# Birds: how does egg mass vary with body mass?

One can imagine that the larger the bird, the larger the egg. This is not always true. Consider the dataset for [1]. Examining it, we see that the Wild Turkey female has an average mass of 4222g and an egg mass of 78.8g, whereas the Malleefowl female has an average mass of 1830g and egg mass of 175g, a relatively much larger egg! The relative sizes of the egg can be visualized by this diagram:

The reason why bigger birds do not always produce bigger eggs is due to the variation in bird strategies in producing successful adults and the interactions with bird physiology. A major point here is that of precocity. You may have seen birds that feed their young in the nest such as this Barn Swallow family:

However, not all birds adopt this strategy. One example is the Australian Brush-turkey:

The Brush-turkey and its relatives in the Megapodes family bury their eggs in mounds of various types and just leave them there. Newly hatched birds are fully ready to face their environment without parental care. The spend more time developing in the egg, and need a larger egg with more nutrients for that development.

These differences between different birds are less pronounced when restricted to single orders, and within orders, more of the variance in egg weight is actually explained by female body weight. This was the perspective taken by Rahn, Paganelli, and Ar in their paper [2], where they (among other things) fit the following model within individual orders:
$$W = aB^b,$$ where $W$ is the egg weight and $B$ is the female body weight. The coefficients of $a$ and $b$ are then different for each family. They also fitted this equation within families for Passerines, since the Passerines is the largest order of birds and so contains quite a diverse set of families. A quick summary of their results is in this graph, reproduced from their paper:

You might notice that the slopes of these lines (as we are plotting on a log-scale) are all similar, suggesting that the $b$ coefficient might be similar across orders. The authors suggest that $b= 2/3$ is a constant value and that $a$ is the coefficient that varies across orders. Therefore, we can say that within any given order, at least a good first approximation to the egg-body weight relationship is that egg weight is proportional to $B^{2/3}$.

## References

[1] Terje Lislevand, Jordi Figuerola, and Tamás Székely. 2007. Avian body sizes in relation to fecundity, mating system, display behavior, and resource sharing. Ecology 88:1605.

[2] Rahn, Hermann, Charles V. Paganelli, and Amos Ar. Relation of avian egg weight to body weight. The Auk 92.4 (1975): 750-765.

All bird photographs in this post are taken and copyrighted by Jason Polak.

Also, this post is part of a new series on science writing with an emphasis on quantitative approaches. If you would like to see more posts like this, please comment below. If you're just interested in birds, you might also like to visit Bad Birding, which is a more birding-oriented blog but occasionally also has scientific material which is less mathematical.