Fission-fusion dynamics and group-size-dependent composition in heterogeneous populations

Many animal groups are heterogeneous and may even consist of individuals of different species, called mixed-species flocks. Mathematical and computational models of collective animal movement behavior, however, typically assume that groups and populations consist of identical individuals. In this paper, using the mathematical framework of the coagulation-fragmentation process, we develop and analyze a model of merge and split group dynamics, also called fission-fusion dynamics, for heterogeneous populations that contain two types (or species) of individuals. We assume that more heterogeneous groups experience higher split rates than homogeneous groups, forming two daughter groups whose compositions are drawn uniformly from all possible partitions. We analytically derive a master equation for group size and compositions and find mean-field steady-state solutions. We predict that there is a critical group size below which groups are more likely to be homogeneous and contain the abundant type or species. Despite the propensity of heterogeneous groups to split at higher rates, we find that groups are more likely to be heterogeneous but only above the critical group size. Monte Carlo simulation of the model show excellent agreement with these analytical model results. Thus, our model makes a testable prediction that composition of flocks are group-size-dependent and do not merely reflect the population level heterogeneity. We discuss the implications of our results to empirical studies on flocking systems.

Figure 1

All the possible transition events are presented here graphically. Each pie chart represents a group with the white region corresponding to type-I and the colored region (blue or red) to type-II. The two regions have been labeled with the number of type-I and type-II individuals in each group. Underneath every group we label it with $(m,l)$ where $m$ is the size of the group and $l$ is the number of type-I individuals in the group. We focus our attention on $(n,k)$ groups, which use the red-white color scheme, while other groups are blue-white. The notation for rates $P_{\alpha }$ and $Q_{\alpha }$ in the second column are defined in Sec. 2b. Note that the above figures are purely representational; in this model, we do not consider geometry of groups or how the two types of species are structured within groups.

Figure 2

Flock composition depends on flock size in heterogeneous populations. We find qualitative agreement between simulation results [(a)–(e), top row] and the iterative solutions [(f)–(j), bottom row] of the analytical model Eqs. (6) and (7). We represent flock composition for each group size $n$ by the probabilities [$W_n$; see Eq. (8)] as a function of the relative composition of type-I individuals ($k/n$). Here, we assumed that two types are equally abundant in the population ($N_1=N_2=N/2$). For small $n$, shown in (a–c) and (f–h), $W_n$ is a bimodal function with modes occurring away from the population ratio of type-I to type-II (which is 0.5); this suggests that small group sizes are dominated by one or the other type/species. For large $n$ above a critical value, shown in (e) and (j), $W_n$ is unimodal at $k/n=0.5$, suggesting that large group sizes represent population ratio of two types. Parameters: $s=10000, N=10000, N_1=5000, p_0=1.0, q=5.0, \delta =8.0$.

Figure 3

Most frequently found composition for flocks, i.e., modes of $W_n$ [see Eq. (8)], changes with group size and shows a qualitative feature similar to a pitchfork bifurcation. There is qualitative agreement between results of Monte Carlo simulations [top row (a)–(c)] and iterative solutions of the analytical model [bottom row (d)–(f)]; specially, the agreement is even quantitatively reasonable for smaller values of excess split rate, $\delta$ [(a), (b) and (d), (e)] but not so for higher values of $\delta$, as seen from comparing (c) versus (f). Parameters of simulations: $s=10000, N=10000, N_1=5000, p_0=1.0, q=5.0$.

Figure 4

Dependence of $n_c$, the critical group size, on $\delta$, the excess split rate. The results of Monte Carlo simulations shows that $n_c$ increases approximately linearly with $\delta$. The semi-analytical result also seems to show an increasing trend for $\delta \ge 3$. However, for smaller values of $\delta$ the trend seems to be reversed. Since there is no accurate way to find the value of $n_c$ from the Monte Carlo simulations (due to slow convergence near criticality), the blue curve must be thought of as an approximate trend line, rather than an exact one. Parameters: $s=10000, N=10000, N_1=5000, p_0=1.0, q=5.0$.

Figure 5

Flock composition for different group sizes, when the number of type-I individuals ($N_1$) is not same as that of type-II ($N_2$). These figures are obtained from the iterative solutions to the analytical model; the color difference in the last column ($n=100$) is an artefact of high density of data. For the top row, the ratio $N_1/N=0.40$ and for the bottom row, $N_1/N=0.15$. The $x$ axis runs from 0 to 1 in all the panels. In the top row, we find that results are qualitatively similar to the case of the ratio being 0.5 (i.e., Fig. 2). In the bottom row, when the population ratio is skewed toward one type or species (bottom row), we find that the distribution is always unimodal. Nevertheless, the biological interpretation is broadly the same: smaller groups are likely to be homogeneous but also contain the abundant type/species. Larger groups, like in Fig. 2 and top row of this figure, reflect the population ratio of the two types.

Figure 6

Flock compositions in a population where of type-I and type-II individuals appear in equal proportions. Here, we look at the compositions of groups of fixed size $n=16$ for different values of excess split rate for heterogeneous groups ($\delta$). This shows that there is a critical split rate beyond which heterogeneous groups are less likely but group composition is bimodal. As before, panels (a)–(e) are results of Monte carlo simulations, while panels (f)–(j) are from the iterative solution of the analytical model. Parameters: $s=10000, N=10000, N_1=5000, p_0=1.0, q=5.0, n=16$.

Figure 7

Most frequently found composition for flocks, i.e., location of modes of $W_n$ [see Eq. (8)], as a function of excess split rate $\delta$, for different values of group size $n$. As before, panels (a)–(e) are results of Monte Carlo simulations, while panels (f)–(j) are from the iterative solution of the analytical model. $s=10000, N=10000, N_1=5000, p_0=1.0 q=5.0$.

Figure 8

Group size distributions plotted in log-log scale. Panel (a) is from Monte Carlo simulations while panel (b) is from the analytical model. The two figures show qualitative agreement. The probability as a function of group size decays faster for higher values of $\delta$.