A communication-based spatial model of antipredator ... - CiteSeerX

A communication-based spatial model of antipredator vigilance C.J.Proctor1 & M.Broom2 Centre for Statistics and Stochastic Modelling School of Mathematical Sciences University of Sussex G.D.Ruxton3 Division of Environmental & Evolutionary Biology University of Glasgow email [email protected] [email protected] [email protected] November 15, 2000 M.Broom and C.J.Proctor are also members of the Centre for the Study of Evolution at the University of Sussex.

1

Corresponding author Mark Broom School of Mathematical Sciences University of Sussex Falmer Brighton BN1 9QH Tel 01273 877243 Fax 01273 678097 email: [email protected]

2

Abstract Many animals have to spend their lives performing two often mutuallyexclusive tasks; feeding and watching out for predators (anti-predator vigilance). There have been many theoretical and empirical studies investigating this trade-o, especially for birds. An important factor for real birds is that of the area occupied by the ock. Individuals feeding close together experience increased competition so that the feeding rate decreases. Widely spaced individuals may suer a loss in vigilance eciency, since communication between individuals is more dicult, such that the predation risk increases. A vigilance model is developed which allows birds to control their spacing (and so the area of the ock) as well as their vigilance rate. The best strategy for the birds is found under a variety of environmental conditions, under two dierent assumptions; rstly the birds cooperate to make the group perform as well as possible; secondly each individual acts sel shly to maximise its own tness. Keywords foraging, predation, communication, spatial model, strategy, vigilance.

3

1 Introduction 1.1 Antipredator vigilance The lives of many animal species are principally divided between foraging for food, and avoiding predation by other animals. These activities are often mutually exclusive, so that extra eort in one reduces the eort available to the other. Thus when an animal forages for food, it must divide its time between feeding and being vigilant for predators. Many animals, especially birds, form groups to feed, and it has been observed that there is an inverse relationship between individual vigilance and group size. This can be explained by two separate eects. 1) The group vigilance hypothesis: when an individual detects a predator, it may pass on this information to its group mates either through an alarm call or sudden ight (Davis, 1975; Lazarus, 1979; Pulliam, 1973). 2) The individual risk hypothesis: members of the ock bene t from the `dilution' eect. The larger the group, the less likely a particular animal is to be killed (Dehn, 1990; Hamilton, 1971). The combination of these two eects increases the bene ts of group feeding even more (Dehn, 1990; Roberts, 1996) and the time saved through reduced vigilance can thus be devoted to foraging (or other activities). The tendency of birds to forage in groups has produced a large body of theory (for example, Broom & Ruxton, 1998; Lima, 1987; McNamara & Houston, 1992; Pulliam et al., 1982). Proctor and Broom (2000) developed a model which introduced the idea of the area used by the ock as a strategic variable under natural selection. This model generated a number of predictions. Under the model, there is always an optimal combination of ock area and vigilance level, which depends upon the environmental parameters chosen. The optimal area is hardly aected at all by some factors, such as the ability to spot predators, although this has a large eect upon the vigilance level. Other parameters, such as food density, aect the area greatly. A simple rule was developed to adapt the model for various bird species. A key assumption of this previous model was that the length of a vigilance scan increased as the area used by the ock increased, being proportional to the length of the perimeter of this region. Whilst this is plausible under certain circumstances (Bahr & Beko, 1999), it is not in others (Lima & Bedneko, 1999) where attacks come from only a single direction. However, 4

there is likely to be another cost to being spread out over a large area, namely poor communication between individuals of imminent attack. Poysa (1994) found that near individuals make better vigilance mates. In this paper we explore the consequences of such a cost.

1.2 Cooperative or sel sh behaviour? Pulliam et al. (1982), found the vigilance rate that maximises the probability of avoiding either predation or death through starvation. They considered two dierent situations: 1. Cooperative Optimum Here, all birds scan at the rate that maximises the tness for the group. This optimum is thought to be unstable as an individual can \cheat" by feeding more than its ockmates, unless the group is composed of close relatives (Grafen, 1979). 2. Sel sh Optimum This is the Evolutionarily Stable Strategy (ESS) (see Maynard Smith & Price, 1973), the scanning rate for which any deviating individual has a lower tness and so \cheating" does not pay. Pulliam et al. (1982)'s model predicted that sel sh birds scan less frequently than cooperative birds. But when they compared their model to data, they found that the observed scanning rate was closest to the cooperative optimum. This surprising result will be discussed in more detail later. Since Pulliam et al. (1982) most authors have considered either the group optimum (e.g. Lima, 1987) or the sel sh optimum, (e.g. McNamara & Houston, 1992) but not both. However, in a more general context, many authors have recognized that cooperation can evolve among unrelated, sel sh individuals if they repeatedly meet. Axelrod & Hamilton (1981) showed that in the Prisoner's Dilemma Game, the strategy `tit-for-tat' is evolutionarily stable if and only if the interactions between the individuals have a suciently large probability of continuing. However, this game only allows the discrete choice of cooperating or defecting, and so cannot be applied to vigilance behaviour. Roberts & Sherratt (1998) consider variable investment and show that cooperation can thrive through a new strategy which they call `raise-the-stakes'. This 5

strategy oers a small amount on rst meeting and then, if matched, raises its investment. This idea could be applied to vigilance behaviour. Suppose we start o with birds that scan at a low (sel sh) level, and then slowly increase this bit by bit only if they nd others around them doing the same. This raises the question - why should birds increase their vigilance? There are two possible explanations. Firstly, individuals may wish to preserve their ockmates, so that they are safer from predators in future attacks as a result of the dilution eect, see Lima(1989). Secondly, if predators are successful, they are more likely to attack the same species in the future (W.Cresswell, personal observation); therefore it is best for individuals to cooperate to prevent the attack rate from increasing. Proctor and Broom (2000) considered only cooperative equilibria, however in this paper we compare such equilibria with the equivalent sel sh ones. The second objective of our paper will be to identify the conditions under which the bene ts of cooperation are greatest.

