An evolutionary maximum principle for density-dependent population dynamics in a fluctuating environment.

The evolution of population dynamics in a stochastic environment is analysed under a general form of density-dependence with genetic variation in r and K, the intrinsic rate of increase and carrying capacity in the average environment, and in sigma(e)(2), the environmental variance of population growth rate. The continuous-time model assumes a large population size and a stationary distribution of environments with no autocorrelation. For a given population density, N, and genotype frequency, p, the expected selection gradient is always towards an increased population growth rate, and the expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Long-term evolution maximizes the expected value of the density-dependence function, averaged over the stationary distribution of N. In the theta-logistic model, where density dependence of population growth is a function of N(theta), long-term evolution maximizes E[N(theta)]=[1-sigma(e)(2)/(2r)]K(theta). While sigma(e)(2) is always selected to decrease, r and K are always selected to increase, implying a genetic trade-off among them. By contrast, given the other parameters, theta has an intermediate optimum between 1.781 and 2 corresponding to the limits of high or low stochasticity.