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Biol. Bull. 202: 275-282. (June 2002)
© 2002 Marine Biological Laboratory

Herd Size in Large Herbivores: Encoded in the Individual or Emergent?

Jean-François Gerard1,*, Eric Bideau1, Marie-Line Maublanc1, Patrice Loisel2 and Carole Marchal1

1 Institut de Recherche sur les Grands Mammifères, INRA, BP 27, 31326 Castanet Tolosan Cedex, France
2 Laboratoire d’Analyse des Systèmes et Biométrie, INRA, 2 Place Viala, 34060 Montpellier Cedex 1, France

* To whom correspondence should be addressed. E-mail: jfgerard{at}toulouse.inra.fr


   Abstract
TOP
 Abstract
Introduction
 Models Assuming Optimum-Size...
 Fusion-Fission Models
 Checking Assumptions and...
 Evolutionary Considerations
 Appendix
 Literature Cited
 
In large mammalian herbivores, the increase of group size with habitat openness was first assumed to be an adaptive response, encoded in the individual. However, it could, alternatively, be an emergent property: if groups were nonpermanent units, often fusing and splitting up, then any increase of the distance at which animals perceive one another could increase the rate of group fusion and thus mean group size. Dynamical models and empirical data support this second hypothesis. This is not to say that adaptive modifications of mean herd size cannot occur. However, this changes the way in which we can envisage the history of gregariousness in large herbivores during the Tertiary.


   Introduction
TOP
 Abstract
 Introduction
Models Assuming Optimum-Size...
 Fusion-Fission Models
 Checking Assumptions and...
 Evolutionary Considerations
 Appendix
 Literature Cited
 
Large mammalian herbivores, such as ruminants or kangaroos, make up groups that are easily recognizable in the field: they consist of individuals located a short distance from one another and most often engaged in a common activity, for example, feeding, traveling, or resting. The size of these groups is very variable and has been a matter of study for ethologists and ecologists for about 40 years.

Two general trends were early identified. First, within a species, group size tends to increase with population density (Spinage, 1969; Johnson, 1983; Wirtz and Lörscher, 1983; Table 1). Second, herd size increases with habitat openness: whereas groups are small in forested habitats, they are much larger in grassland and other open landscapes. This second trend was initially reported in African antelope taken as a whole, considering the typical habitat and herd size of each species (Estes, 1974; Jarman, 1974). It was then recorded within species using habitats of varying openness (Leuthold, 1970; Evans, 1979; LaGory, 1986; Hillman, 1987; Table 2; Fig. 1).


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  		Table 1 Variation of group size with population density in different species of large herbivores (Macropus spp. are kangaroos, which are marsupials; the other species are ruminants—i.e., eutherians)

 

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  		Table 2 Variation of group size with habitat openness in different species of large herbivores (Macropus giganteus is a kangaroo, which is a marsupial; the other species are ruminants—i.e., eutherians)

 


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  		Figure 1. Relative frequencies of roe deer (Capreolus capreolus) group sizes recorded in winter in two study areas, and for two levels of population density supported by the two areas some years apart (n: number of groups sighted, solitary animals included; after Gerard et al. 1995, with permission of the Revue d’Ecologie Terre et Vie). Group size increases with both population density (Mann-Whitney U test: in forest, P < 0.0001; in open plain, P = 0.001) and habitat openness (at low density, P < 0.0001; at high density, P < 0.0001).

 
These two general trends were rapidly explained in two diverging ways. As early as 1964, Caughley hypothesized that groups of large herbivores were nonpermanent units that often fused and split up. On this basis, the author suggested that any increase in population density should increase the rate of group meeting, and thus the average group size. This purely mechanistic proposal is equivalent to saying that group size is an emergent property, resulting from multiple fusion and fragmentation events, and that it is sensitive to variations of population density.

