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On the advantages of flocking
J. theor. Biol. (1973) 38, 419-422
On the Advantages of Flocking Two recent articles (Hamilton, 1971; Vine, 1971) appearing in this Journal consider peripheral predation as an important factor leading to the evolution of flocking behavior. Both authors show that an animal feeding in the midst of an existing aggregation may minimize its domain of danger by moving towards its neighbors. This is consistent with the observation that when attacked by a predator many kinds of animals form tighter aggregations. These authors also point out that an animal on the periphery of such an aggregation would be at a relative disadvantage as compared to an animal in the center, and Hamilton indicates that the total predation on the group might be greater than if the individuals did not tend to aggregate. However, neither author considers the relative merits of remaining on the disadvantaged periphery as compared to leaving the flock and feeding alone. If an individual in the center of a flock is less likely to be eaten than an individual feeding alone, it stands to reason that an individual on the periphery of a flock must stand a greater risk of predation than one feeding alone if the total predation per individual is greater for flocking birds than for solitary birds. This could result in a peeling off of the individuals on the periphery exposing new individuals to the periphery and thus could eventually lead to disbanding of the group. Since neither author considers the conditions under which the disadvantages of being on the periphery are less than the disadvantages of feeding alone, it is impossible to evaluate the importance of peripheral predation as a factor leading to flocking. It, therefore, seems appropriate to consider another factor which could lead to flocking and which can explain many of the observations discussed by Hamilton. In my studies on the flocking behavior of finches, I have noted that these birds frequently stop and cock their heads to the side. Such a behavior could obviously be adaptive to the extent that it leads to the detection of predators. When a bird does detect a predator the entire group tlies off together; often one or more individuals has given a chirp. In such a flock there appears to be no correlation between the time at which one bird cocks its head and the time at which a neighbor cocks its head, unless they are both responding to the same external stimulus such as a sharp noise. Thus, an advantage from feeding near another individual accrues from the increased probability that a predator will be detected. This idea is by no 41Y
420
H.
R.
PULLIAM
means new and indeed is considered briefly by both Hamilton and Vine. However, ideas backed up by precise mathematical models tend to gain more acceptance in biology than those ideas only formulated verbally. For this reason T offer a simple model depicting an advantage in Rocking due to increased predator detection. Suppose each flock member cocks its head instantaneously at a mean rate of 2 cocks per minute and that there is no correlation between the time that one individual cocks its head and the time that any neighbor cocks its head. For the simplest situation let the length of the inter-cock times be random, that is the cocks occur according to a Poisson probability law. Then the probability that the time passing between cocks for any one individual is less than or equal to some constant t is given by P[T
< f] = l-e-“‘.
Thus, the probability that at least one individual of length t for a flock of size n is given by P[Vl
I MJ(T2
I OU(T3
5 t> u. . .u (T, I t)].
This is equal to one minus the probability in the interval t, which is
1-m
cocks its head in an interval
that no individual
cocks its head
> 0 n v2 > 4 n v3 > t) n. . . n CT,> 01,
which in turn equals l-P[T,
> t]xP[T,
> t]xP[T,
> f]X...XP[T”>
t]
because of the independence of cocking between individuals. Since the birds are all cocking their heads at a rate A, this last expression is equal to 1-
he-“’ i=l
which equals 1 -em”“‘. Suppose z is the time necessary for a predator to make his final uncovered approach to within striking range of the flock. The formula just derived giving the probability that the flock does not detect the predator in a given amount of time, shows that the safety of the flock depends on the number of birds in it. Table 1 shows how the probability of detecting a predator varies with flock size for various assumptions about the magnitude of Jr. A reasonable value for r might be 10 seconds for a moderately fast predator. A bird feeding alone and cocking its head at a rate of 12 cocks per minute would have a probability of detecting a predator of 0.865 (2 = 12 per minute, T = 0.17 minutes and
LETTERS
TO
THE
421
EDITOR
iz = 2). The same bird feeding in a flock of just 4 birds would enjoy the same protection by cocking its head only three times per minute. If each cock requires say 2 seconds, the bird would have only 36 seconds each minute to feed in the first case, but 54 seconds each minute to feed in the second case ! The results shown in Table 1 indicate that the probability of detecting a predator levels off very rapidly as flock size increases. This is the probability of TABLE
