John M. McNamara

The Common Currency for Behavioral Decisions

The study of optimal life histories involves the maximization of lifetime fitness but usually ignores the details of behavioral sequences. In contrast, the study of optimal behavioral sequences usually looks at the details of a sequence in isolation and not as part of the whole life history of the animal. The currency that is maximized is assumed to be related to lifetime fitness, but this relationship is rarely explored. In this paper we develop a common currency for behavioral decisions that is directly related to lifetime fitness. The common currency makes it possible to compare the benefits of qualitatively different behaviors. We show that many different costs can be used to explain a given behavioral sequence. Most of these costs have no biological interpretation. From these we single out a particular cost, the canonical cost, which measures the reduction in fitness that results from choosing a suboptimal action. Our general framework is illustrated by the example of a small bird in winter. We quantify the value of energy in terms of fitness and show how this value depends on energy reserves and time of day. As a result of this dependence, optimal foraging decisions depend on energy reserves and time of day, as does the optimal trade-off between foraging and looking around for predators.

Starvation and Predation as Factors Limiting Population Size

We consider a simple model in which an animal can control both its prob- ability of starvation and its probability of predation. Probability of starvation is decreased by increasing the mean amount of food obtained in the day, but this increases the probability of predation. The optimal mean gain minimizes the total mortality. It is shown that as the amount of food that is required per day increases, the probabilty of starvation does not necessarily increase, and may actually decrease. This result arises because as the food requirement increases, the animal increases its predation risk in order to avoid starvation. The results suggest that it is inappropriate to argue that food alone or predation alone limits the size of a population when there is a strong interaction between them. Furthermore, the number of animals that die from starvation may not provide a reliable indication of the importance of food.

The sexual selection continuum

The evolution of mate choice for genetic benefits has become the tale of two hypotheses: Fishers ‘run–away’ and ‘good genes’, or viability indicators. These hypotheses are often pitted against each other as alternatives, with evidence that attractive males sire more viable offspring interpreted as support for good genes and with a negative or null relationship between mating success of sons and other components of fitness interpreted as favouring the Fisher process. Here, we build a general model of female choice for indirect benefits that captures the essence of both the ‘Fisherian’ and ‘good–genes’ models. All versions of our model point to a single process that favours female preference for males siring offspring of high reproductive value. Enhanced mating success and survival are therefore equally valid genetic benefits of mate choice, but their relative importance varies depending on female choice costs. The relationship between male attractiveness and survival may be positive or negative, depending on life–history trade–offs and mating skew. This relationship can change sign in response to increased costliness of choice or environmental change. Any form of female preference is subject to self–reinforcing evolution, and any relationship (or lack thereof) between male display and offspring survival is inevitably an indicator of offspring reproductive values. Costly female choice can be maintained with or without higher offspring survival.

Incorporating rules for responding into evolutionary games

Evolutionary game theory is concerned with the evolutionarily stable outcomes of the process of natural selection. The theory is especially relevant when the fitness of an organism depends on the behaviour of other members of its population. Here we focus on the interaction between two organisms that have a conflict of interest. The standard approach to such two-player games is to assume that each player chooses a single action and that the evolutionarily stable action of each player is the best given the action of its opponent. We argue that, instead, most two-player games should be modelled as involving a series of interactions in which opponents negotiate the final outcome. Thus we should be concerned with evolutionarily stable negotiation rules rather than evolutionarily stable actions. The evolutionarily stable negotiation rule of each player is the best rule given the rule of its opponent. As we show, the action chosen as a result of the negotiation is not the best action given the action of the opponent. This conclusion necessitates a fundamental change in the way that evolutionary games are modelled.

The value of fat reserves and the tradeoff between starvation and predation

It is shown that in a range of models, the probability that a forager dies from starvation is, to a good approximation, an exponential function of energy reserves. Using a time and energy budget for a 19g passerine, we explore the consequences, in terms of starvation and predation, of various levels of energy reserves. It is shown that there exists an optimal level L of reserves at which total mortality (starvation plus predation) is minimized. L increases when the environment deteriorates as a result of a decrease in either temperature or mean gross gain or an increase in the mean search time. The effect of combined deteriorations is greater than the sum of their individual effects. At L, the probability of predation is much higher than the probability of starvation. A simple analytic model suggests that this result will be fairly general, but also indicates conditions under which the result might not hold.

Optimal patch use in a stochastic environment

Abstract This paper considers an animal foraging on prey which are distributed in well-defined patches. It is assumed that the environment may be stochastic and that the animal can gain information on patch type as it forages. The foraging policy which maximises mean reward rate for the environment is characterised in terms of a function of state called the potential function. This policy is shown to be given by the rule: continue foraging on the present patch while the potential is positive, when the potential falls to zero move on to the next patch. Let r denote the current reward rate on a patch and let γ denote the maximum mean reward rate for the environment. It is shown that r ⩽ γ if it is optimal to leave. Conditions which ensure r γ are also given. For a large class of environments the optimal policy is stated in terms of a revised reward rate r , and is given by the rule: continue on the present patch while r > γ , when r falls to γ move on to the next patch. Finally, it is shown that the stay time on a patch is a decreasing function of γ.

