A first principles derivation of animal group size distributions

doi:10.1016/j.jtbi.2011.04.031

Journal of Theoretical Biology

Volume 283, Issue 1, 21 August 2011, Pages 35-43

https://doi.org/10.1016/j.jtbi.2011.04.031 Get rights and content

Abstract

Several empirical studies have shown that the animal group size distribution of many species can be well fit by power laws with exponential truncation. A striking empirical result due to Niwa is that the exponent in these power laws is one and the truncation is determined by the average group size experienced by an individual. This distribution is known as the logarithmic distribution. In this paper we provide first principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. In particular, we investigate two such models. Firstly, we look at a model in which groups merge whenever they meet but split with a constant probability per time step. This generates a distribution similar, but not identical to the logarithmic distribution. Secondly, we propose a model, based on preferential attachment, that produces the logarithmic distribution exactly. Our derivation helps explain why logarithmic distributions are so widely observed in nature. The derivation also allows us to link splitting and joining behavior to the exponent and truncation parameters in power laws.

Highlights

► First principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. ► Link the truncation size to the merging and splitting behaviors of animal groups. ► The logarithmic distribution and truncated power laws can be obtained by preferential attachment mechanisms. ► The logarithmic distribution is a limiting case of truncated power law and negative binomial distribution.

Introduction

Animals are often found in groups. Fish school, birds flocks and insects swarms are ubiquitous examples. Being in groups benefits individuals in several ways. Amongst other things, it can enhance their foraging efficiency, reduce their chance of being captured by a predator, and help to conserve energy (Foster and Treherne, 1981, Parrish, 1989, Barbraud and Weimerskirch, 2001, Sumpter, 2010). However, animal groups do not always have a typical size, and group sizes often have large variation both among and within species (Gerard et al., 2002, Bonabeau et al., 1999). This property brings up several interesting questions: How are animal group sizes distributed? Are there any common patterns for these distributions? How do different distributions form?

A wide range of models has been suggested for distribution of animal group sizes. The earliest such models proposed a single stable group size, around which the size of the groups may fluctuate (Sibly, 1983, Beauchamp and Fernández-Juricic, 2005, Clark and Mangel, 1986, Mottley and Giraldeau, 2000). For example, Caraco proposed the Poisson and negative binomial distributions as one or, respectively, two parameter models of group size distributions (Caraco et al., 1980). Such distributions have a single peak at a group size somewhat larger than one and a narrow variation around this maximum.

Observed group size distributions of many animal species do not follow such Poisson or negative binomial distributions (Gerard et al., 2002, Bonabeau et al., 1999). Most importantly, the variation in group sizes is usually much wider than predicted by these distributions. The geometric distribution is a special case of the negative binomial, obtained by maximizing the variance. In his influential review, Okubo predicted that group sizes should follow a geometric distribution and presented a number of empirical cases where this relationship held (Okubo, 1986). However, even the geometric distribution fails to capture the large variation in group sizes observed for many species. In particular, several studies have shown that many species follow power law distributions over a number of orders of magnitude (Bonabeau and Dagorn, 1995, Bonabeau et al., 1999, Sjöberg et al., 2000).

A natural question emerges about how these distributions arise from interactions between individual animals. Bonabeau and Dagorn proposed a model for animal grouping based on a single assumption: if groups meet they always merge to form a larger group (Bonabeau and Dagorn, 1995, Bonabeau et al., 1999). Their model predicts power law distributions of group sizes, which again appeared consistent with some observational data of fish and mammals. However, in their model individuals need to be continuously added in order to get a power law. Furthermore, unless they add spatial structure to the model the power law exponent is always −2. Even with spatial structure, where such models give power laws with exponents between −4/3 and −3/2 (Takayasu et al., 1988), the dynamics of their model are difficult to motivate from a biological perspective. Although they suggest that power laws should be truncated by faster decreasing functions such as an exponential function, no natural explanation on the cutoff is given. In particular, Bonabeau and Dagorn did not provide a method for relating their model assumptions to the point in the distribution at which the cutoff should occur.

Niwa (2003) proposed a site-based model, which can be described as follows. Assume that space is divided into s sites on which a total of $Φ$ individuals are initially randomly distributed. Every site is either empty or occupied by one group. At each discrete time step, (1) each group with size larger than 1 has a probability p of splitting into a pair of groups. When a group splits, the size of the two components is chosen uniformly at random; (2) all groups move to a new randomly chosen site. If two groups move to a same site, they merge to a new group with a size equal to the sum of the two groups. The same rule holds if three or more groups meet.

Niwa predicted the following stable group size distribution for his model: $W (n) \sim n^{- 1} \exp (- \frac{n}{N_{P}} (1 - \frac{\exp (- \frac{n}{N_{P}})}{2}))$ where W(n) is the probability density function of group of sizes. Niwa noted that the final term $(1 - \exp (- n / N_{P}))$ makes little difference to the quality of data fitting compared to a simpler distribution $W (n) \sim n^{- 1} \exp (- \frac{n}{N_{P}})$ Here, if n is considered to be a continuous variable, then the single parameter N_P denotes the expected size of groups in which an arbitrary individual engages, i.e. $N_{P} = \frac{\int n^{2} W (n) dn}{\int nW (n) dn}$ This is a spectacularly simple yet powerful result. Simply by measuring the average group size experienced by an individual, Niwa is able to recreate the whole distribution of group sizes. Furthermore, Niwa showed that this distribution fits well with data of several fish species, and to a lesser degree that of several mammal species. More recently, Griesser et al. (in submission) have shown that it fits data for house sparrows too.

