Graeme D. Ruxton Division of Environmental & Evolutionary Biology, Graham Kerr Building, University of Glasgow, Glasgow G12 8QQ, UK. Tel: +44 141330 6617; fax: +44 141330 5971; e-mail: g.ruxton@bio.gla.ac.uk

Abstract

Previous models of kleptoparasitism (resource stealing) assume that contests over resource items are of fixed duration. Here we suggest that such contests will often be well represented as a war of attrition, with the winner being the individual who is prepared to contest for the longer time. Given that time spent in contests cannot be used to search for other resource items, we provide an analytical expression for the evolutionarily stable distribution of contest times. This can be used to investigate the circumstances under which we would expect kleptoparasitism to evolve. In particular, we focus on situations where searching for conspecifics to kleptoparasitize can only be achieved at a cost of reduced resource discovery by other means; under such circumstances we show that kleptoparasitism is not evolutionarily stable.

Kleptoparasitism (stealing resources harvested by another) is widespread among vertebrates, especially birds (see Brockman & Barnard (1979) and Furness (1987) for reviews). Recently, Broom & Ruxton (1998) described a mechanistic simple model of foraging, and found the optimal kleptoparasitic strategy as a function of parameter values. A key assumption of the model, and of other recent models of kleptoparasitism by Ruxton et al. (1992 ), Holmgren (1995), Ruxton & Moody (1996) and Stillman et al. (1997 ), is that all contests last for an amount of time outside the control of the two contestants. This may be a reasonable approximation for some types of interaction – for example, where contests are settled by performance in a ritual display of defined moves – but will not be realistic for others. Many types of contests could be considered as trials of endurance, where the winner is the individual who is able, or prepared, to last the longest; examples of these include the pushing contests seen in many horned ungulates, in contests between male elephants, and in male–male conflict in ploughshare tortoises. Such contests do not last a fixed length of time, but rather are under the control of the contestants, the trial lasting for the lesser of the times which the two are prepared to invest in it. This is often called a ‘war of attrition’. We would expect that selection will act so that individuals invest an appropriate amount of time in such contests, since not fighting for long enough will deny them access to resources, but fighting for too long when they encounter a similarly stubborn individual would waste time and energy which could be better spent in other activities. Here we will consider this situation for the first time and quantitatively describe the evolutionarily stable strategy (ESS) for a simple kleptoparasitic situation analogous to previous models. The ESS will be a combination of the probability of entering into a contest over a resource item, and the distribution of the lengths of time for which the individual is prepared to compete. In our model, the resource is food, although the theory applies equally well to other resources (e.g. mating opportunities).

The foraging model

We consider a population of foragers with constant population density. At any point in time, each individual within the population will be engaged in one of three mutually exclusive activities: searching for food items or opportunities to kleptoparasitize, handling an item or fighting over a discovered item. The rates at which searchers encounter prey and handling conspecifics are, respectively, fv_{f} and Hv_{H}, where f and H are the population densities of food items and handlers, respectively (dimensionally an inverse area), and v_{f} and v_{H} are constant rates at which area is searched for these food and handlers, respectively (dimensionally an area divided by time). We assume that both f and H can be considered as approximately constant over the behavioural time-scale of interest to us. Effectively, we assume that prey depletion is negligible in the situation under study. Upon finding a food item, a searcher switches to being a handler. It now requires an uninterrupted period in which to handle this food item before being able to ingest it. In common with previous workers, we will assume that this time is drawn from an exponential distribution with a fixed mean value (our results do not critically depend on this assumption, but see Lendrem, 1983; Lendrem et al., 1986 ; for a biological justification for this distribution). If, during the handling time, a searching individual challenges the handler for its food item, then there is a contest, which lasts for a time drawn randomly from a probability distribution with mean t_{a}/2 (the factor of one half is introduced to keep subsequent calculations as tidy as possible). After the contest, one individual (the loser) returns to searching, whilst the other (the winner) returns to handling. We assume that previous handling of the item has had no effect on it; whenever a handling bout is interrupted by a contest, afterwards handling must begin again. An implicit assumption made when adopting an exponential distribution of handling times is that the probability of finishing handling is constant per unit time, and so is unaffected by previous handling. First we will review the case where t_{a} is fixed, and each participant in a fight is equally likely to emerge the winner or loser. We then consider the novel case where contest time is a phenotypic trait which can be controlled by individuals. In each case, we find the evolutionarily stable kleptoparasitic strategy assuming individuals seek to maximize their food intake rate.

The best strategy when fights are of fixed duration

Broom & Ruxton (1998) examined a similar model with the simplifying assumption that t_{a} was fixed, and that the distribution of contest durations was exponential. They demonstrated that when t_{a}fv_{f} > 1, searchers should always turn down opportunities to attempt kleptoparasitism. In contrast, when t_{a}fv_{f} < 1, searchers should always take opportunities to fight for food. In the boundary case, where t_{a}fv_{f} = 1, accepting or turning down the chance to steal food both result in acquiring equivalent items of food, in exactly the same average time, and so it does not matter which strategy is employed. This makes intuitive sense, as increasing the duration of contests (increasing t_{a}) should make kleptoparasitism a less attractive strategy, and similarly increasing the ease with which food can be found independently (increasing fv_{f}) should also make investing time in contests less attractive.