1.3 Spatial factors Consider a ock of birds feeding in a region of area A. Just as birds have to nd the right balance between feeding and scanning, so there is a tradeo between utilising large and small areas. If the area is small, there will be less food available and so food gains will be small; however, individuals will be closer together and therefore safer from predators, because alarm calls will be easier to detect (but also see Proctor and Broom (2000) for an alternative explanation). Unlike Pulliam (1973), we will not assume that if one bird spots the predator then the rest of the group is also informed and all escape. Instead we will assume that any individual who spots the predator will successfully pass on the information to another individual with a probability PA , all such communications being independent, and that this probability will decrease as the area increases. The area of the ock will depend on the size of the species being considered and so it is necessary to specify this. We chose to develop our model for a

ock of lapwings; however, it is easy to adapt the model for dierent sized birds. We initially assume that the shape of the region is circular, but we include a shape parameter in the model which enables us to also consider elliptical regions. 6

2 The Model We assume that the predator targets a particular individual with each individual having the same probability of being chosen. The predator requires a certain time, ta, to initiate an attack and will be spotted by any individual in the ock which scans during this interval, scans being of length ts. For the probability that an individual does not spot the predator, denoted by Pns , we follow Pulliam et al. (1982)

Pns = ue?b u1 ? (

(1)

1)

where u is the proportion of time spent feeding, and b = ta=ts . If one individual spots the predator, then the probability that it is able to inform another of the attack is a decreasing function of the area occupied by the ock (A). Any particular individual may itself spot the predator with probability P = 1 ? Pns , or may be informed of an attack by another member of the ock with probability PPA , where PA is the probability of passing on the information. We now de ne PA as: PA = 1 + (1 A)d where and d are positive real numbers. In our model, the cost of feeding is the product of the attack rate, the cost of being targeted and being unaware of the attack, and the probability that the targeted bird is unaware of the predator Ptu . This probability is the product of the probability that an individual does not spot the predator and the probability that the individual is also not informed by any other member of the group. The rst term is simply Pns = (1 ? P ); the second term is (1 ? PPA )N ? , assuming that all individuals scan independently, so that 1

Ptu = (1 ? P )(1 ? PPA )N ?

1

Two separate models will now be considered; the group pay-o optimisation model or cooperative model and the individual pay-o optimisation or sel sh model.

7

2.1 The Group Payo Function In the cooperative model we will maximise the net gains from feeding for the group. Following Broom and Ruxton (1998), this is given by: Group payo = Fg ? yKPtu

(2)

where Fg is the rate of food gain for the ock; y is the predator attack rate and K is the cost of an attack if the target is unaware of the predator. This cost may be assumed to be equivalent to death; a rationale for the default value of K is given in Proctor & Broom (2000). Now Fg = Nu, where  is the rate of food uptake which will increase asymptotically with A but decrease with N . Following Proctor and Broom (2000), we de ne  as follows: 1  =  1 ? 1 + (A=NZ )m

!

where  is the maximum possible feeding rate, Z is the area required for a single bird to feed at half capacity and m is a constant. The expected payo is now given by: ! 1 f (u; A) = Nu 1 ? 1 + (A=NZ )m ? yK (1 ? P )(1 ? PPA )N ?

(3)

1

2.2 The Sel sh Pay-o Function We will use the approach of McNamara & Houston (1992) and consider the behaviour of a `mutant' individual which spends a proportion of time u feeding when other group members spend a proportion of time v. The mutant's best response is to feed for a proportion of time u that maximises the pay-o function for this individual. The payo for the mutant is: ! 1 Fv (u) = 1 ? 1 + (A=NZ )m u ? yK N1 (1 ? P )(1 ? QPA )N ? (4) 1

where Q = 1 ? ve?b v1 ? . (

1)

To maximise Fv (u) we dierentiate (4) with respect to u and set equal to zero, giving: 8

Fv0 (u) =

!

1 1 yKP 0(1 ? QP )N ? 1 ? 1 + (A=NZ  + A m ) N

1

!

1 1 yK (1 ? QP )N ? e?b u1 ? = 1 ? 1 + (A=NZ  ? A )m N 1

(

1)

b +1 u

!

(5)

At the ESS if the group is feeding for a proportion of time u, then the mutant's best response is also to feed for a proportion of time u. So we have: ! ! 1 1 ? b 1 N ? ? b  0 Fu (u ) = 1 ? 1 + (A=NZ )m  ? N yK (1 ? PPA ) e u u + 1 (6) 1

(

Rearranging equation (6), we obtain:

 = 1 + (A=NZ )m (1 ? PP )N ? e?b u1 ? A yK N (A=NZ )m 1

(

1)

b +1 u

1)

!

