A discrete-time model of phenotypic evolution

A model is proposed for the phenotypic evolution of a very large population under sustained environmental change and non-overlapping generations, with a single trait considered. Due to an extension of the standard law of quantitative inheritance, each evolutionary mechanism corresponds to a function between random variables associated with distinct stages of the life cycle. Such an approach leads to a two-dimensional map where the dynamics of the phenotypic mean and variance are directly connected. Then, the declining population paradigm is explored in terms of the critical rate of environmental change and using the techniques of the dynamical systems theory. Our results first reveal the opposing pressures on the phenotypic variance due to the conflict between phenotypic load and the ability to pursue the optimum, translated into an optimal value for maximizing the critical rate. Secondly, the introduction of development, through the particular case of linear plasticity, leads to a decreasing degree of stability with the magnitude of plasticity, which means that the recovery time from disturbances is harmed as the plastic effect intensifies, even though no constitutive costs have been assumed, a feature almost as important as the mean fitness to the viability of populations subject to persistent changes. Notwithstanding, the system is stable, and the growth rate benefits from increased plasticity, as expected.