*Heredit*y **35,** 67–74 (1975)

**Polymorphisms in cyclically-varying environments**

Thomas Nagylaki

**Summary**

We analysed both continuous and discrete two-allele models of cyclically-varying environments with an arbitrary degree of dominance. In continuous models, the gene frequency fluctuates with the period of the environmental oscillation. For the discrete case, the calculations were carried out to second order in selection. In contrast to the continuous models, and depending on the amount of dominance and the initial gene frequency, fixation is possible as well as polymorphism.

**1. INTRODUCTION**

THE maintenance of genetic variability in natural populations is a subject of considerable interest in population genetics and evolutionary theory. At any given time, some proportion of the polymorphisms in a population are transient. Many mechanisms have been proposed which preserve genetic diversity in the equilibrium state. Recurrent mutation will keep deleterious genes in a population at low frequencies, preventing their elimination by natural selection. Kimura and Ohta (1971, Chs. 8 and 9) discuss the amount of genetic variability maintained by mutation and random drift in panmictic and geographically structured populations. These authors also review some of the pertinent experimental data.

While mutation is obviously the ultimate source of all genetic heterogeneity, polymorphic equilibria may be stable without recurrent mutation. The well-known overdominant equilibrium is probably the most important example of such a situation. If a population occupies multiple ecological niches, under suitable conditions a stable polymorphism may occur (Levene, 1967). Heterogeneity will be maintained if rare genotypes are favoured by selection. Numerous cases of this type of frequency-dependent selection are analysed by Wright (1969, Ch. 5), who also treats many other selective mechanisms which can produce stable equilibrium, e.g. unequal selection in the two sexes, sex linkage, and meiotic drive (1969, Ch. 3).

Balanced polymorphisms may be due to time-dependent selection. Haldane and Jayakar (1963) proved that if one of two alleles is completely dominant to the other, both alleles will remain in the population if the arithmetic mean of the fitnesses of recessives (relative to those of dominants) in different generations exceeds unity and the geometric mean is less than one. Recently, Hartl and Cook (1973) have shown in a two-allele model that if the fitnesses are linear functions of a random variable and have the same mean, then natural selection favours the genotype with the smallest variance in fitness. In particular, there will be a stable polymorphism whenever twice the standard deviation of the heterozygote fitness is less than the sum of the corresponding homozygote standard deviations.

In this paper, we shall study the consequences of periodic variations in selection coefficients on a single autosomal diallelic locus in a population sufficiently large to permit neglect of random drift. An example of this type of phenomenon is the seasonal variation in the frequencies of different gene arrangements of Drosophila pseudoobscura discussed by Dobzhansky (1971, pp. 109-133). Larval crowding appears to favour the homokaryotype ST/STrelative to CH/CH, while its absence does the reverse. Thus, seasonal fluctuations in population size will induce regular oscillations in the selection coefficients. Allard and Workman (1963) have observed cyclic changes in selection coefficients in lima bean populations. Powell (1971) has reported that the average heterozygosity and the average number of alleles maintained at a locus in 13 experimental populations of Drosophila willistoni were higher in heterogeneous than in homogeneous environments. It is easy to see from his table 1 that this effect still exists if only the cage temperature is varied at weekly intervals between 19° and 25°C, while the other two factors, which provided spatial heterogeneity, are kept constant.

This problem was investigated theoretically some time ago by Dempster (1955) and Kimura (1955). Our aim is to test the generality of their results by analysing broader classes of models with continuous random births and deaths and with discrete, non-overlapping generations.

**Concluson**

The main qualitative conclusion of this paper is that in a continuous model with random births and deaths and regularly oscillating selection coefficients, polymorphism is maintained with the gene frequency following the environmental cycles through values depending on the initial gene frequency, while if generations are discrete and non-overlapping, cyclicallyvarying selection coefficients lead to stable or unstable equilibrium points independent of the initial gene frequency. The reason for the very different behaviour of the two models is that in the first case the gene frequency can "track" the environment continuously while in the second each environmental cycle operates on the population after a discrete change in gene frequency. The continuous time model applies accurately to some lower organisms such as bacteria, whereas the discrete, non-overlapping generations scheme represents well populations like annual plants. If a population with overlapping generations reproduces continuously, as do human beings, for example, we would expect the first model to be a better approximation. If generations overlap but there is a definite breeding season, as in many mammals, gene frequencies will change discretely, but in closer phase with the environment than in the second model. For detailed predictions, one must know the nature of the environmental fluctuation as well as the demographic structure of the population.

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