(7)

3 The Model Parameters The probability of passing on information about a predator, PA We need to estimate d and . Due to a lack of appropriate data, the choice of d was necessarily rather arbitrary. Figure 1 shows a plot of PA against A for d = 1; 2; 3; 4 for = 0:04, our eventual choice. d = 3 seemed to give a plausible shape to the graph (however, it turns out that the model is very robust to changes in d, see Figure 8). FIGURE 1 Hilton et al. (in press) studied escape ights of ocks of redshanks from sparrowhawk attacks. The times of ight of each member of the ock, after the rst bird to y, were recorded. Assume that every bird which ies within the rst 0.08 seconds has seen the predator itself, and that every bird which

ies between 0.08 and 0.2 seconds after the initial ight has been warned directly by the ight of one of these. The average ock size was 24 and the groups covered an area of about 60m . During an attack the proportion of birds which y in the rst 0.08 seconds is 0.18, so that on average 4 birds 2

9

spot the predator. The proportion ying between 0.08 and 0.2 seconds after the rst ight is 0.32. This proportion should be equal to the probability of being warned by another bird, which we estimate as 1 ? (1 ? PA ) where

4

1 PA = 1 + (60

)

3

This gives us an approximate value for of 0.04, so that PA = 0:5 when A = 25.

The parameter b

b is the ratio of the time taken for the predator to attack ta and the length of scan ts. The length of scan is generally taken to be one second (Pulliam et al., 1982), and we have selected the attack time ta to be two seconds, in accordance with Proctor and Broom (2000). Choice of the other model parameters and their default values again follows Proctor and Broom (2000). Table 1 summarises all the parameters and chosen default values. Note that u and A are the parameters that we are trying to optimise and so do not have default values.

Table 1: The model parameters parameter description default value u proportion of time feeding A area of region occupied by group N number of birds in the group 10  maximum feeding rate 10Js? K cost of an attack 10 J y attack rate 5  10? s? ta time taken for an attack 2s ts time taken for a bird to scan for predators 1s b ta=ts 2 Z area required for one bird to feed at half its capacity 1m m a measure of how fast feeding rate increases with area 5

1/area for PA = 0:5 0.04 d a measure of how information transfer decreases with area 3 1

7

5

2

10

1

4 Results Both of our models maximise a function which is a linear relationship between food consumption and predation risk. In reality birds will need a certain amount of food to survive per day, and there is a limited amount of food that they can use. Our model describes what happens when the birds are feeding in the open. Thus if food is plentiful, they may feed as rapidly as possible up to their physiological gut capacity, and then go to cover, so exposing themselves to risk for a short period only. If food is rare they may have to spend a great deal of time reaching the level that they need. All payos are shown as a percentage of the theoretical maximum rate of energy gain.

4.1 A xed area In this section we nd the best vigilance strategy for a group which are occupying a speci ed area. This is shown in Figure 2. In very small areas there is little food, so that it is not worth feeding (i.e. u is low) - realistically the birds would just go elsewhere, of course. For medium-sized areas, there is a fair level of food and communication is good, thus the feeding level is high since birds can be relied upon to inform others of a predator. Large areas give poor communication, but not that much more food - it therefore pays to be more vigilant, and so u is lower. Note that the pay-o is always higher for the cooperative birds; if cooperation could be forced it would be bene cial. Vigilance is higher for the cooperative birds, since each is willing to 'do its bit' for the group; however cheats can exploit this by feeding more and the sel sh optimum has less vigilance and more feeding. These are general features of the comparison between the two models. FIGURE 2

4.2 The default parameters Table 2 shows the optimal strategy and the pay-o per bird for a ock of 10 birds, when all the default parameters are used. The models predict that sel sh birds spend more time feeding and less time scanning than cooperative 11

birds. This is in agreement with the ndings of Pulliam et al.(1982). Sel sh birds also require larger areas, possibly due to the increased feeding; and receive a smaller pay-o as a result of the increase in predation costs.

Table 2: Comparison of the Group and Sel sh Optimum u A Payo per bird Cooperative 0.797 18.5 7.25 Sel sh

0.858 20.8

5.87

Note that the percentage of successful attacks is very dierent for the two models. For the cooperative model only 1.4 % of attacks are successful, whereas for the sel sh model 5.5 % are successful. The cooperative birds have to feed longer, to obtain the same level of food, and thus would encounter more attacks, but there is still a large dierence between the two predation levels.

4.3 Varying the parameters In the previous section we chose a set of 'default' parameters which we considered reasonable. However dierent parameters may prevail in reality either due to natural variation or to the inaccuracy of our choice. Therefore the parameters will be changed to see the eect upon model predictions. Each of the parameters was varied in turn, while the rest were kept at their default values in order to see how the model predictions were aected. The results are shown in the gures below.

Varying m The results in Figure 3 show that as m increases, u increases and the pay-o increases for both the cooperative and sel sh model. The area decreases for both models. The increase in pay-o can be explained as follows. The feeding rate has the same value when A = NZ for all values of m. For smaller areas, the lower m, the higher the feeding rate, but for larger areas the reverse is true. The optimal area is greater than NZ = 10, and so the larger m is, the higher the feeding rate. The birds can thus feed at the same rate in a smaller area, and since this improves communication they do not need to be so vigilant, and so u can be higher. The values of u and A are larger, and the pay-o smaller, for the sel sh model than for the cooperative model. 12

FIGURE 3

Varying b

In our model b depends on the attack time and the time taken for one scan. An increase in b can be interpreted as either an increase in the attack time or a decrease in the time taken for a scan. Figure 4 shows that as b increases, there is an increase in both u, and the pay-o for both models, the area staying roughly the same. As b increases attacks are relatively slower, so the rate of vigilance does not need to be as high, so increasing the overall pay-o. Again the values of u and A are larger, and the pay-o smaller, for the sel sh model. The advantage of the cooperative model is larger for small b. If predators are more dicult to spot, it becomes more important to help your ockmates. FIGURE 4

