Adaptation in Stochastic Environments

What is maximized by natural selection? A general answer is the expected reproductive success of the individual; i.e. mean individual fitness. This basic fitness maximization principle underlies much of the modern evolutionary theory of adaptation (Fisher 1930, Williams 1966, Maynard Smith 1978). Criticism of the hypothesis of fitness maximization has, however, also been widespread (e.g. Dupré 1987).

The effect of random environmental changes on fitness is modelled for a wide range of life histories and patterns of environmental and demographic variability with the following main conclusions:

Environmental fluctuation may not only alter the life history optimization problem but also query optimization itself. Under density regulation annual growth rate is influenced by the direct effect of fluctuation as well as by an indirect effect due to fluctuating population density. For a weak fluctuation there is an optimal strategy which is slightly different from the stable environment optimum, since (a) it should adapt to an average density altered by fluctuation, (b) it should diminish the fluctuation in annual growth rate caused by direct and indirect effect of environmental fluctuation, and (c) it should exploit an increase, but avoid a decrease in average annual growth rate caused by fluctuation. The “optimal” strategy becomes meaningless if the fluctuation is strong, because long run growth rates are not independent of the established population. Coexistence (or exclusion of the rare strategy) may be mediated by a sufficiently strong fluctuation, which is illustrated by a simple model elucidating the connection with resource-competition models. Moreover, some other consequences of strong fluctuation are demonstrated by the example of a lottery model, such as multiple ESS, ESS which cannot invade an established population, and historical events which determine the outcome of the evolution.

Most models of selection operating in fluctuating environments consider only rigid phe-notypes. Genetic models (Haldane & Jayakar 1963, Gillespie 1973, Karlin & Liberman 1974) are also incomplete because they refer only indirectly to the phenotype. Strategic (optimality) analyses, on the other hand, usually look for an optimal fixed phenotypic compromise, presumably established by selection when each of the possible environmental states would favour a different type (Levins 1968a, Schaffer 1974, Oster & Wilson 1978, Real 1980, León 1983, Brown & Venable 1986, Yoshimura & Clark 1991). This optimum is obtained as follows. Any given phenotype X would exhibit an array of fitnesses W _y (X) corresponding to different environmental states Y. The type endowed with maximal geometric mean M (or logarithmic expectation E(ln W) of the fitnesses, would eventually prevail (Gillespie 1973, León 1985, Frank & Slatkin 1990, Yoshimura & Clark 1991). This outcome is valid only in populations without age structure or density dependence since the presence of these factors would complicate the analysis (Caswell 1983, Buhner 1984, 1985; Elmer 1985a, 1985b; Goodman 1984, Tuljapurkar 1982, 1989). The second order Taylor expansion of the logarithmic expectation gives (Gillespie 1977, León 1983, Real 1980) the approximation: where W is fitness, E(W) its expectation and V(W) its variance. Therefore, the constrained maximum of E(ln W) will be a compromise between maximizing the mean E(W) and minimizing the variance V(W).

The interplay of demographic and environmental variance is considered in a behavioral context involving foraging in the presence of uncertain, fluctuating risk of predation. The evolutionary contest between two strategies — high risk and high fecundity versus low risk and low fecundity — is considered. Often the risky strategy will be dominant when it is sufficiently abundant in the population. Demographic (sampling) variance, however, implies that either strategy may be at a disadvantage when rare. In a phenotypic model the stable strategy may be determined by initial conditions and by chance. In a genetic model, polymorphism may be maintained if the risky strategy is heterozygotic.

We review some methods for quantifying environmental predictability and discuss their advantages and disadvantages. Colwell’s (1974) index can be used to give an absolute measure of predictability, which can be partitioned into constancy and contingency (seasonality), and is applied to time series of categorical data. Spectral analysis is useful for measuring periodicity or seasonality of environments, and requires time series of continuous data. We compare the behavior of these measures for 15 wetland sites in tropical and subtropical portions of North and South America. Estimates of seasonality and predictability based on spectral densities were related positively to estimates of predictability and contingency, and to some extent negatively with constancy, derived from Colwell’s index. These methods performed better than simple statistical measures of variation such as the standard deviation or coefficient of variation. Studies seeking to characterize environmental variation and explore its implications for life-history evolution should employ these time series approaches.

Conditional strategies, in which individuals can exercise different tactics depending on the circumstances, are believed to be adaptive when the environment varies stochastically. We discuss a genetic model that treats such strategies as threshold traits. Examples are provided that illustrate how the model can be used to examine the effects of selection on conditional strategies when the environment varies spatially. The implications of the model for maintenance of conditional strategies in stochastic environments are discussed.

The optimal allocation of time and effort by foraging animals for searching, sampling and learning the distribution of food in patches is modelled. The optimal effort maximises the net gain, which is the difference between the benefit and the cost of searching. The optimal search effort increases as a function of the variance of quality between the patches, and the turnover of new patches. It decreases as a function of the cost of searching, e.g. the movement cost between the patches.

The variance of quality in the patches decreases as a function of the total search effort by all the foragers. A joint stable equilibrium of search effort and patch variance is reached when the variance reaches the level which is generated by the search effort which is optimal for all the foragers. The joint equilibrium search effort is a decreasing function of the population density, and of the search cost. It is an increasing function of the inherent variance in patch quality, and of the patch renewal rate. The equilibrium variance in patch quality is a decreasing function of the populaton density of the foragers, and of the search cost.

A rare type with a different optimal search as a function of the variance, behaves according to the variance equilibrium with the common type. The optimal search effort of rare types diverges from that of the common type more than it would if they were on their own. In heterogeneous populations with several different types, the equilibrium search effort of each type is the optimal search effort at the variance generated by the total search effort of all the types.

Coexistence is possible between species with high information gathering and high energy reqirements, which utilise the richer newly discovered food sources, and species with low information gathering and low energy demands, which utilise poorer depleted food sources.

A small average number of foraging visits per patch generates a stochastic distribution of patch quality even in inherently uniform patches. In such cases, the equilibrium search effort and patch variance may be high, and may be determined entirely by this stochastic generation of variance.