Background/Question/Methods Properly accounting for population structures is essential for understanding population dynamics. Though this has been shown repeatedly (e.g., in metapopulation ecology and epidemiology), there has been less theoretical development for socially-structured populations (i.e., populations in which individuals are members of stable social groups). Nevertheless, socially-structured populations are likely to have rich dynamics because social groups themselves can rearrange, multiply, and collapse due to between-group processes such as fissions, fusions, and migration. These processes are known to be important for socially-structured populations, but a general, flexible mathematical framework that can account for a wide range of nonlinear between-group processes is lacking. We develop such a framework in the form of two mathematical models that are adapted from the group selection literature. The models track the number of groups of a given size and detail the way that births, deaths, and between-group processes change the group-size distribution. Formally, the models are a Markov chain that incorporates demographic stochasticity, known to be important for socially-structured populations, and its deterministic limit. We use these models to show the way that between-group processes alter 1) group and population growth rates, 2) the distribution of group sizes, and 3) extinction risk. Results/Conclusions We show that both between-group processes and within-group density dependence (e.g., whether groups grow according to logistic growth or an Allee effect) are vital for understanding the dynamics of socially-structured populations. Between-group processes predictably alter the group size distribution. The effect of this altered distribution is determined by within-group density dependence. Ultimately, this interaction facilitates the way that between-group processes affect population growth and extinction risk. A particularly illuminating example of this feedback is the case of group fissions. With groups growing according to logistic growth, fissions produce more, smaller groups that have greater potential to grow and thus lead to an increase in population growth and a decrease in extinction risk. However, when groups have an Allee effect, fissions can increase the proportion of groups below the Allee threshold, resulting in a heightened extinction risk.