How should individuals decide how long to fight for?

Previously, we defined a certain distribution of contest durations: exponential with mean t_{a}/2. We also assumed that at the end of each contest, the winner was decided by a simple coin-toss. Let us now imagine that each individual has control over how long it is prepared to continue each fight. For each contest, the two contestants select the time for which they are prepared to fight. The one that selects the longer time wins the contest. If individuals pick too low a time, then they rarely win contests; conversely if they pick too big a value, then they often win but at the cost of investing a large amount of time (when they encounter other ‘stubborn’ individuals) which could be spent searching for food. Here we wish to find the optimal value. As we have set it up, each contest falls into the well studied context of a ‘war of attrition’ ( Maynard Smith, 1974; Bishop & Cannings, 1978). Formally, the war of attrition is formulated as follows. Two animals compete for a resource of value V. Each selects the time for which it is prepared to fight, the contest being resolved when the animal which selected the shorter time quits; the other animal gains the reward. If each selects the same time, then each has an equal chance of obtaining the reward. Thus the expected net payoff to an individual which selects time m_{1} in contests where the opponent selects time m_{2} is

where k is a constant which converts time invested in a contest to the same currency as the reward. The evolutionarily stable strategy (ESS) of such a game is unaffected if the pay-off matrix is divided by a constant, since all the pay-offs are being rescaled in the same way. Hence for notational convenience, we choose to work with the equivalent payoff matrix, obtained by dividing eqn 1 by k,

Bishop & Cannings (1976) give the ESS of the above game as a probability distribution of playing for a time t given by

That is, the evolutionarily stable strategy is to select a time (independently for each contest) from an exponential distribution with mean time (V/k).

Consider a searching individual which has just discovered another handling a food item. It has the option of attempting to steal the handler’s food item. If it does so, then both individuals enter a war of attrition. The reward for winning this contest is to become a handler; we define this as having value V. The cost of this interaction is the lost opportunity to gain food by finding it independently. Food items are found at rate fv_{f}, so the cost is the value of the food that could be found in a time t which is fv_{f}Vt. Thus k = fv_{f}V, and so the ESS of the game is an exponential distribution of times with mean

The lengths of contests will be distributed as the shorter of independent pairs of times drawn from this distribution. A well-known mathematical result (e.g. Grimmett & Welsh, 1986) is that this itself is an exponential distribution with mean equal to one half of that of the original distribution. Thus contest durations will be exponentially distributed with mean 1/(2 fv_{f}). This is a special case of the model we described earlier, since the contests follow the same (exponential) distribution and both players have an equal chance of winning. In the model definition, the mean duration of contests was defined as t_{a}/2. This means that the optimal strategy is the one for which

i.e. when t_{a}fv_{f} = 1. This corresponds to situations where accepting or declining the chance to steal food results in the same long-term feeding rate.

It is interesting to note that in our original model, we arbitrarily chose the distribution of encounter durations to be exponential. A similar choice was made by Ruxton et al. (1992 ), Holmgren (1995) and Ruxton & Moody (1996). In contrast, in the derivation above we make no such assumption, but find that the exponential distribution arises naturally as the optimal solution for individuals with a free choice over the time for which they are prepared to contest for food items.

What if there is a trade-off between finding food and finding handlers?

Until this point, we have implicitly assumed that searching for food items and for handlers are separate activities which can be varied independently, since v_{f} and v_{H} are separate constants. In fact, it seems likely that the effectiveness of these related activities will often be linked, so that individuals will only be able to enhance their effectiveness at one at the expense of the other (see Broom & Ruxton, 1998, for a fuller discussion). Let us now assume that both v_{f} and v_{H} are bounded above, and further that v_{H} can only be increased at the expense of decreasing v_{f}: Mathematically, we express this by fixing two constants, α and β, such that

We now assume that individuals are free to vary v_{f}, so that an individual’s strategy is now defined by the triple {v_{f}, p, g(t)}, where p is the probability of attempting kleptoparasitism when a handler is discovered and g(t) is the probability density function of the time an individual is prepared to fight for. Once v_{f} is chosen, v_{H} is obtained from the equation above.