In contrast to Caughley’s proposal, the variation of group size with habitat openness was assumed to be a biological adaptation, encoded in the individual. A central argument, developed by Estes (1974) and Jarman (1974), was that in closed habitat, a herbivore can easily reduce the probability of being detected by predators by being discreet and, especially, by living in small groups. By contrast, in open habitat, it is more difficult to escape notice. Being surrounded by many conspecifics should then ensure the best protection against predators because, in the event of an attack, there is a high probability that the victim will be another group member ("selfish avoidance of predators by aggregation"; Hamilton, 1971). As a consequence, natural selection should have retained individuals preferring to be within small groups when in closed habitat, and within large groups when in open landscape.

Because they relate to the effect of two different ecological factors, these contrasting hypotheses seem to have been implicitly considered as compatible, and the opposition between emergence and individual encoding long passed unnoticed. Moreover, the plausibility of the two hypotheses remained unquestioned for a long time. The situation has changed owing to theoretical and empirical works carried out during the 1980s and 1990s.


   Models Assuming Optimum-Size-Seeking
TOP
 Abstract
 Introduction
 Models Assuming Optimum-Size...
Fusion-Fission Models
 Checking Assumptions and...
 Evolutionary Considerations
 Appendix
 Literature Cited
 
An important step was made when Sibly (1983) had the idea to develop and examine the properties of a dynamical model of group formation formalizing the idea that individuals exhibited preferences for group sizes shaped by natural selection.

The basic assumptions of this model are the following. First, there is a relationship between fitness and group size such that fitness is maximized for a given size (according to the rationale of Estes and Jarman, the optimal size should be small in closed habitat and large in open habitat). Second, the individuals behave as if they know which group size, in the actual environment, will give them better fitness than another: in the course of encounters, each individual leaves a group for another as soon as the size of the latter will enhance its fitness.

Although these assumptions seem to prescribe the group size that should be obtained, Sibly’s model exhibits emergent properties that are rather puzzling.

1. Mean group size at equilibrium is, unexpectedly, generally larger than the optimum size. The reason for this is not very difficult to understand. Suppose that a solitary animal encounters a group of optimal size s*. None of the group members will leave, but the solitary individual will join the group, provided being within a group of size s* + 1 entails a better fitness than being alone. Groups still greater can form in the same way, provided their members keep a fitness higher than that of a solitary individual.

2. At equilibrium, no group shows a size lower than the optimum size, and the standard deviation of group sizes is always extremely limited (<=0.5). This is inconsistent with the group size distributions actually observed in large herbivores, where small groups are rarely lacking, even when average herd size is large, and the standard deviation of group sizes can be very great, especially when the mean is large (Table 1, Table 2, and Fig. 1).

3. Mean group size at equilibrium depends on the initial group size distribution. However, it does not depend on population density. The absence of effect of population density contrasts with the effect recorded in the populations of large herbivores.

4. Groups no longer fuse or split up once the equilibrium is reached: they become permanent units. This is inconsistent with the high lability of groups revealed in an increasing number of large herbivore species since the beginning of the 1980s (Murray, 1981; Schaal, 1982; Southwell, 1984b; Fichter, 1987; Hillman, 1987; Putman, 1988; Barrette, 1991; Estes, 1991; Le Pendu et al., 1995, 2000). Furthermore, this shows that Sibly’s model is incompatible with the hypothesis proposed by Caughley for the increase of group size with population density.

Sibly’s model was further developed during the 1980s and 1990s, by introducing altruism towards relatives and/or the possibility for group members to limit the increase of the size of their group by repelling joiners (see Giraldeau and Caraco, 2000, for a review). Indeed, the initial model ignored kin selection. Moreover, by joining a group whose size is larger than or equal to the optimal size, an individual enhanced its fitness but lowered the fitness of the group members, so that the latter could be assumed to repel the joiner. Some of these modifications of the initial model improve the first property described above in that they lead, at equilibrium, to a mean group size that tends to be closer to the optimal size. However, they do not improve the other properties of the model. So, they remain both inconsistent with the data recorded in large herbivores and incompatible with Caughley’s hypothesis.


