Ecological Archives E094-051-A1

A. W. Bateman, A. Ozgul, J. F. Nielsen, T. Coulson, T. H. Clutton-Brock. 2013. Social structure mediates environmental effects on group size in an obligate cooperative breeder, Suricata suricatta. Ecology 94:587–597. http://dx.doi.org/10.1890/11-2122.1

Appendix A. Details of model formulation and assessment.

For a group of size nt at the beginning of any observation period, t, we can calculate the group’s size at the beginning of the next period by accounting for observed recruitment of juveniles, rt; mortality, mt; immigration, it; and emigration, et, in period t:

.(A.1)

Allowing for different rates in each class, c, we get:

(A.2)

Now, if we treat nt+1 as the value of a random variable, Nt+1, that has yet to be observed by the beginning of period t (and do similarly for each demographic rate in period t), we can take the expectation, E, across all possible values of Nt+1 to get:

(A.3)

In this context, we use the convention that an uppercase Latin character represents a random variable, while the corresponding lowercase character represents an observed value of that random variable. 

We assumed it possible to describe the expected value of each component rate from (Eq. A.3) using a smooth function of local conditions (usually) in period t:

(A.4)

where Xt = {x1, t, x2, t, ..., xk, t} is the set of relevant conditions in period t. To estimate functions for the expected demographic rates, we used generalized additive models (GAMs) fit to demographic data corresponding to the appropriate two-month windows. A GAM fits additive combinations of linear and nonlinear functions of given predictor variables to achieve a parsimonious description, via a link function, of a univariate response variable (see Wood 2006). We fit each GAM using data available from all groups throughout the course of our study.

The predictors we considered for each demographic rate included both intrinsic and extrinsic variables. We included group size, measured as the number of group members older than two months (recruitment age – see below), to assess group-level density dependence, and we included population density, measured as meerkats per square kilometer across the study population, to assess population-level density dependence. To assess annual patterns in demographic rates we included season, as indicated by the two-month time period, which, though technically a discrete-valued measure, we considered as a continuous variable to avoid overly flexible models. Finally, we included rainfall in a given two-month period and, separately, rainfall in the preceding ten months, both measured as deviations (standardized by the season-specific standard deviations) from the appropriate seasonal averages taken over the course of the study.

Given the potentially important predictor variables, we used an information-theoretic approach to select the most parsimonious model from a candidate set of plausible models.  We developed the candidate model set (Table A1) to include models that incorporated reasonable combinations of smooths of the predictor variables and their two-way interactions (fit as tensor product smooths; Wood 2006). After fitting each GAM, we used Akaike’s Information Criterion (for AIC in the context of GAMs, see Wood 2006) to compare the models and adopted the minimum-AIC model for future use.  In fitting the models, we used a cyclic cubic spline basis for season and a standard cubic spline basis for all other predictors; three knot locations for each predictor except season, for which we used four knot locations to allow more reasonable cyclic smooths; and a degrees-of-freedom inflation factor (the gamma argument of the fitting function – see Methods: Statistical software) of 1.4, to avoid overly flexible smooths (Wood 2006).

We used a similar generalized additive modelling approach for part or all of each component demographic rate. Unless otherwise specified, we fit GAMs to per-capita demographic rates (see below), using a binomial error structure with a logit link function. Because each component demographic rate has slightly different properties, however, the details of our approach differed slightly for each.  

Recruitment

We treated the number of juveniles recruited as the result of a multi-step process dependant on litter production (by dominant or adult subordinate females only), the size of any litter produced, and pup survival to recruitment age (Kendall and Wittmann 2010). We assumed that each female of class c gave birth to an emergent litter in period t - 1 with probability E(Bc, t - 1), that litter size for mothers of class c followed a zero-truncated generalized Poisson distribution (Kendall and Wittmann 2010) with expectation E(Lc), and that pups born to a mother of class c in period t - 1 survived to at least recruitment age to be counted at the start of period t + 1 with probability E(Sc, t). The generalized Poisson distribution has been proposed for modelling litter sizes, because it avoids attributing likelihood to excessively large litters, and truncating (and rescaling) the distribution to omit zero allowed the probability of a litter of size zero to be incorporated into the litter-production stage (Kendall and Wittmann 2010). We set recruitment age to two months, which meant that pups born in period t - 1 reached recruitment age in period t and were counted as group members at the beginning of period t + 1 at between two and four months of age (three months – the midpoint of this range – is the approximate age of nutritional independence for meerkats; Clutton-Brock et al. 2002). Under the above conditions, the expected per-capita recruitment attributable to a female of class c in period t becomes (Kendall and Wittmann 2010)