We know that for any fixed v_{f}, the ESS distribution g(t) is given by

So we need only consider finding the optimal combination of p and v_{f}. Consider the case where p = 0, then individuals never encounter handlers, and so should maximize their own food finding abilities by playing v_{f} = α. Is this an ESS strategy? Firstly we note that when v_{f} = α, then (since handlers are never detected) all values of p yield the same return. Now consider a situation where every individual plays v_{f} = α, except one mutant individual who plays some v_{f} < α. How does this individual fare compared with the rest? We assume that the population is large, so that, for any one individual, contests with the mutant have a negligible effect on the performance, and average time required to find each food item is 1/fα. Clearly, since it plays some v_{f} < α, the mutant’s ability to find food for itself is inferior to the others. However, providing it selects some p≠ 0, then it can also obtain food through contests. We can calculate the mean time that it must invest in contests to obtain a food item in this way (assuming the mutant plays a time τ in a given case) as the expected contest duration divided by the probability of winning:

Notice that this holds for all τ, and so no matter what probability distribution the mutant individual uses to generate its contest times, its average time to obtain a food item through contests is 1/fα. Hence the mutant’s ability to obtain food from contests is identical to that of the other individuals finding food for themselves; however, the mutant is less efficient than the others at finding food of its own. Since the mutant’s mean time to obtain a food item when it is in a contest is the same as the others’, but its mean time to obtain food when it is not in a contest, i.e. when it is searching for food and/or other individuals to challenge is longer, it does worse overall. Thus v_{f} = α is an evolutionarily stable strategy (ESS). Further, it is easy to see that it is the only ESS. The only other possible candidates are strategies with v_{f} < α and g(t) = fv_{f}exp(–fv_{f}t). We know that for this value of g(t), all choices of p yield the same return, i.e. {v_{f}, p, g(t)} gives the same return as {v_{f}, 0, g(t)} for all p. But we have already shown that the return from playing {α, p, g(t)} is greater than that of playing any other strategy {v_{f}, p, g(t)}. Hence v_{f} = α is the only ESS, and we conclude that when contests are settled by wars of attrition and there is a trade-off between discovering food items and discovering handlers, individuals should devote themselves purely to searching for food and so kleptoparasitic interactions will not occur.

If we have a resident population with v_{f} = α, then, in the absence of mutants, aggressive encounters never occur. Thus there is no selection pressure on behaviour in aggressive interactions. Hence, under such circumstances, we might expect drift in the population in traits such as those governing giving-up times in contests. Thus, occasional contests sparked by mutant individuals are required to provide the selection pressure for individuals to retain traits leading to optimal behaviour in contests.

Discussion

We have shown here that the form of kleptoparasitic contests in a given species has fundamental implications for the fitness benefits of this behaviour. If kleptoparasitic contests are of fixed duration, then, as we have shown earlier, ( Broom & Ruxton, 1998) under almost all conditions individuals should either take every opportunity to kleptoparasitize or take none at all. In contrast, here we have shown that if the contest takes the form of a trial of endurance which can be well modelled by a war of attrition, then we predict that contests will last (on average) for the same amount of time as it takes on average for an individual to find a food item. Under such circumstances, accepting or declining an opportunity to kleptoparasitize will yield the same benefit. Most importantly, our model predicts that if individuals can increase their food finding ability at the expense of their ability to find kleptoparasitic opportunities, then it is optimal for them to do so as much as possible. Hence if individuals of a given species are subject to such a trade-off, and contests are well-described as a war of attrition, then we would not expect to observe kleptoparasitism. However, if such a trade-off does not occur, then kleptoparasitism may well be in evidence.

The model presented here is the first attempt to investigate the trade-offs which individuals will face when choosing to invest time or energy between finding resources of their own and stealing from others. Although simple, our model has produced a number of interesting (and testable) predictions, as discussed above. However, in common with the other models of kleptoparasitic behaviour described earlier, it makes several restrictive simplifying assumptions. Further extensions to the model which relax these assumptions would be interesting and biologically realistic. We have assumed that there are no costs to contests other than the time investment, although other potential costs (e.g. increased energetic expense or risk of injury) could be added to the model relatively easily. Our assumptions that all individuals are intrinsically identical and that the population is well mixed could also be relaxed, but only at the expense of making analysis of the model considerably more difficult. However, between individuals differences are to be expected in nature, and so such extensions would be particularly appropriate. Technical difficulties notwithstanding, such extensions to the model, allied to carefully controlled experimentation, would be worthwhile, and should considerably increase our understanding of the evolution and operation of kleptoparasitism as a life-history strategy.

Another useful extension of individual-level models of kleptoparasitism such as the one presented here would be to build a bridge between them and the considerable body of literature on group-level ‘producer–scrounger’ models ( Ranta et al., 1998 , and references therein). Producer–scrounger models consider a finite group of identical individuals, each of which can adopt one of two fixed strategies: producer or scrounger. Producers find food for themselves but have to share their discoveries with scroungers. Workers generally specify the characteristics of both strategies, then search for the equilibrium composition such that no individual could improve their fitness by switching strategy. Introducing explicit consideration of trade-offs faced by individuals, such as that considered here, would allow evolutionarily stable strategies to be found for foragers free to choose from a wide suite of options, rather than simply between two fixed ones.

Footnotes

Mark Broom is also a member of the Centre for the Study of Evolution, University of Sussex.