1
The probability of detecting a predator vs. flock size for three values of IT. I is the mean rate of head cocking per minute and z is time necessary for a predator to make his final uncovered approach on the flock. Flock
size 1 2 3 4
12 = fr
lz = 1
lJ = 2
0.394 0.632 0.171 0.865 O-918 0.950
0.632 0.865 0.950 0.982 0.993 0.998
0.865 0.982 0.998 1.000 1WO 1WO
detecting the predator on only one approach. Since presumably the probabilities of detection on different approaches would be independent of each other if the time between approaches is large, the probability of detecting a predator on each of r approaches is given by the probability of detection for a single approach raised to the power r. Thus, for the case where AZ = 3, the probability of detecting a predator on each of ten approaches becomes (O-95)” = 0.60 for a flock of 6 birds but only (O-63)l” = 0.01 for a solitary bird. Thus, even very large flocks might be adaptive if predation is frequent and if individual birds are so pressed by their feeding demands as to have little time for predator detection. Hamilton has demonstrated that in an aggregation of animals those in the center are more protected than those on the periphery. Regardless of whether or not this is an important factor leading to the evolution of flocking, it is no doubt an important factor in determining the dynamics of flock structure. Even if animals form aggregations for reasons such as predator detection, there should still be an advantage to being in the center of the aggregation. Thus, one would expect more dominant individuals to be located in the center of an aggregation, leaving the periphery to their subordinates. Indeed, it is quite possible that peripheral predation was an important factor leading to the evolution of dominance behavior if dominant individuals have greater
422
H.
R.
PULLIAM
access to the protected center. This would partially explain the simuitancous occurrence of the conflicting tendencies of aggregating and fighting in flocking species with dominance hierarchies. Department of Biological Sciences. University of Arizona, Tucson, Arizona 85721, U.S.A. (Received 17 May 1972, and in revised form 8 September 1972) REFERENCES theor. Biol. 30, 405. VINE, I. (1971). J. theor. Biol. 31, 295. HAMILTON,
W. D. (1971).
J.
H. RONALD
PULLIAM
On the Advantages of Flocking Two recent articles (Hamilton, 1971; Vine, 1971) appearing in this Journal consider peripheral predation as an important factor leading to the evolution of flocking behavior. Both authors show that an animal feeding in the midst of an existing aggregation may minimize its domain of danger by moving towards its neighbors. This is consistent with the observation that when attacked by a predator many kinds of animals form tighter aggregations. These authors also point out that an animal on the periphery of such an aggregation would be at a relative disadvantage as compared to an animal in the center, and Hamilton indicates that the total predation on the group might be greater than if the individuals did not tend to aggregate. However, neither author considers the relative merits of remaining on the disadvantaged periphery as compared to leaving the flock and feeding alone. If an individual in the center of a flock is less likely to be eaten than an individual feeding alone, it stands to reason that an individual on the periphery of a flock must stand a greater risk of predation than one feeding alone if the total predation per individual is greater for flocking birds than for solitary birds. This could result in a peeling off of the individuals on the periphery exposing new individuals to the periphery and thus could eventually lead to disbanding of the group. Since neither author considers the conditions under which the disadvantages of being on the periphery are less than the disadvantages of feeding alone, it is impossible to evaluate the importance of peripheral predation as a factor leading to flocking. It, therefore, seems appropriate to consider another factor which could lead to flocking and which can explain many of the observations discussed by Hamilton. In my studies on the flocking behavior of finches, I have noted that these birds frequently stop and cock their heads to the side. Such a behavior could obviously be adaptive to the extent that it leads to the detection of predators. When a bird does detect a predator the entire group tlies off together; often one or more individuals has given a chirp. In such a flock there appears to be no correlation between the time at which one bird cocks its head and the time at which a neighbor cocks its head, unless they are both responding to the same external stimulus such as a sharp noise. Thus, an advantage from feeding near another individual accrues from the increased probability that a predator will be detected. This idea is by no 41Y
420
H.
R.
