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Optimal foraging of a reproducing animal as a discounted reward problem

Published online by Cambridge University Press:  14 July 2016

S. Merad*
Affiliation:
University of Newcastle upon Tyne
J. M. McNamara*
Affiliation:
University of Bristol
*
Postal address: Department of Mathematics and Statistics, University of Newcastle, NEI 7RU, UK.
∗∗ Postal address: School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK.

Abstract

A model in which a foraging animal can reproduce if its energy reserves reach a critical level is presented. The animal's reserves are modelled as a finite-state Markov chain. The animal has the choice between foraging options which may have the same mean net gain but differ in their variances. In addition to starvation, animals are subject to death because of predation, bad weather, and so on. We focus on the case where the rate of mortality due to these sources is the same under all options. We investigate policies that maximise the expected lifetime reproductive success. It is found that the optimal value function is concave at low reserves (low-variance action region) and convex at high reserves (high-variance action region). The value function under the low-variance action has also the same shape and the same inflexion point. This result allows us to compute optimal policies just by looking at the low-variance value function. The result is also used to show that increasing the mortality rate increases the high-variance region under the optimal policy. The pattern of risk-sensitive behaviour predicted by this model is in contrast to that predicted by a similar model in which no reproduction occurs and the optimality criterion is to minimise the probability of death (McNamara (1990)).

MSC classification

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1994 

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References

Blackwell, D. (1967) Positive dynamic programming. Proc. 5th Berkeley Sym. Math. Statist. Prob. 1, 415418.Google Scholar
McNamara, J. M. (1990) The policy that maximizes long-term survival of an animal faced with the risks of starvation and predation. Adv. Appl. Prob. 22, 295308.Google Scholar
McNamara, J. M. and Houston, A. I. (1992) Risk-sensitive foraging: a review of the theory. Bull. Math. Biol. 54, 255378.Google Scholar
McNamara, J. M., Merad, S. and Houston, A. I. (1991) A model of risk-sensitive foraging for a reproducing animal. Anim. Behav. 41, 787792.Google Scholar
Ross, S. M. (1982) Stochastic Optimization. Wiley, New York.Google Scholar
Whittle, P. (1983) Optimization Over Time, Volume 2. Wiley, Chichester.Google Scholar