Richard F. Green

Bayesian birds: A simple example of Oaten's stochastic model of optimal foraging

Abstract Allan Oaten (1977 , Theor. Pop. Biol. 12 , 263–285) has argued that stochastic models of optimal foraging may produce results qualitatively different from those of the analogous deterministic models. Oatens model is very general and difficult to understand intuitively. In this paper a simple, tractable model is considered in which the predator searches each patch systematically (without going over the same area twice) until he exhausts the patch or decides the patch is not very good. It is assumed that each patch contains a fixed number of bits, each of which may contain a prey. The number of prey per patch is assumed to have a binomial distribution with n equal to the number of bits and p being a random variable having a beta distribution. After searching each bit the predator decides whether to leave the patch or not according to how many prey it has found. In this paper the best strategy is determined and the long-term rate of feeding is compared with that of the naive animal that searches each patch completely. The advantage of being a Bayesian is determined for a variety of environmental conditions.

Stochastic Models of Optimal Foraging

For many years, one of the central concerns of natural history was the food habits of animals; that is, what food animals eat, and how they go about obtaining it. The question was, How do animals forage? More recently, optimal foraging theory has posed the question, How should animals forage? This question may be asked in many forms, and the answers have cast light on the old subject of animals’ food habits. Optimal foraging theory has helped make the study of foraging more interesting, which may account for the theory’s remarkable popular success noted by Krebs, Stephens and Sutherland (1983).

Optimal Egg Distribution Among Host Patches for Parasitoids Subject to Attack by Hyperparasitoids

Using an aphid-parasitoid-hyperparasitoid system as an example, we study the optimal oviposition strategy of a parasitoid whose offspring are subject to attack by hyperparasitoids. We assume that hyperparasitoids, which search aphid colonies for aphids that have been infected by parasitoids, decide to leave a colony when they have searched some fixed number of aphids consecutively without finding an infected one. We use a simulation model to investigate how many hosts the parasitoid should infect per colony to maximize the long-term average rate of producing eclosing offspring. We consider three different variables and deal with them one at a time: (1) N = the number of aphids in a colony, (2) H = the average number of perparasitoids visiting each colony, and (3) τ = the parasitoid travel time between colonies. The optimal number of aphids to infect in a colony is sometimes much less than the total number of aphids available. The optimal number of aphids to infect within a colony decreases with a decrease in the colony size, with an increase in the average number of perparasitoid visits, and with a decrease in travel time between aphid colonies.

Outlier-Prone and Outlier-Resistant Distributions

Abstract Neyman and Scott [3] have considered the ideas of outlier-proneness and outlier-resistance of families of distributions. Under their definition, individual distributions (one-member families) cannot be outlier-prone. This paper offers definitions of outlier-proneness and outlier-resistance that apply to individual distributions, and theorems are given showing the connection with the classical laws of large numbers for maxima.

Selection for Fertility and Development Time

Volterras equation is solved explicitly for the growth rate in the case where reproduction is described by a gamma function, and the effects of small changes in development rate and fertility are derived in this special case. It is shown that the effects of such changes can be derived from Volterras equation without making specific assumptions about the schedule of reproduction. This analysis of Volterras equation is applied to correlations between development rate and fertility, and a model for r selection in bacteria is developed where correlations are precisely formulated.

Optimal foraging and sex ratio in parasitic wasps

Abstract Unlike many other animals whose sex ratios have been studied, parasitic wasps are able to determine the sex of their offspring. It is known that parasitic wasps sometimes produce different offspring sex ratios on different sized hosts. A model is constructed which includes the choice of accepting or rejecting a host as well as the choice of sex of offspring. The best reproductive strategy satisfies MacArthurs “product rule” for sex ratios and Charnovs “marginal value theorem” for optimal foraging. The model can be used to show that optimal sex ratio may vary with host density and size distribution.

Central-place foraging in a patchy environment

Early models of central-place foraging treated animals that search for prey in identical, homogenous patches. If patches vary in quality, then optimal foraging requires strategies based on time spent in a patch, and not simply on the type or number of prey found. In particular, a forager that takes no more than one prey from a patch should leave a patch after searching unsuccessfully for a certain fixed time. When patches are more variable, the forager should stay a shorter time in each patch, and the resulting rate of delivering prey to the central place will be lower. This implies that aggregation should be favored by prey faced with a single-prey-loading predator.

Partial attraction of maxima

There exist three classes of probability laws that are stable for maxima. A number of well-known distributions lie in the domains of attraction of these laws. This fact is sometimes exploited by fitting the distribution of maxima with one of the stable laws. Such a procedure may well be misguided, however, since distributions exist which produce maxima having any desired distribution and not just a stable type. In this paper partial attraction of maxima is defined and it is shown that all distributions have a non-empty domain of partial attraction of maxima. In fact, there exists a distribution that lies simultaneously in the domain of partial attraction of maxima of all distributions. STABLE LAWS; DOMAIN OF ATTRACTION; PARTIAL ATTRACTION; PARTIAL ATTRACTION OF MAXIMA

A simple Markov model for the assessment of host patch quality by foraging parasitoids

Abstract Insect parasitoids search for their hosts using a method that may be broken into three parts. First, they locate plants which may harbor their hosts, then they assess the quality of these plants to decide whether to search them further for hosts and, finally, if they decide to accept a plant for further search, they exploit the plant by searching for hosts and attacking them when they are found. We study the way that parasitoids assess plant quality by developing a mathematical model based on behavioral observations of foraging parasitoids that attack aphids which infest crucifers. Assessment of plants is based on the concentration of cues produced by hosts that inhabit them. Parasitoids are more likely to exploit plants on which more host cues are detected, and the willingness of a parasitoid to exploit a given plant depends on the quality of other plants that have been visited recently. Plants whose quality exceeds a certain threshold will be accepted for exploitation. The threshold for plant acceptance will change with the experience of the parasitoid, increasing when plants heavily infested with hosts are encountered, decreasing when uninfested plants are encountered. We analyze several rules that might describe how the acceptance threshold changes with parasitoid experience, and for each rule we show how the number of parasitoids willing to accept plants with various levels of infestation depends on the number of plants with various levels of infestation. We then consider different rules for exploitation of hosts on plants and find how the proportion of hosts attacked depends on host density.