Varying Z

Z is the area required per bird to feed at half its maximum feeding rate. As Z increases, A increases quite considerably. This is to be expected, since increasing Z means increasing the area required per bird. At the same time u decreases, since birds need to be more vigilant when the area increases due to a decrease in the probability of being informed of an attack. Hence the pay-o also decreases. Low Z means that there is a lot of food available, so the birds only need to feed quickly for a short time and then go to cover. If Z is high then the relative predation risk is high, so they should be very vigilant but feed for a long period of time. The cooperative model gives a much larger pay-o than the sel sh model when Z is small. The birds are closer together, so that communication is good, so that it pays to cooperate. For small Z the values of u and A are larger for the sel sh model, but when Z = 4 the optimal pair is identical in both cases. Large values of Z correspond to large areas per bird and hence large distances between birds. The model may be predicting that for Z = 4 the birds behave as if they are feeding alone rather than as part of a ock (compare with Figure 9, low N ). FIGURE 5

Varying =yK

Since increasing =yK means increasing food availability  (or decreasing risk due to predators yK ) it is fairly obvious that both u and the pay-o must increase. In fact A also increases as birds take account of the reduction in risk or the increase in food to feed quickly and minimise their time in the 13

open. We varied =yK by varying the value of K , so that it was easier to make comparisons for the group pay-os. If we varied  instead, then the same optimal pair was obtained but the pay-os were scaled according to . Figure 6 shows the model predictions. The cooperative model does especially well relative to the sel sh model when K is small. Paradoxically, the sel sh individuals over-compensate for their relative safety by choosing a large area and low vigilance, whereas cooperative individuals leave their area almost unchanged. FIGURE 6

Varying

The value of corresponds to PA = 1=2 if the area of the ock of birds is 1= m , so that the lower is, the better the communication between the birds. Thus if is small the birds are relatively safe in a large area and so can spend more time feeding. Conversely, if is large, then the probability of being informed of an attack becomes very small if the area is large, and so the best strategy is to choose a smaller area and spend less time feeding in order to increase the chance of spotting a predator. Figure 7 shows that increasing results in both u and A decreasing. For small cooperation is especially advantageous over sel shness; communication is good, so there is a better chance of being warned by your ockmates (which occurs more in cooperative groups since their vigilance levels are higher). 2

The value of may depend on the size of bird, and will also depend upon the shape of the feeding group. These factors are considered separately in sections 4.4 and 4.6. FIGURE 7

Varying d

It can be seen from Figure 8 that in fact varying d does not have much eect on the optimal pair for either model. A large d gives a slightly larger u and pay-o. This is reassuring since our choice of the value of d was fairly arbitrary. FIGURE 8

Varying N We now consider the most variable of the parameters, N , the number of birds. The results are shown in Figure 9. The value of u increases with N up to a ock size of 18 but then decreases again with another increase at size 14

100. The area per bird slowly declines up to a ock size of 50, but then starts to increase. These results can be explained as follows: for small ock sizes the total area is relatively small and so there is a reasonable probability of passing on information about a predator and therefore the birds can spend more time feeding. However, there comes a point when the total area is so large that communication between birds becomes very inecient, so that they must become more vigilant again. The model predicts that when the group reaches a size of 80 the area per bird increases and birds spend more time feeding. This is probably due to the increasing importance of the dilution eect; as N becomes large, if an attack is successful, the probability that you are the bird eaten gets very small. It is interesting to note that the pay-o per bird increases with ock size up to a ock size of 12 and then decreases with ock size. This suggests that the best ock size for this model is 12. This is a very interesting feature of the model as all previous models have predicted that the bigger N is, the better. The pay-o starts to increase again for large N due to the increasing in uence of the dilution eect, since for large N the chance that any given bird is targeted is small. However, in reality, it is probable that the attack rate will be larger for very large ocks, so that the results may become less valid for large N . FIGURE 9

4.4 The probability of being informed of an attack depends on N The probability of being informed of an attack has so far only depended on the total area of the group regardless of how many birds are within that area. For instance if the total area occupied by a ock of birds is 20m , then in our previous model PA = 0:661 regardless of the number of birds in the area. This is probably unrealistic, since if there are a lot of birds within a given area they will be closer together than if there were only a few. An individual is more likely to obtain information from another individual who is close by and it is then likely to pass on the information to other individuals who are close to it. This process would rapidly continue until the whole ock are noti ed of the attack. This is very like a contact-process epidemic model. This means that PA should be an increasing function of N . If another bird spots the predator, then there are N ? 1 other birds that it can inform. If it informs just one other bird then this individual may pass on the information 2

15

and the process cascades until our particular individual is informed. So we now de ne PA as follows:

PA =

1 d 1 + ( N A ? ) 9

1

The factor 9 is included to give identical results for various for the default

ock size N = 10. Figure 10 shows that for a constant area of 20 m , PA now increases as N increases. 2

FIGURE 10 We then varied N , to see how the predictions of the model are aected. The results of this new model are in Figure 11. As N increases, both u and the area per bird steadily increase. This means that vigilance declines with group size and the birds are bene ting by being in a larger group with the individual payo increasing. The reason for this result is that, as ock size increases, PA increases for a given area, and so we obtain larger areas for which the probability of passing on information about a predator is still high. FIGURE 11