   Fusion-Fission Models
TOP
 Abstract
 Introduction
 Models Assuming Optimum-Size...
 Fusion-Fission Models
Checking Assumptions and...
 Evolutionary Considerations
 Appendix
 Literature Cited
 
According to Caughley (1964), an increase in population density should increase the mean size of groups that frequently fuse and split up, by enhancing the rate of group encounter and thus fusion. Following the same rationale, it could be hypothesized that any increase of habitat openness, and thus of the distance at which groups can perceive one another, should increase the rate of group fusion, and thus mean group size (Gerard et al., 1993). This should at least be the case, provided group fusion results from an attraction between groups. Clearly, if group fusion results from simple " collisions," then the distance at which animals can perceive one another should be without influence.

The plausibility of this hypothesis was confirmed in the mid-1990s, when Bonabeau and Dagorn (1995), Gueron and Levin (1995), and two of the authors of the present paper (Gerard and Loisel, 1995) developed new dynamical models of group formation. These models contrasted with Sibly’s model in that groups were assumed to fuse and split up without any group size being preferred by the individuals. They were in fact generalizations of a previous model by Cohen (1971), in which "casual groups" were assumed to increase or decrease by a single individual (see also Okubo, 1986: pp. 45–49).

The model by Bonabeau and Dagorn (1995) is probably the simplest. First, the groups (solitary individuals included) that compose the population are assumed to move at random. Second, two groups arriving within the same portion of space systematically merge, then behave as a single group. Third, at each time step, a fraction p of the individuals leave the groups they are in by temporarily becoming solitary.

In its basic version, the model by Gueron and Levin (1995) differs from the latter in two aspects. First, two groups arriving in view of each other merge with a probability that is independent of their sizes, but not necessarily equal to 1. Second, the groups are assumed to split into two (and not only to lose single individuals) with a probability ßs, where ß is a constant and s, the group size. Here, ß can be interpreted as the probability with which any individual adopts a trajectory differing from that of the other group members and is possibly followed by some of them. In practice, the size distribution of splitting groups is assumed to be uniform.

The assumptions of our model (Gerard and Loisel, 1995) are more complicated than those considered by Bonabeau and Dagorn, and Gueron and Levin. First, as is more or less implicit in the two latter models, each individual is assumed to be able to detect any conspecific present inside an area {alpha}, characterizing habitat openness. Second, each individual oscillates in a probabilistic way between a "social" state and an " individualistic" state. When in the social state, an animal joins every perceived conspecific, then behaves in such a way as to stay with it. By contrast, when in the individualistic state, an animal moves without taking conspecifics into account. As a consequence, groups fuse through attraction and split up. When two individuals (or groups of individuals) in the social state perceive one another, the individuals merge and form a single group. When an individual (or a group of individuals) in the social state perceives an animal in the individualistic state, it joins it. In this case, the resulting group actually includes a leader, which is the animal in the individualistic state. However, if, within a group of this kind, a second individual turns out to be individualistic, then the group includes two leaders moving independently of each other; as a consequence, the other group members distribute themselves at random (with probability 1/2) near the two leaders, and the group splits up. The probability {varepsilon} of shifting from the social state to the individualistic state and the probability µ of the reverse shifting are fixed, so that the individual’s behavior is independent of habitat openness, population density, and group size.

Though they rely on different assumptions, the Bonabeau and Dagorn model (1995), the Gueron and Levin model (1995), and our model exhibit remarkably similar emergent properties.

1. The first property that the three models have in common is that the group size distributions obtained at equilibrium resemble those ordinarily recorded in large herbivore populations: the group frequency exhibits a single maximum for isolated individuals or a small group size, then monotonously decreases with group size; moreover, the standard deviation of group sizes tends to be large when the mean is large.

