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• #### Index 275-285

Understanding equilibria is an essential step toward a nonequilibrium theory and should provide greater insights than we now possess.

I first discuss different theoretical approaches to the definition of evolutionary equilibria. Next I consider the classical theory and define two special equilibria: the "salmon" limit, and the "bacterium" limit. These equilibria reveal critical issues concerning the assumptions and structure of the theory. The consequences of some specific generalizations of the classical theory will then be reviewed. Finally, I outline a program of theoretical work that should lead to a more useful evolutionary theory of senescence.

#### Evolutionary Equilibria

Consider the dynamics of a population phenotype under the action of selection and mutation. At any time, there is some frequency distribution of the phenotype (among individuals) and an underlying distribution of genotypes. Genotypes map into phenotypes, and phenotypes map into fitness. Fitness differences, mutation, and random drift can lead to a change in the relative numbers of different genotypes and thus to changes in phenotype distribution. Any theoretical analysis makes assumptions about each step in this process. Given such assumptions, an evolutionary equilibrium (EE) is defined as a phenotypic distribution for a population that remains unchanged under the above dynamic process.

For an age-structured population, the individual phenotype of interest is a vector z = (µ, m). The components µ(x) and m(x) of the two vectors listed on the right are, respectively, the mortality rate and fertility at age x. For a size-structured population, we would add size-specific growth rate g(s) at size s, and make µ(s) and m(s) size-dependent to get z = (µ, m, g). Suppose that time is measured in discrete units (e.g., generations) and that the frequency distribution of the phenotype in the population is given by ft(z) at time t. The evolutionary dynamic process above changes this distribution, in one time step, into a new one, ft+1(z). An evolutionary equilibrium is a distribution F( z) that remains unchanged under the dynamics. We are mainly interested in stable equilibria: the analysis of stability properties can be difficult, although this paper does consider the stability of some special equilibria. Unstable equilibria can also be interesting, typically in situations where one asks if a newly introduced phenotype can increase in frequency in a population from which it was previously absent. Such invasion analyses are used here to examine the effects of random environments on the evolution of mortality patterns.

There are three ways of modeling the dynamics of a phenotype distribution, and I consider them in turn. In each case, I direct attention to the assumptions and limits of the method. The reader should note that the results of the theory depend not only on the assumptions within each method below but also that different methods can produce different answers.

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