EVOLUTION IN A SEASONAL ENVIRONMENT: SIMPLICITY LOST?

The evolution of many of the parameters describing population growth in a constant environment is now quite well understood. Recent papers by Roughgarden (1971, 1977), Charlesworth ( 1971), King and Anderson (1971), and Ginzburg (1977) have demonstrated that the correct measure of fitness in a constant environment (without frequency dependence) is the maximum equilibrium population size, which in a logistic model is described by the carrying capacity, K. Recent papers have discussed the evolution of population parameters where the population growth rate is governed by a stochastic process, allowing consideration of either sampling effects or random environmental variability (Heckel and Roughgarden, 1980; Turelli and Petry, 1980). In these papers a more complex picture of fitness emerges: no simple average of population size can be shown to be maximized although most simple averages do seem to increase. Many organisms grow in an environment with at least some seasonal component, so the study of growth in a seasonal environment is extremely important. Many early successional species can be viewed as exploiting a periodic environment. Evolution of parameters describing population growth in a seasonal environment was first discussed by Roughgarden (1971) and King and Anderson (1971). Their results will be discussed more fully below. There recently have been many studies of population growth in a periodic environment using the logistic model and more complex models (Sonneveld and van Kan, 1979; Boyce and Daley, 1980; Coleman, 1981). One question addressed by these authors has been the evolution of tracking ability, which in a logistic model is controlled by the natural rate of increase, r. For all of these models to be interesting, the species must complete several generations during one environmental cycle. Hence, the results are probably of most interest in the study of insect populations. This paper has three major goals. The first is to explore further the evolution of tracking ability within the context of the models for a periodic environment introduced above. This will be done by studying the evolution of r in a logistic equation and of other parameters in more complex models. The second goal is to consider quantities which are maximized by selection in a periodic environment, as a contrast to the simple case of population size in a constant environment. The final goal of the paper is to compare and contrast evolution when the carrying capacity is seasonally (or periodically) varying versus the case where an additional seasonal (or periodic) density dependent mortality is superimposed on a population. For all these questions, I will use the approach of evolutionarily stable strategies (Maynard Smith, 1976), but similar answers would emerge from an analysis based on single locus polymorphisms (Charlesworth and Leon, 1976; Hastings, 1978). There is an additional complication that I will not consider here. All the models in this paper are of necessity best thought of as models for populations with overlapping generations. A next step in the analysis beyond the one considered here would be to consider the importance of age structure in these models. For purely genetic models excellent analyses including the effects of age structure have been presented by Charlesworth (1980). The organization of the remainder of the paper is as follows. I will first consider the logistic model and the univariate evo-