4.5 Varying the attack rate So far we have assumed that the predator attack rate is constant for all group sizes. However, studies have shown that predators are attracted to large groups as they are more easily located (e.g. Page & Whitacre 1975, Sullivan, 1984). Therefore, we will now allow the attack rate to be proportional to N , with y = 0:00005s? when N = 10. So y = 0:00005  N=10. We will consider two cases. Firstly when PA only depends on A, and secondly when PA also depends on N . 1

Attack rate depends on N , PA only depends on A From Figure 12, we can see that both the cooperative and the sel sh model predict that there is an optimal group size of 10. This is close to the predicted optimal group size when the attack rate is constant (see Figure 9). In both models, u initially increases with N , until N = 14, and then decreases. When ock sizes are small, individuals can bene t from the vigilance of their 16

ockmates and so can spend more time feeding; however, when the ock is larger than 14, the total area for the ock becomes very large and so the probability of being informed becomes very small. Therefore, it is necessary to become more vigilant yourself, especially as attacks are becoming more frequent as N increases. If we compare Figure 12 with Figure 9, where the attack rate was constant for all ock sizes, we see that if the ock is larger than 10, individuals spend more time being vigilant when the attack rate increases with group size, as intuition would predict. Note that both u and the pay-o per bird tend to a constant level as N increases in Figure 12. FIGURE 12

Attack rate and PA depend on N The cooperative model predicts that the pay-o increases continually with

ock size (see Figure 13), and so large groups are best. In the sel sh model, the pay-o initially increases, then levels o , and nally slightly decreases when N > 200. Although the attack rate is increasing with group size, the probability of being informed also increases with N and when ocks are large there is a high probability that at least one bird will spot the predator and pass on the information. This is particularly true for cooperative groups, where individuals within large ocks are still being reasonably vigilant (e.g. in a ock of 100 birds, it is predicted that each individual spends about 4 % of the time being vigilant). In sel sh groups, individuals spend far less time being vigilant (< 2% for a ock of size 100), and so there is a much greater risk of a successful attack and lower pay-os. The model predicts that the area per bird increases with group size for cooperative groups. Large groups are as safe as small groups despite the increase in attacks with ock size and so individuals can space themselves out and increase their food intake. For the sel sh model, birds initially spread out as N increases, probably in response to the large increase in u. Then as u becomes large, so that very little vigilance is taking place, it is better for birds to move closer together in order to increase PA . FIGURE 13

17

4.6 Varying the shape of the region The default value of corresponded to PA = 1=2 if the area of the ock of birds is 25m . This assumes that the area of the region is circular. Thus the greatest distance p between any two birds is the diameter of the circle which in this case is 10= . If the region was elliptical in shape,p then we can assume that PA = 1=2 if the length of the major axis is 10=  ' 2:82m. As the region became thinner, then the area would decrease and hence value of would increase. Therefore we can think of as the shape parameter. 2

Table 3 shows how the area corresponding to PA = 1=2 varies as the length of the minor axis of an ellipse increases from 0.5m to 2.82m, when the length of the major axis is 2.82m. It also shows the corresponding value of and the optimal strategy for the cooperative case. The model predicts that the best shape is circular, as would be expected. As the shape of the ellipse becomes thinner, the area required for PA = 1=2 becomes smaller and when the length of the minor axis is 1.0m the area is less than 10m , the area required for 10 birds to feed at half their maximum capacity. The optimal strategy is greatly aected by the shape of the region. The model predicts that birds should spend more time being vigilant as the region becomes more elongated. This seems a reasonable prediction, as we would expect that birds in a long thin region would not make such good vigilant mates as birds in a circular region, visually obstructing each other more in certain directions. Studies carried out by Beko (1995) show this to be the case. The model predicts that there is a turning point in the optimum area which initially decreases as the region becomes thinner and then increases again. In fact the turning point occurs when = 0:10 which corresponds to an area of 10m when PA = 1=2. At this point the predicted optimal area is 13.60m ; after this the probability of passing on information about a predator becomes very small, so that the bene ts of increasing the area outweigh the costs. 2

2

2

18

Table 3 Varying the shape of the region Length of minor axis Area of ellipse 0.5 4.43 1.0 8.86 1.5 13.29 2.0 17.72 2.5 22.15 2.82 25.00

0.23 0.11 0.08 0.06 0.05 0.04

u 0.366 0.562 0.685 0.747 0.773 0.797

A payo for the group 38.53 30.90 13.62 38.51 14.35 52.45 15.87 62.95 17.00 67.93 18.49 72.46

5 Applying the Model for Dierent Bird Species Our model was developed for a medium-sized bird, such as a lapwing. Note that it is fairly easy to adapt the model for smaller or larger birds. We shall use a similar procedure to Proctor and Broom (2000). It is not clear whether large birds can communicate better over similar size areas than small birds. Therefore we will adapt the model for both the case when PA is dependent upon and independent of bird size. The relationship between metabolic rate and the body size of an animal is well known; a plot of metabolic rates against the logarithm of body mass, produces a straight line (Schmidt-Nielsen, 1984). Metabolic rate R / M y where M is the mass of the bird, and 2=3  y  3=4. We assume that a bird needs to gain energy proportional to its metabolic rate. Thus Z = CM y , for some constant C . The mass of a lapwing is about 200g and this corresponds to Z = 1. Therefore C = 1=200y . This gives

1 My Z = 200 y and so we can calculate Z for dierent species of bird if we know their mass. Table 4 shows some approximate masses for four dierent species of bird and the corresponding values of Z for when y = 2=3 and 3/4.