A framework for the functional analysis of behaviour

We present a general framework for analyzing the contribution to reproductive success of a behavioural action. An action may make a direct contribution to reproductive success, but even in the absence of a direct contribution it may make an indirect contribution by changing the animals state. We consider actions over a period of time, and define a reward function that characterizes the relationship between the animals state at the end of the period and its future reproductive success. Working back from the end of the period using dynamic programming, the optimal action as a function of state and time can be found. The procedure also yields a measure of the cost, in terms of future reproductive success, of a suboptimal action. These costs provide us with a common currency for comparing activities such as eating and drinking, or eating and hiding from predators. The costs also give an indication of the robustness of the conclusions that can be drawn from a model. We review how our framework can be used to analyze optimal foraging decisions in a stochastic environment. We also discuss the modelling of optimal daily routines and provide an illustration based on singing to attract a mate. We use the model to investigate the features that can produce a dawn song burst in birds. State is defined very broadly so that it includes the information an animal has about its environment. Thus, exploration and learning can be included within the framework.

A theoretical investigation of the fat reserves and mortality levels of small birds in winter

We assume that the fat levels of a small bird in winter reflect a tradeoff between starvation and predation. This tradeoff is formalized by finding the level of fat that minimizes total mortality (starvation plus predation) in a given environment. A bird is characterised in terms of its level of energy reserves. The bird starves if these reserves fall to zero. In general, the probability that the bird is killed by a predator increases with increasing reserves. We consider two models. In both at each time unit during a day a bird can choose, as a function of its reserves, to forage or to rest. In the forage vs rest model there is only one foraging option. In the foraging intensity model the bird can choose from a range of options, where options with a high energetic gain also incur a high predation risk. We find the optimal level of reserves for various environments, together with the resulting levels of starvation and predation. Unless food availability is very high, an improvement in overall availability results in a decrease in the mean reserves at dusk (MRD); in the foraging intensity model there is also a trend towards choosing safer foraging options with lower mean gains. An increase in variability, either as a result of a decreased probability of finding food or an increase in interruptions of foraging, increases MRD. The increase in reserves is not sufficient to prevent an increase in starvation. As a result, reserves and starvation can be positively correlated across environmental conditions. The level of starvation tends to be lower than the level of predation, but the ratio of starvation to predation tends to increase as conditions become worse. In the middle of winter, the days are short and the nights are long and cold. The optimal response to a decrease in daylength involves an increase in both dawn and dusk levels of reserves. This pattern is also found when overnight expenditure increases but daylength is constant. In this context we show that when temperatures are very low, a small saving in energy can result in a substantial increase in survival probability. The relevance of this result for evaluating the importance of hypothermia is discussed.

Evolutionarily stable levels of vigilance as a function of group size

An animals level of vigilance is usually interpreted as a trade-off between gaining food and reducing the danger of predation. In the context of a group of animals, vigilance has been analysed using the evolutionarily stable strategy (ESS) concept. In this paper a general ESS model of vigilance as a function of group size is developed. The model is based on an explicit foraging process that may be terminated prematurely by events such as bad weather that are outside the animals control. It can be used to investigate the effect of both environmental parameter, such as the rate of attack by predators and the rate of food intake, and the animals internal state on evolutionarily stable levels of vigilance. The biological assumptions underlying other models of vigilance are explored, demonstrating why some factors influence the level of vigilance in some models but not in others. Pulliam et al. (J. theor. Biol., 1982, 95, 89–103) presented data on vigilance as a function of group size that they found hard to explain in terms of an ESS model. This paper introduces various modifications to their model and shows that a reasonable fit to the data can be obtained.

Optimal foraging and learning

Optimal foraging thoery usually assumes that certain key environmental parameters are known to a foraging animal, and predicts the animals behaviour under this assumption. However, an animal entering a new environment has incomplete knowledge of these parameters. If the predictions of optimal foraging theory are to hold the animal must use a behavioural rule which both learns the parameters and optimally exploits what it has learnt. In most circumstances it is not obvious that there exists any simple rule which has both these properties. We consider an environment composed of well-defined patches of food, with each patch giving a smooth decelerating flow of food ( Charnov, 1976 ). We present a simple rule which (asymptotically) learns about and optimally exploits this environment. We also show the rule can be modified to cope with a changing environment. We discuss what is meant by optimal behaviour in an unknown and possibly changing environment, using the simple rule we have presented for illustrative purposes.