There are, however, a number of limitations with Niwa's derivation of the above distribution. Firstly, he used simulation results to establish how fluctuations in group size change as a function of group size itself. This relationship is then used in a stochastic differential equation approximation of the site-based model. However, since this initial result is obtained by simulation, the further results based on the stochastic differential equation lack a rigorous foundation. Secondly, given the stochastic differential equation, the derivation of the potential function does not appear correct. Specifically, Eqs. (8), (9), (11) in Niwa's paper do not appear to follow. Thirdly, Niwa states self-consistency of the model by calculating N_P in continuous sense, while it is more natural to use a discrete distribution. Animals are discrete entities. The discrete calculation gives different results from the continuous one and Eq. (3) fails to hold.

Given the excellence of Niwa's distribution in explaining observations and its one-parameter simplicity, it is important that we have a clear derivation linking individual behavior of animals to the predicted group size distribution. In this paper we give several such derivations and at the same time address the above issues in Niwa's original paper.

To make this derivation we use a discrete analog of a general continuous split–merge model, also called coagulation–fragmentation processes (CFP), discussed by Gueron and Levin (1995). They study the following general evolution equation for the density distribution of groups: $\frac{\partial \tilde{f} (x)}{\partial t} = - p (x) \tilde{f} (x) - \int_{0}^{\infty} \tilde{f} (x) \tilde{f} (z) ψ (x, z) dz + \frac{1}{2} \int_{0}^{x} \tilde{f} (y) \tilde{f} (x - y) ψ (y, x - y) dy + \int_{x}^{\infty} \tilde{f} (y) φ (y, x) dy$ where $\tilde{f} (x) = f (x, t)$ denotes number of groups of size x, p(x) is the rate of split for groups of size x, $ψ (x, y) = ψ (y, x)$ is a symmetric function denotes merge rate of group of size x and group of size y. The first two terms in (4) account for decrease caused by split of groups of size x and groups of size x merge with another group. $φ (x, y) = φ (x, x - y)$ denotes the rate of a group of size x splits to two groups whose size are, respectively, y and x−y. The last two terms in (4) account for increment caused by merge of smaller groups to become an x-sized group, and larger groups splitting to become size x.

We derive such an evolution equation for a slightly different version of Niwa's merge and split site-based model and find that it gives a distribution similar to but not identical to that proposed by Niwa. We further discuss several other models which produce a discrete version of Niwa's distribution.

Section snippets

The logarithmic distribution

We begin by re-stating equation (2) as a discrete distribution and discuss some of its properties. It is more natural to use a discrete distribution because animal groups consist of integer number of individuals. Eq. (2) is a truncated power law with exponent −1 and an exponential tail from a cutoff at N_P. In discrete form this can be written as $W (n) = \frac{{Ka}^{n}}{n}$ which is now the probability that a group is of size n. The parameter $K = - 1 / \ln (1 - a)$ is a normalization constant and a is a positive constant

Merge and split model

We now investigate a simpler version of the site-based model proposed by Niwa for how groups split and merge (Niwa, 2003, Niwa, 2004). Assume a system with fixed total population $Φ$ and s sites. Each site is either empty or occupied by one group. In time interval $τ$ , each group with size larger than one splits to two groups with probability $p τ$ , i.e. p is the instantaneous rate of splitting. The two component groups both move to randomly chosen empty sites. With rate q, each group moves to a

Other derivations of the logarithmic and related distributions

There are alternative ways of deriving the logarithmic distribution (Eq. (5)) from first principles. The first point to note is that the logarithmic distribution is a limiting case of the negative binomial distribution (Fisher et al., 1943). If an observed variable follows the negative binomial distribution $f (k, p) = \frac{(k + n - 1)!}{(k - 1)! n!} p^{n} (1 - p)^{k}$ then taking the limiting case k=0, and assuming that group sizes are never zero, we recover Eq. (5) with a=p. The negative binomial distribution has been used

Discussion

A large number of recent theoretical studies have derived power law distributions from assumptions about how individuals interact (Newman, 2001a, Newman, 2005, Sornette, 2004, Ma et al., 2008, Takayasu et al., 1988, Barabasi and Albert, 1999). On the whole these studies are concerned with producing distributions with a power law tail rather than, as we have done here, simultaneously deriving both the power law and an exponential decay above some point of truncation. The exponential decay is

Acknowledgments

We thank Michael Griesser for providing the empirical data that drove us to understand more about animal group size models. This research was funded by the European Research Council grant (IDCAB) and the Chinese Scholarship Council.

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A first principles derivation of animal group size distributions

Abstract

Highlights

Introduction

Section snippets

The logarithmic distribution

Merge and split model

Other derivations of the logarithmic and related distributions

Discussion

Acknowledgments

Theor. Popul. Biol.

Theor. Popul. Biol.

Math. Biosci.

Physica A

Anim. Behav.

J. Theor. Biol.

J. Theor. Biol.

Adv. Biophys.

Anim. Behav.

Emergence of scaling in random networks

Science

Emperor penguins and climate change

Nature

The group-size paradox: effects of learning and patch departure rules

Behav. Ecol.

Possible universality in the size distribution of fish schools

Phys. Rev. E

Scaling in animal group-size distributions

Proc. Natl. Acad. Sci. USA

Avian flocking in the presence of a predator

Nature

The equilibrium behavior of reversible coagulation–fragmentation processes

J. Theor. Probab.