The loss of adaptive plasticity during long periods of environmental stasis

Adaptive plasticity allows populations to adjust rapidly to environmental change. If this is useful only rarely, plasticity may undergo mutational degradation and be lost from a population. We consider a population of constant size N undergoing loss of plasticity at functional mutation rate m and with selective advantage s associated with loss. Environmental change events occur at rate theta per generation, killing all individuals that lack plasticity. The expected time until loss of plasticity in a fluctuating environment is always at least tau, the expected time until loss of plasticity in a static environment. When mN > 1 and N theta >> 1, we find that plasticity will be maintained for an average of at least 10(8) generations in a single population, provided tau > 18/theta. In a metapopulation, plasticity is retained under the more lenient condition tau > 1.3/theta, irrespective of mN, for a modest number of demes. We calculate both exact and approximate solutions for tau and find that it is linearly dependent only on the logarithm of N, and so, surprisingly, both the population size and the number of demes in the metapopulation make little difference to the retention of plasticity. Instead, tau is dominated by the term 1/(m+s/2).