2. Whatever the model, the group size distribution obtained at equilibrium for any given values of the parameters is independent of the initial group size distribution, provided population density is left unchanged.

3. Whatever the model and the values of its parameters, any increase of population density entails, at equilibrium, not only an increase of mean group size, as suggested by Caughley, but also an increase of group density (i.e., the number of groups per unit area).

4. Whatever the model and the values of its other parameters, any increase of the distance at which groups can perceive one another increases mean group size at equilibrium (see Fig. 2 for an illustration with our model). Actually, in the three models, multiplying the area perceived by the individuals by a factor k has exactly the same effect on mean group size as multiplying the population density by the same factor (Gerard and Loisel, 1995; Gueron and Levin, 1995; Appendix).



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  		Figure 2. Relative frequencies of group sizes obtained at equilibrium with the model of Gerard and Loisel (1995), for two values of population density and two values of the area perceived by the individual (after Gerard et al., 1997, with permission of Hermes editions). In every case, the probabilities of individual state shifting are µ = 0.9 and {varepsilon} = 0.1.

 

   Checking Assumptions and Predictions
TOP
 Abstract
 Introduction
 Models Assuming Optimum-Size...
 Fusion-Fission Models
 Checking Assumptions and...
Evolutionary Considerations
 Appendix
 Literature Cited
 
The fusion-fission models just described thus exhibited realistic, emergent properties. In addition, they showed that the same mechanism could be both at the origin of the increase of group size with population density and of the increase of group size with habitat openness. What remained to be checked were the main basic assumptions of the models, as well as predictions that could be deduced from their emergent properties. This is what we have begun to do in a population of roe deer (Capreolus capreolus) living in an open cultivated plain in Northern France (Marchal, 1998; Marchal et al., 1998). The data were recorded between November and April, observability being maximum at this season owing to the high proportion of plowed fields and the low height of crops in the cultivated fields (mainly winter cereal and rapeseed).

First, we checked that roe deer groups, like those of many other herbivores, were nonpermanent units. To that aim, we monitored 73 groups during 3 h or more. The proportion of these groups that had not broken up or fused with another group since the beginning of their observation decreased rapidly and regularly through time: after 2 h 38 min of observation, half had their composition altered by at least one fusion or fission event (Marchal et al., 1998).

We then checked that group fusion generally involved inter-attraction, and thus perception at distance. Among the 103 fusion events observed, 3% were simple "collisions": they occurred when two groups met at a wood’s corner, without having previously perceived each other (as shown by the absence of any behavior directed towards the members of the other group before the encounter, and the reactions exhibited on the encounter). In some other cases (21%), fusion followed a human disturbance. However, in 76% of the instances, fusion did result from an attraction between groups whose members indisputably perceived one another: the deer of at least one of two groups looked at the members of the other group and approached them over a distance often exceeding 100 m (Marchal et al., 1998).

We also examined the causes of splitting up. Among the 76 break-up events observed, 11% followed a human disturbance, and 21% were caused by agonistic interactions between group members. Finally, 68% occurred when one individual, alone or followed by others, left the group without any previous visible interactions between group members (Marchal et al., 1998). In other words, most of the splitting-up events were caused by the spontaneous departure of an individual which, until this time, behaved in such a way as to stay with the other group members. So, the mechanism of group fission explicitly included in our model and that suggested in the model by Gueron and Levin (1995) seemed to be rather realistic, at least in the case of roe deer.

Afterwards, we tested some predictions that could be deduced from the emergent properties of the fusion-fission models. First, roe deer being active by day as well as by night (like many other large herbivores), we expected group size to be larger at sunset, after the whole period of daylight during which animals can perceive one another at long distance, than at sunrise, after a period of reduced visibility. This prediction was found to be correct: the mean size ± SD of the groups sampled in the 3 h before sunset was 6.25 ± 4.43 (n = 232) against only 4.66 ± 3.64 (n = 440) in the three hours after sunrise (Mann-Whitney U test, P < 0.0001). In addition, fusion events appeared to be more frequent than break-up events between sunrise and sunset, and more especially during the first daylight hours (Marchal et al., 1998).