(A.5)

for dominant and adult subordinate females and zero otherwise. Expected recruit production for a class of size nc, t at the beginning of period t is therefore

.(A.6)

We modeled expected pup survival, E(Sc, t), according to the standard approach described above, using a GAM incorporating smooths of local conditions in period t, but we treated expected litter production, E(Bc, t - 1), and expected litter size, E(Lc), differently. We assumed that expected litter production depended on local conditions in period t - 2 rather than period t - 1, because gestation in meerkats is approximately 70 days (Clutton-Brock et al. 2008), and pups born in period t - 1 were, therefore, most commonly conceived in period t - 2.  The model forms we fit for litter production were identical to the standard GAM model forms, except that they incorporated predictor variable from period t - 2 rather than period t. For expected litter size, we fit one truncated generalized Poisson distribution to the observed litter sizes for each reproductive class of females. To do this, we used numerical optimization to find the distribution parameter values that maximized the total log likelihood of the appropriate observed litter sizes across time periods. This allowed us to use the appropriate distribution, at the cost of excluding covariate predictors.

Mortality and Emigration

We used our standard GAM approach to model expected per-capita rates of emigration and mortality, E(Ec, s) and E(Mc, s), respectively, so that the expected rates of class-specific emigration and mortality took similar forms:

.(A.7)

Dominant female emigration is almost never observed, so we made the assumption that it does not occur and did not include it in our models. Although resident males may disperse when immigrants arrive (Doolan and Macdonald 1996, Young 2003), the associated close temporal correlation meant we were unable to include the effect in our models.

Immigration

Unlike the other rates, immigration in period t is not readily attributable to any individuals present at the beginning of the period, and we observe immigration almost exclusively in adult males. We therefore modeled expected immigration not as a per-capita rate but as the mean of a count variable.  Because immigration commonly occurs when “coalitions” of males join a group (Doolan and Macdonald 1996, Young 2003), we assumed the number of immigrants to be distributed negative-binomially, to allow for aggregation (overdispersion relative to a Poisson random variable).  We used the standard candidate model set, but with a negative binomial error structure and natural log link, and fit the negative binomial shape parameter using outer iteration (Wood 2006).

Final Model

The final model took the form:

.(A.8)

Within this final model, each function, f, was the maximum-parsimony GAM from the appropriate candidate model set.

Calculation of R2

We calculated R2 based on predicted and observed group sizes, taking total sum of squares to be , where Enull(Nt + 1) is the expected value of Nt + 1 under the null model. In (Eq. A.4), we implicitly treated each observed demographic rate in (Eq. A.2) as the value of a function, f, plus error, e

(A.9)

so that

.(A.10)

Given the high quality of the data involved (in each two-month time step we have near-perfect knowledge of each group’s composition), we make the simplifying assumption that deviations of nt+1from E(Nt+1) represent process error only.  For a population-dynamics model with pure process error, the appropriate null model is a random walk, so that Enull(Nt+1) = nt and total sum of squares = .

Monte Carlo Simulations

We initially seeded 2000 trajectories with the conditions from the first three time steps for each of five groups present at the beginning of the study. To attain each group trajectory, we used the final models for each component demographic rate, with their associated error distributions, to simulate individual recruitment, mortality, immigration, and emigration within a group, and thereby group dynamics, in two-month time steps over the course of the study period. To simulate any individual demographic rate for a given time step, we made a pseudorandom draw from the distribution defined by the appropriate model’s predicted mean and error structure, taking group sizes from the simulation data but all other predictor variables from the true population values for the given time step.

While the demographic rate models defined the probabilities associated with rates in each time step, we needed additional rules and assumptions to produce a functional individual-based model. As we defined them, the probabilities of mortality and emigration were mutually exclusive; therefore we determined whether each individual of class c remained in its group in time step t by drawing from a single binomial distribution with P(death or emigration) = P(death) + P(emigration) = E(Mc, s) + E(Ec, s). We drew the ages of immigrants from the distribution of immigrant ages observed across all groups in the field data. We kept track of each simulated individual’s age throughout the course of the simulations, advancing its age class as appropriate. When a dominant individual died (or a dominant male emigrated), we “promoted” the oldest same-sex individual within the group. We assigned pup sex stochastically, with a 50% chance of each sex. We stopped simulating a group’s trajectory when its group size fell below two or it contained only males (since our assumptions about demography did not allow for female influx in such a situation).

