Dynamic optimization in fluctuating environments

McNamara, J.M., Webb, J.N. and Collins, E.J.

Department of Mathematics, University of Bristol

Summary

We consider the problem of finding an optimal strategy for an organism in a large population when the environment fluctuates from year to year and cannot be predicted beforehand. In fluctuating environments, geometric mean fitness is the appropriate measure and individual optimisation fails. Consequently, optimal strategies cannot be found by stochastic dynamic programming alone. We consider a simplified model in which each year is divided into two non-overlapping time intervals. In the first interval, environmental conditions are the same each year; in the second, they fluctuate from year to year.

During the first interval, which ends at time of year $T$, population members do not reproduce. The state and time dependent strategy employed during the interval determines the probability of survival till $T$, and the probability distribution of possible states at $T$ given survival. In the interval following $T$, population members reproduce. The state of an individual at $T$ and the ensuing environmental conditions determine the number of surviving descendants left by the individual next year.

In this paper, we give a general characterisation of optimal dynamic strategies over the first time interval. We show that an optimal strategy is the equilibrium solution of a (non-fluctuating environment) dynamic game. As a consequence, the behaviour of an optimal individual over the first time interval maximises the expected value of a reward $R^{*}$ obtained at the end of the interval. However, $R^{*}$ cannot be specified in advance and can only be found once an optimal strategy has been determined. We illustrate this procedure with an example based on the foraging decisions of a parasitoid.

Appeared as McNamara, J.M., Webb, J.N. and Collins, E.J. (1995). "Dynamic optimization in fluctuating environments". Proceedings of the Royal Society (Lond.) B, 261 pp 279-284.

John McNamara

School of Mathematics
University of Bristol
University Walk
Bristol. BS8 1TW

Telephone: +44 (0)117 928 7986
Fax: +44 (0)117 928 7999
Email: John.McNamara@bristol.ac.uk