According to the models’ properties, we also expected that an increase of population density would entail not only an increase of mean group size, but also an increase of group density. This was tested taking advantage of the fact that, in the study area, local population density varied strongly because the fields (20 hectares each, on average) were more or less attractive for roe according to the kind of vegetation they supported. As expected, both mean group size (Kendall’s {tau} = 0.71; n = 6; one-tailed P = 0.042) and group density ({tau} = 0.73; n = 6; one-tailed P = 0.028) appeared to be positively correlated to local population density over the six main types of fields found in the study area (Marchal, 1998). It must be pointed out that the increase of group density with population density is certainly a very general law in large herbivores. Group density, in a population, can be estimated by dividing the population density by the mean group size, and making this calculation in each of the cases described in Table 1 invariably reveals an increase of group density with population density.

More quantitative investigations must now be performed. In particular, the individuals’ aggregative and locomotor behavior must be quantified and compared between closed and open habitats. However, it seems reasonable to say even now that group size in large herbivores is emergent and not encoded in the individual.


   Evolutionary Considerations
TOP
 Abstract
 Introduction
 Models Assuming Optimum-Size...
 Fusion-Fission Models
 Checking Assumptions and...
 Evolutionary Considerations
Appendix
 Literature Cited
 
On this basis, it is now interesting to return to evolutionary considerations. According to paleontologists, continents supported almost exclusively forest ecosystems during the first 40 million years of the Tertiary (Janis, 1993). Afterwards, because of large-scale climatic changes in the course of the Miocene (23 to 5 million years ago), open landscapes began to develop, and several lineages of large herbivores colonized them. In this context, the hypothesis of the individual encoding of group size implied that the colonizers of the open plains made up large herds only after natural selection fitted them to their new environment. The hypothesis of the emerging group size suggests another possibility: provided the behavior of ancestral forms was similar to that of extant forest-dwelling species, it suggests that the colonizers of open lands spontaneously made up large groupings.

The roe deer actually provides a modern example (Gerard et al., 1995). The species is a forest-dwelling herbivore. This is attested by the habitats it typically uses (Eisenberg, 1981; Putman, 1988), as well as those it formerly used. Indeed, paleontological data show that the distribution of roe deer followed that of woodland during the late glacial and postglacial times (Hufthammer and Aaris-Sørensen, 1998). Furthermore, it is clear that the roe deer’s digestive system does not allow it to feed on rough forage, such as that provided by the grasses during most of the year in open habitats (Duncan et al., 1998). Nevertheless, as the size of its populations increased in the forest habitats 40 or so years ago, the species began to colonize the large fields, almost devoid of any human presence, that have resulted from the recent mechanization of agriculture. And as soon as it left the woodland, the roe deer made up much larger groupings (Fig. 1).

This is not to say that the size of the groups in which the individual is found on average cannot undergo adaptive modifications. If an individual had less tendency to spontaneously leave the group it is in than did the other individuals of the population, it would be at the origin of fewer group-splitting events and thus would find itself in groups on average a little larger. Accordingly, it can be imagined that natural selection may fine-tune the probability of spontaneous departure, that is, parameter p in the model of Bonabeau and Dagorn, parameter ß in that of Gueron and Levin, or parameter {varepsilon} in our model. Of course, such an adaptive modification might occur only if the parameters involved really are affected by mutations (and, if possible, by mutations not having effects on other aspects of the phenotype). In addition, what remains to be proved is that the herd size of large herbivores actually does influence the fitness of the individual. When a predator successfully attacks the herd, an increased group size certainly decreases the probability for any given group member of being the victim. Nonetheless, the frequency at which a herd is attacked probably increases with its size. Indeed, a larger size should increase both the risk of detection (be this at short distance in closed habitat, or at great distance in open habitat) and the chance, for the predator, that the herd includes a vulnerable prey. Furthermore, the panic occurring in a large herd might increase the probability of the predator’s attack being successful.