Once simulation was complete for a group trajectory, we used the predictive model to calculate the expected group size for each period, given the simulated conditions for the previous periods. With this information, we calculated 10 000 R2 values (one for each trajectory), representing the distribution of goodness of fit when the model was used to predict the individual-based stochastic version of itself.

Table A1. Candidate model set for generalized additive models. S = two-month “season” of period t*, R2 = normalized rainfall in period t (relative to the long-term seasonal mean), R10 = normalized rainfall in the ten months prior to period t (relative to the long-term seasonal mean), GS = group size at the start of period t, PD = estimated population density for period t; s() indicates a cubic regression spline smooth of a single variable, while te() indicates a tensor product smooth of two variables using cubic regression spline bases .

model ID model form for fRATEc, d
1 s(S)
2 te(S,R2)
3 te(S,R10)
4 te(S,R2) + te(S,R10)
5 te(S,R2) + te(S,R10) + te(R2,R10)
6 s(S) + s(GS)
7 te(S,R2) + s(GS)
8 te(S,R10) + s(GS)
9 te(S,R2) + te(S,R10) + s(GS)
10 te(S,R2) + te(S,R10) + te(R2,R10) + s(GS)
11 te(S,GS)
12 te(S,R2) + te(S,GS)
13 te(S,R10) + te(S,GS)
14 te(S,R2) + te(S,R10) + te(S,GS)
15 te(S,R2) + te(S,R10) + te(R2,R10) + te(S,GS)
16 te(S,GS)  + te(R2,GS)
17 te(S,R2) + te(S,GS)  + te(R2,GS)
18 te(S,R2) + te(S,R10) + te(S,GS)  + te(R2,GS)
19 te(S,R2) + te(S,R10) + te(R2,R10) + te(S,GS)  + te(R2,GS)
20 s(S) + te(GS,PD)
21 te(S,R2) + te(GS,PD)
22 te(S,R10) + te(GS,PD)
23 te(S,R2) + te(S,R10) + te(GS,PD)
24 te(S,R2) + te(S,R10) + te(R2,R10) + te(GS,PD)
25 te(GS,PD) + te(S,GS)
26 te(S,R2) + te(GS,PD) + te(S,GS)
27 te(S,R10) + te(GS,PD) + te(S,GS)
28 te(S,R2) + te(S,R10) + te(GS,PD) + te(S,GS)
29 te(S,R2) + te(S,R10) + te(R2,R10) + te(GS,PD) + te(S,GS)
30 te(GS,PD) + te(S,GS)  + te(R2,GS)
31 te(S,R2) + te(GS,PD) + te(S,GS)  + te(R2,GS)
32 te(S,R2) + te(S,R10) + te(GS,PD) + te(S,GS)  + te(R2,GS)
33 te(S,R2) + te(S,R10) + te(R2,R10) + te(GS,PD) + te(S,GS)  + te(R2,GS)

* Models for litter production incorporated conditions from period t - 2 instead of period t (see main text).
 All smooths of season used cyclic cubic regression spline bases.
 As the smooth components of GAMs are, by default, centered, each model also includes a constant parameter, omitted here for brevity.


Literature Cited

Clutton-Brock, T. H., S. J. Hodge, and T. P. Flower. 2008. Group size and the suppression of subordinate reproduction in Kalahari meerkats. Animal Behaviour 76:689–700.

Clutton-Brock, T. H., A. F. Russell, L. L. Sharpe, A. J. Young, Z. Balmforth, and G. M. McIlrath. 2002. Evolution and development of sex differences in cooperative behavior in meerkats. Science 297:253–256.

Doolan, S. P. and D. W. Macdonald. 1996. Dispersal and extra-territorial prospecting by slender-tailed meerkats (Suricata suricatta) in the south-western Kalahari. Journal of Zoology (London) 240:59–73.

Kendall, B. E., and M. E. Wittmann. 2010. A stochastic model for annual reproductive success. American Naturalist 175:461–468.

Wood, S. N. 2006. Generalized additive models: an introduction with R. CRC Press, Boca Raton, Florida, USA.

Young, A. J. 2003. Subordinate tactics in cooperative meerkats: breeding, helping and dispersal. Dissertation, University of Cambridge, Cambridge, UK.


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