19

Table 4: Values of Z for dierent species bird when the exponent of the body mass is varied Species blue tit redshank lapwing oystercatcher

mass (g) 12 100 200 500

Z y = 2=3 y = 3=4 0.153 0.121 0.630 0.595 1.000 1.000 1.842 1.988

5.1 PA depends on the size of bird First we will assume that PA depends on the size of bird using the model in which PA is independent of N . We can nd values of Z which corresponds to dierent sized birds by using the previous argument. We assumed that PA = 1=2 when the total area for a ock of 10 birds is equal to 25m , which gave a value of 0.04 for . This applied for lapwings. It would seem reasonable that the ratio of the area required for say a blue tit and a lapwing would be in the same ratio as Z , the area required to feed at half capacity. Since Z = 1 for a lapwing, we simply need to multiply the values of Z for each bird by 25 to nd the area needed for PA = 1=2 and then calculate the corresponding value of . We can then nd the optimal strategy and payo for dierent species of bird. The results for four dierent species of bird are shown in Tables 5 and 6, when the exponent y is 2/3. 2

It can be seen from Table 5 that for all species of bird the optimal value of the proportion of time spent feeding is the same at 0.797. The area required for a ock of 10 birds increases with size of bird but the payo is identical for all species. In all cases, all species of birds manage to obtain about 72.5 per cent of their energy requirement. As the pay-o is related to the mass of the bird, then the total amount of energy gained increases with bird size.

Table 5 Optimal cooperative strategy when exponent = 2/3 Species blue tit redshank lapwing oystercatcher

Z A when PA = 1=2 0.15 3.83 0.63 15.75 1.00 25.00 1.84 46.00 20

0.26 0.06 0.04 0.02

u 0.797 0.797 0.797 0.797

A Group payo 2.8 72.5 11.6 72.5 18.5 72.5 34.0 72.5

As in the cooperative solution, the sel sh model predicts that the proportion of time spent feeding is the same for all species. The area increases with bird size, but the pay-o is the same for all species being about 58.7 percent of their energy requirement. Comparing the predictions of the sel sh and cooperative models for birds of the same size, we nd that sel sh birds feed more, use larger areas, and receive smaller payos in all cases.

Table 6 Optimal sel sh strategy when exponent = 2/3 Species blue tit redshank lapwing oystercatcher

Z A when PA = 1=2 0.15 3.83 0.63 15.75 1.00 25.00 1.84 46.00

0.26 0.06 0.04 0.02

u 0.858 0.858 0.858 0.858

A Group payo 3.2 58.7 13.1 58.7 20.8 58.7 38.0 58.7

5.2 PA is independent of bird size The assumption that the probability of a bird being informed of an attack depends on the size of the bird may well not be true. Although smaller birds need smaller areas to feed, a ock of, say, 10 birds in a given area might be able to see one another just as well whatever their size. Therefore, we repeated the above calculations with PA constant. The results are shown in Tables 7 and 8. This time we nd that the model predicts that u decreases with increasing bird size, so that smaller birds spend more time feeding and so smaller birds also obtain a larger percentage of the maximum payo. This result is borne out by studies as small birds generally do spend more time feeding than large birds (Newton, 1998).

Table 7: Optimal cooperative strategy when PA is constant and the exponent = 2/3 Species blue tit redshank lapwing oystercatcher

Z 0.15 0.63 1.00 1.84

u 0.845 0.825 0.797 0.706 21

A Group payo 5.7 81.8 13.9 77.9 18.5 72.5 27.1 55.8

Table 8 Optimal strategy when PA is constant and the exponent = 2/3 Species blue tit redshank lapwing oystercatcher

Z 0.15 0.63 1.00 1.84

u 0.910 0.888 0.858 0.746

A Group payo 6.6 62.5 15.8 60.9 20.8 58.7 30.1 50.0

Note that birds of dierent species but of the same size will not necessarily feed for the same proportion of time, as other factors such as type of food eaten and food handling times also need to be considered. The optimal group size for each of the species was obtained. We found that the optimal size was 17 for redshanks; 12 for lapwings; and 7 for oystercatchers in both models. The sel sh model for blue tits showed that the pay-o increased until the group size was about 50 and then remained fairly constant. In the cooperative model, blue tits obtained the best pay-o in groups of about 60. Hilton et al. (in press) studied raptor attacks on ocks of common redshanks (Tringa totanus) wintering at the Tyninghame estuary on the Firth of Forth in 1998. They noted that ock size ranged from 7-61 with median ock size 23.5 (95% C.I.: 16-30). Cresswell & Whit eld (1994) have also observed redshanks in the same area over 3 winters (1989-1992), and noted that there are regularly between 200-400 redshank feeding in the area. This data suggests that redshanks do prefer to feed in medium-sized ocks. So the prediction of an optimal group size is a desirable feature of our model and the prediction of an optimal group size of 17 for redshanks compares well with the data.