PULLIAM
means new and indeed is considered briefly by both Hamilton and Vine. However, ideas backed up by precise mathematical models tend to gain more acceptance in biology than those ideas only formulated verbally. For this reason T offer a simple model depicting an advantage in Rocking due to increased predator detection. Suppose each flock member cocks its head instantaneously at a mean rate of 2 cocks per minute and that there is no correlation between the time that one individual cocks its head and the time that any neighbor cocks its head. For the simplest situation let the length of the inter-cock times be random, that is the cocks occur according to a Poisson probability law. Then the probability that the time passing between cocks for any one individual is less than or equal to some constant t is given by P[T
< f] = l-e-“‘.
Thus, the probability that at least one individual of length t for a flock of size n is given by P[Vl
I MJ(T2
I OU(T3
5 t> u. . .u (T, I t)].
This is equal to one minus the probability in the interval t, which is
1-m
cocks its head in an interval
that no individual
cocks its head
> 0 n v2 > 4 n v3 > t) n. . . n CT,> 01,
which in turn equals l-P[T,
> t]xP[T,
> t]xP[T,
> f]X...XP[T”>
t]
because of the independence of cocking between individuals. Since the birds are all cocking their heads at a rate A, this last expression is equal to 1-
he-“’ i=l
which equals 1 -em”“‘. Suppose z is the time necessary for a predator to make his final uncovered approach to within striking range of the flock. The formula just derived giving the probability that the flock does not detect the predator in a given amount of time, shows that the safety of the flock depends on the number of birds in it. Table 1 shows how the probability of detecting a predator varies with flock size for various assumptions about the magnitude of Jr. A reasonable value for r might be 10 seconds for a moderately fast predator. A bird feeding alone and cocking its head at a rate of 12 cocks per minute would have a probability of detecting a predator of 0.865 (2 = 12 per minute, T = 0.17 minutes and
LETTERS
TO
THE
421
EDITOR
iz = 2). The same bird feeding in a flock of just 4 birds would enjoy the same protection by cocking its head only three times per minute. If each cock requires say 2 seconds, the bird would have only 36 seconds each minute to feed in the first case, but 54 seconds each minute to feed in the second case ! The results shown in Table 1 indicate that the probability of detecting a predator levels off very rapidly as flock size increases. This is the probability of TABLE
1
The probability of detecting a predator vs. flock size for three values of IT. I is the mean rate of head cocking per minute and z is time necessary for a predator to make his final uncovered approach on the flock. Flock
size 1 2 3 4
12 = fr
lz = 1
lJ = 2
0.394 0.632 0.171 0.865 O-918 0.950
0.632 0.865 0.950 0.982 0.993 0.998
0.865 0.982 0.998 1.000 1WO 1WO
detecting the predator on only one approach. Since presumably the probabilities of detection on different approaches would be independent of each other if the time between approaches is large, the probability of detecting a predator on each of r approaches is given by the probability of detection for a single approach raised to the power r. Thus, for the case where AZ = 3, the probability of detecting a predator on each of ten approaches becomes (O-95)” = 0.60 for a flock of 6 birds but only (O-63)l” = 0.01 for a solitary bird. Thus, even very large flocks might be adaptive if predation is frequent and if individual birds are so pressed by their feeding demands as to have little time for predator detection. Hamilton has demonstrated that in an aggregation of animals those in the center are more protected than those on the periphery. Regardless of whether or not this is an important factor leading to the evolution of flocking, it is no doubt an important factor in determining the dynamics of flock structure. Even if animals form aggregations for reasons such as predator detection, there should still be an advantage to being in the center of the aggregation. Thus, one would expect more dominant individuals to be located in the center of an aggregation, leaving the periphery to their subordinates. Indeed, it is quite possible that peripheral predation was an important factor leading to the evolution of dominance behavior if dominant individuals have greater
422
H.
R.
PULLIAM
access to the protected center. This would partially explain the simuitancous occurrence of the conflicting tendencies of aggregating and fighting in flocking species with dominance hierarchies. Department of Biological Sciences. University of Arizona, Tucson, Arizona 85721, U.S.A. (Received 17 May 1972, and in revised form 8 September 1972) REFERENCES theor. Biol. 30, 405. VINE, I. (1971). J. theor. Biol. 31, 295. HAMILTON,
W. D. (1971).
J.
H. RONALD
PULLIAM