Be this as it may, it should be noted that the control exerted by natural selection, if any, will remain limited. Indeed, the dynamics of group fusion and fission imply that each individual is found in groups whose size greatly varies and is thus often suboptimal. In addition, the fusion-fission process makes the average size of the groups in which the individual is found sensitive to the fluctuations of population density, which can only aggravate the problem.


   Appendix
TOP
 Abstract
 Introduction
 Models Assuming Optimum-Size...
 Fusion-Fission Models
 Checking Assumptions and...
 Evolutionary Considerations
 Appendix
Literature Cited
 
Mean group size at equilibrium in the model by Bonabeau and Dagorn
We here correct an error made by Bonabeau and Dagorn (1995) when deriving the mean group size from their fusion-fission model. We further show that once the correction is made, increasing the population density or the area perceived by the individuals by a multiplicative factor k has exactly the same effect on mean group size.

Analytical expression of mean group size
In the model by Bonabeau and Dagorn, space is divided into N sites, the whole population included n individuals (n << N), and the individuals simultaneously present within any given site are considered as the members of a single group. At each discrete time step, a fraction p of the n individuals of the population leave the group they are in as solitary animals, and are reinjected at random into the N sites. In addition, each group moves towards a randomly selected site, and the groups (and solitary individuals) entering the same site aggregate to form a single group.

As a consequence of these assumptions, the expected number of groups (i.e., the number of sites occupied) N+ varies between two successive time steps according to:

The denominator of the right part of the equation is N, and not N2 as written by Bonabeau and Dagorn (1995). At equilibrium, N+ (t + 1) = N+ (t), so that the expected number of groups is approximately

and the mean size of groups

Mean group size therefore varies with {surd}n/N, and not with {surd}n/N as found by Bonabeau and Dagorn. Furthermore, once corrected, the analytical expressions of the number and mean size of groups at equilibrium become strictly equivalent to those obtained by Gueron and Levin (1995) with their own model.

Effect of population density and habitat openness
In the model of Bonabeau and Dagorn, groups entering the same site aggregate into a single group. So, a site can be considered as an area in which any individual perceives its conspecifics. If A designates the area available to the whole population, then the area of each site is {alpha} = A/N. It follows that the mean size of groups at equilibrium is

where d = n/A is the population density. It then appears that multiplying the area perceived by the individuals ({alpha}) or the population density (d) by a factor k has exactly the same effect on the mean size of groups at equilibrium. The same is true with the model by Gueron and Levin (1995) and ours (Gerard and Loisel, 1995: appendix B).


   Acknowledgments
 
We dedicate this paper to Francisco Varela (1946–2001) and Stephen Jay Gould (1941–2002). We acknowledge Diana Blazis and Scott Camazine for their invitation to the workshop they organized at Woods Hole on the limits to self-organization in biological systems. We thank the two anonymous referees for their constructive comments on the first version of the manuscript. Finally, we are grateful to Odile Rollin, David Murray, and Peter Winterton for checking the English language of the original communication and of the different versions of the present paper.


   Footnotes
 
This paper was originally presented at a workshop titled The Limits to Self-Organization in Biological Systems. The workshop, which was held at the J. Erik Jonsson Center of the National Academy of Sciences, Woods Hole, Massachusetts, from 11–13 May 2001, was sponsored by the Center for Advanced Studies in the Space Life Sciences at the Marine Biological Laboratory, and funded by the National Aeronautics and Space Administration under Cooperative Agreement NCC 2-896.


   Literature Cited
TOP
 Abstract
 Introduction
 Models Assuming Optimum-Size...
 Fusion-Fission Models
 Checking Assumptions and...
 Evolutionary Considerations
 Appendix
 Literature Cited
 

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