Mathematical Analysis of a Model Describing Evolution of an Asexual Population in a Changing Environment

M. Broom*, Q. Tang* and D. Waxman‡

Mathematical Biosciences 186: 93-108 (2003)

*School of Mathematical Sciences, University of Sussex, Brighton BN1 9QH, UK
‡Centre for the Study of Evolution, School of Life Sciences, University of Sussex, Falmer, Brighton BN1 9QG, United Kingdom.

We investigate a mathematical model for an asexual population with non-overlapping (discrete) generations, that exists in a changing environment. Sexual populations are also briefly discussed at the end of the paper. It is assumed that selection occurs on the value of a single polygenic trait, which is controlled by a finite number of loci with discrete-effect alleles. The environmental change results in a moving fitness optimum, causing the trait to be subject to a combination of stabilising and directional selection. This model is different from that investigated by Waxman and Peck [23] where overlapping generations and continuous effect alleles were considered. In this paper, we consider non-overlapping generations and discrete effect alleles. However in both the work of [23] and the present work, there is the same pattern of environmental change, namely a constant rate of change of the optimum. From the work of [23], no rigorous theoretical conclusion can be drawn about the form of the solutions as t grows large. Numerical work carried out in [23] suggests that the solution is a lagged travelling wave solution, but no mathematical proof exists for the continuous model. Only partial results, regarding existence of travelling wave solutions and perturbed solutions, have been established (see [18] and [21]). For the discrete case of this paper, under the assumption that the ratio between the unit of genotypic value and the speed of environment change is a rational number, we are able to give rigorous proof of the following conclusion: the population follows the environmental change with a small lag behind, moreover, the lag is represented using a calculable quantity (Section 6). A remark about this assumption is made at the end of Section 4.