A quadratic correlation between long-term mean group size and group density in a cooperatively breeding passerine

Both mean group size (MGS) and mean group density (MGD) are critical indices to characterize a population of cooperatively breeding birds. When a population reaches its carrying capacity, both long-term MGS and long-term MGD will remain relatively stable. However, there has been little study of how these two variables relate. The Masked laughingthrush Garrulax perspicillatus is a cooperatively breeding bird living in fragmented habitats. During 2010 and 2012-2016, we used song playback to observe and confirm the group sizes and territory ranges of the birds and the data of bird presence to determine habitat suitability. By grouping the nearest territories according to their geographical coordinates, we divided the whole study area into 12 subareas and the whole population into 12 subpopulations. Then, we calculated both MGS and MGD for different time durations for each subpopulation. Finally, using MGD as independent variable and MGS as the dependent variable, we explored the correlations between MGS and MGD by fitting quadratic functions and modeling quadratic regression. Both MGS and MGD were averaged for different time durations and were cross-related. Our results show that the MGS for more than 2 years significantly correlated with MGD for more than 3 years in a reverse parabolic shape, differing from that of short-term effects. Our findings suggest that long-term MGD is a better predictor of long-term habitat quality and that long-term MGS is determined by long-term habitat quality in Masked Laughingthrushes. Based on above findings, we can infer that: (1) Long-term habitat quality determines the long-term MGS, but it sets no prerequisite for the status and source of group members; (2) Long-term MGS in certain populations is adapted to the corresponding level of long-term habitat quality, it facilitates us to predict the helper effects on current or future survival or reproduction in different situations. These findings and inferences are both helpful for us to understand the evolution of cooperative breeding.