6 Discussion The models developed in this paper are a modi cation of the spatial model of Proctor and Broom (2000). The main dierence between the models is in how the dierence in the eectiveness of vigilance for large and small areas is incorporated. Proctor and Broom (2000) assumed that the time taken to make an antipredatory scan increased with the circumference of the group area. This is not always a reasonable assumption, and in this paper we have 22

introduced a more generally realistic eect; that of worsening communication over longer distances. We now compare the two models (note that Proctor and Broom, 2000 only considered a cooperative model). Both models predict that a ock of ten lapwings should spend about 80% of the available time feeding. However the predictions for the area are dierent, with the optimal area for our model being 18:5m compared to that of Proctor and Broom (2000) being 24:3m . In general the cost of a large area is more severe in our model, so that areas tend to be smaller. But this also means that the area has more of an eect in our model; for instance the group pay-o varies much more with the area than in Proctor and Broom (2000) (see Figure 2), as does the optimal area for various parameters (e.g. see Figure 5). The most striking dierence between the two models, is that the model described in this paper predicts an optimal group size, whereas that of Proctor and Broom (2000) does not. In contrast to our model, the overwhelming majority of theoretical works conclude that bigger groups are always better, although birds commonly feed in medium sized ocks. 2

2

Feare (1984) studied ocks of starlings feeding on grass elds at a farm and noted that although up to 2000 starlings fed each day, 58% of ocks contained less than 50 birds and only 6% of ocks contained over 250 birds. He observed larger ocks in the autumn and winter months and only small ocks in the spring. He also noted that ock size varied according to the time of day and the food source. Caraco (1979) and Caraco et al. (1980) also found that changes in temperature and changes in the risk of predation aected the predicted optimal group size. Sibly (1983) showed that ocks of around optimal size may be unstable and will tend to increase in size. He argued that if we have a ock which is of optimal size in terms of individual tness, individuals might still do better by joining the group than by feeding alone. Therefore group size will grow beyond its optimal size, and we might expect that in reality ock sizes would be larger than our predictions. We considered both cooperative and sel sh behaviour of birds. Previous work suggests that vigilance behaviour might be cooperative (Pulliam et al., 1982) but cooperation is subject to cheating. Sel sh behaviour generates lower pay-os than cooperative behaviour, so that it is better for birds to cooperate, if non-cheating could be enforced. Since the amount of time spent being vigilant can take any value between zero and one, we might be able to apply Roberts and Sherratt (1998)'s strategy of `raise-the-stakes' to vigilance behaviour. Although cooperation is better for the group, the question of how to stop cheating still remains. Individuals may decide not to cheat because 23

they want to preserve their ockmates, so that in a future attack, they are less likely to be the targeted bird (Lima, 1989). Another possible explanation is that if a predator is successful then it may be more likely to attack again in the same area. Therefore individuals may bene t from cooperating. Our model predicts that cooperation has a particular advantage over sel sh behaviour when predators are hard to detect (Figure 4), when resources are plentiful (Figure 5), when the relative predation risk to feeding ability is small (Figure 6) or when communication is good (Figure 7). Thus in these circumstances it may be more likely for cooperation to emerge in practice. Most models, including the ones presented in this paper, assume that predators select their prey at random. However, it is possible that predators preferentially select an individual that is not being vigilant at the time it launches its attack (e.g. Rudebeck 1950). Packer & Abrams (1990) modelled this situation and found that sel sh birds should scan more frequently than cooperative birds. Another assumption of the model is that all birds have the same chance of escaping but it has been suggested that birds which are actually vigilant at the time of the attack will escape more quickly (Broom and Ruxton, 1998; Lima, 1995). Again, this would lead to increased vigilance for sel sh groups. When we compared the predictions for the cooperative and sel sh models, we found that sel sh birds are less vigilant, feed in slightly larger areas and receive a smaller payo. When we varied each of the model parameters in turn we found that both the cooperative and sel sh models were aected in almost an identical manner with just a shift in the absolute values. Therefore in the discussion below about varying the parameters, it is not necessary to consider the cooperative and sel sh models separately. Note, however, that this implies that it is very dicult to distinguish between cooperative and sel sh behaviour in real populations since both types of population would react to the changing of any (possibly controlled) factors in a similar way. On the other hand, it may help explain why past theoretical work has produced sensible results whether a cooperative or sel sh model was assumed, and that the question of which type of behaviour is truly occurring, though of great interest in itself, may not be crucial for predictive purposes. We were particularly interested in what happens when we vary the group size. Many studies have shown that individuals spend less time being vigilant as group size increases and most previous models predict this to be the case. However, there is data which suggests that vigilance does not decrease inde nitely with group size. For instance, in their study of house sparrows, 24

Elgar and Catterall (1981) found that there was no appreciable decrease in scanning rates with ock size for ocks with more than ve individuals. The model predictions for varying N depend on whether the probability of passing on information about a predator, PA , is independent or dependent on N . In the former case, our models predict that vigilance initially decreased with group size, but for groups sized 12-60, vigilance increased with group size, and nally for groups larger than 60, vigilance again decreased with group size. In terms of the average payo per bird, groups of about 12 individuals did best and so this model predicts an optimal group size. When PA , depends on N , then our models predict that vigilance decreases with group size and that large ocks are best. Since we obtain such dierent results according to whether PA depends on N or not, it is necessary to discuss which case is more realistic. If PA is independent of N , then when groups get large, the probability of passing on any information is close to zero. This might be the case if the targeted bird has to be informed by the actual individual that spots the predator. If PA depends on N , then this allows for the possibility that information can be passed along, so that the targeted bird can be quickly informed even if it is far away from the individual that has spotted the predator. Studies have shown that birds respond more quickly to their nearest neighbours (Hilton et al., in press) and suggests that PA may not increase with N to the extent given by our model, but our results suggest more empirical research is warranted.

