should be extended to a predator-prey-plant patch model, but this greatly modified the list of potential stabilizing mechanisms. Several of the mechanisms that have a stabilizing effect on a ditrophic model lose this effect in a tritrophic model and may even become destabilizing; for example, the dispersal phase of the prey confers stability to the predatory-prey model, but destabilizes the steady state in the predator-prey-plant model in much the same way as the dispersal phase of the predator destabilizes the steady state in the predator-prey model. Other mechanisms retain their stabilizing effect in a tritrophic context; for example, dispersal of prey from predator-invaded prey patches has a stabilizing effect on both predator-prey and predator-prey-plant models.

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