## Introduction

Wright's adaptive topography is one of the conceptual foundations of evolutionary biology (Wright, 1932, 1937, 1969; Simpson, 1953; Lande, 1976, 1979, 1982; Kauffman, 1993; Arnold *et al.*, 2001; Gavrilets, 2004). Representing a population as a point on the surface of mean fitness as a function of allele frequencies at one or more genetic loci, assuming random mating, weak selection and loose linkage (promoting approximate Hardy–Weinberg equilibrium within loci and linkage equilibrium among loci), natural selection in a constant environment causes a population to evolve uphill on this adaptive topography, increasing the mean fitness in the population. The utility of this concept has been questioned, particularly in fluctuating environments when the population appears at best to be chasing a continually moving optimum (Fisher, 1958, pp. 44–45; Crow & Kimura, 1970, p. 236; Gillespie, 1991, p. 305).

The theory of population growth in a random environment was developed by demographers ignoring genetic variation. For a large density-independent population subject to a stationary distribution of environmental states, the expected rate of increase in ln *N*, the natural logarithm of population size, is known as the long-run growth rate (Cohen, 1977, 1979). With small fluctuations in age-specific vital rates of reproduction and mortality, this can be approximated as , in which *r* is the population growth rate in the average environment, and is the environmental variance in population growth rate (Appendix 1). Under these conditions, all population trajectories eventually achieve the same slope, (Cohen, 1977, 1979; Tuljapurkar, 1982; Caswell, 2001; Lande *et al.*, 2003).

Combining the above genetic and demographic theories, I previously demonstrated for allele frequency in a single-locus haploid (asexual) population, *p*, and for the mean phenotype of quantitative characters in a sexual population, , that the expected evolution is governed by the gradient of the long-run growth rate of the population, , with respect to *p* or (Lande, 2007). Stochastic demographic theory describes density-independent growth of an age-structured population, but the main theories of fluctuating selection (reviewed in Gillespie, 1991) do not incorporate age structure. For simplicity, the present theory describes a population without age structure undergoing density-dependent population growth, but assuming that selection does not depend on population density. Extensions to density-dependent selection and age-structure will appear elsewhere.

For diploid genetic systems in a randomly mating population, I demonstrate here that the long-run growth rate again provides an adaptive topography governing the expected evolution of allele frequencies. The evolution of such a population represented as a point on this surface is expected to produce a stochastic maximization of , which therefore constitutes an adaptive topography that does not change with time despite environmental fluctuations. However, the actual dynamics and stability of the process depend on the genetic system, the form of selection, and the amount of stochasticity in particular cases (Turelli, 1981; Gillespie, 1991; Lande, 2007). This is established by the theory developed below, and illustrated by particular examples including the Haldane–Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.