References Axelrod, R. & Hamilton, W.D. 1981. The evolution of cooperation. Science 211 1390-1396

Bahr, D.B. & Beko, M. 1999. Predicting ock vigilance from simple passerine interactions: modelling with cellular automata. Anim. Behav. 58 831839. Beko, M. 1995. Vigilance, ock size and ock geometry: information gathering by western evening grosbeaks. Ethology 99 150-161 Broom, M. & Ruxton, G. D. 1998. Modelling responses in vigilance rates to arrival to & departures form a group of foragers. IMA Journal of Mathematics Applied in Medicine and Biology 15 387-400. Caraco, T. 1979. Time budgeting and group size: a test of theory. Ecology 60 618-27 Caraco, T., Martindale, S., & Pulliam, H. R. 1980. Avian ocking in the presence of a predator. Nature 285 400-1 Cresswell, W., & Whit eld, D. P. 1994. The eects of raptor predation on wintering wader populations at the Tyninghame estuary, southeast Scotland. 25

Ibis 136 223-232 Davis, J. M. 1975. Socially induced ight reactions in pigeons. Anim. Behav. 23 547-601 Dehn, M. M. 1990. Vigilance for predators: detection and dilution eects. Beh. Ecol. Sociobiol. 26 337-342 Elgar, M.A. & Catterall, C.P. 1981. Flocking and Predator Surveillance in House Sparrows: Test of an Hypothesis. Anim. Behav. 29 868-872. Feare , C. 1984. The Starling Oxford University Press Grafen, A. 1979. The hawk-dove game played between relatives. Anim. Behav. 27 905-907. Hamilton, W. D 1971. Geometry for the sel sh herd. Jour. of Theor. Biol. 31 295-311 Hilton, G. D., Cresswell, W., & Ruxton, G. D. Intra- ock variation in the speed of escape- ight response on attack by an avian predator (in press). Lazarus, J. 1979. The early warning function of ocking in birds: an experimental study with captive quela. Anim. Behav. 27 855-865 Lima, S. L. 1987. Vigilance while feeding and its relation to the risk of predation Jour. of Theor. Biol. 124 303-316 Lima, S.L. 1989. Iterated Prisoner's Dilemma: an approach to evolutionarily stable cooperation. Am. Nat. 134 828-834. Lima, S. L. 1995. Back to basics of anti-predatory vigilance: the group size eect Anim. Behav. 49 11-20. Lima, S. L. & Bedneko, P. A. 1999. Back to basics of antipredatory vigilance: can non-vigilant animals detect attack? Anim. Behav. 58 537-543. Maynard Smith, J. & Price, G.R. 1973. The logic of animal con ict. Nature 246 15-18. McNamara, J. M., & Houston, A. I. 1992. Evolutionarily stable levels of vigilance as a function of group size. Anim. Behav. 43 641-658 Newton, I. 1998. Population limitation in birds. Academic Press Packer, C. & Abrams, P. 1990. Should cooperative groups be more vigilant than sel sh groups? Jour. of Theor. Biol. 142 341-359 Page, G. & Whitacre D. F. 1975. Raptor predation on wintering shorebirds. Condor 77 73-83 Poysa, H. 1994. Group foraging, distance to cover and vigilance in the teal, Anas crecca. Anim. Behav. 48 921-928 Proctor, C. J. & Broom, M. 2000. A spatial model of antipredator vigilance IMA Journal of Mathematics Applied in Medicine and Biology (in press). Pulliam, H. R. 1973. On the advantages of ocking Jour. of Theor. Biol. 38 419-422 Pulliam, H. R., Pyke, G. H, & Caraco, T. 1982. The scanning behaviour of juncos: a game-theoretical approach. Jour. of Theor. Biol. 95 89-103

26

Roberts, G. 1996. Why individual vigilance declines as group size increases. Anim. Behav. 51 1077-1086 Roberts, G. & Sherratt, J.N. 1998. Development of cooperative relationships through increasing investment. Nature 394 175-179 Rudebeck, G. 1950. The choice of prey and modes of hunting of predatory birds with special reference to their selective eect. Oikos 2 65-88 Schmidt-Nielsen, K. 1984. Scaling: why is animal size so important? Cambridge University Press Sibly, R. M. 1983. Optimal group size is unstable. Anim. Behav. 31 947-948 Sullivan, K.A. 1984. Information exploitation by downy woodpeckers in mixed species ocks Behaviour 91 294-311

27

Captions for Figures

Figure 1: The probability of being informed of a predator as a function of area for dierent values of d. Figure 2: The best strategy for u when the area is xed for the cooperative and sel sh models. Figure 3: How the strategy for u and the area changes with m. Figure 4: How the strategy for u and the area changes with b. Figure 5: How the strategy for u and the area changes with Z . Figure 6: How the strategy for u and the area changes with =yK . Figure 7: How the strategy for u and the area changes with . Figure 8: How the strategy for u and the area changes with d. Figure 9: How the strategy for u and the area changes with N when PA = 1=(1 + ( A)d) and the attack rate is independent of N . Figure 10: How the probability of being informed changes with ock size when PA = 1=(1 + (9 A=(N ? 1))d ). Here A = 20 and = 0:04. Figure 11: How the strategy for u and the area changes with N when PA = 1=(1 + (9 A=(N ? 1))d ) and the attack rate is independent of N . Figure 12: How the strategy for u and the area changes with N when PA = 1=(1 + ( A)d) and the attack rate depends on N . Figure 13: How the strategy for u and the area changes with N when PA = 1=(1 + (9 A=(N ? 1))d ) and the attack rate depends on N .

28