Aggregating a Leslie matrix model: the power of simplicity

“As one introduces more and more detail, breaking species into age classes and age classes into even smaller groups, one loses the principal advantages of aggregation: the ability to treat statistical ensembles of diverse elements as groups for which mean properties convey the essence of individual behavior.” 

-Yoh Iwasa, Viggo Andreasen, and Simon Levin (1987)

One lesson I learned about simplicity came from a recording session where, seeking a big sound, I arranged a pop song for a multitude of instruments from voice to trumpets to tuba. The mastering engineer, Rick Fisher, who worked with the Steve Miller Band, told me that if I wanted a big sound, I should use fewer instruments. Less is more. Thus began my fascination with simplicity, not just in music, but also in mathematical biology. My heroes could capture biological phenomena with few equations. 

Applying simplicity to Leslie matrix models, I grouped individuals into broader classes to reduce the size of the matrix, a process called aggregation (or “collapse”). In a paper published last week in Population Ecology (www.doi.org/10.1002/1438-390X.12149), I present several reasons to aggregate:

• To increase sample sizes of vital rate estimates, with a possible gain in precision and accuracy.
• To compare two matrices of different sizes. One aggregates the larger matrix so that its classes coincide with the smaller matrix and vital rates are comparable.
• To summarize a matrix. For example, the dominant eigenvalue is a highly aggregated 1x1 matrix. 
• To study how matrix size influences elasticities, and asymptotic and transient dynamics.

Does size matter? Here is a video abstract of the paper: https://youtu.be/RL0zde01wp0.

Figure 1. To aggregate a Leslie matrix A of size n into a smaller matrix B of size m, one raises the original matrix to the power of k=n/m, then applies weighted least squares. 

The machinery to aggregate is known in economics, but the method must be modified to accommodate two constraints of Leslie models: the projection interval equals age class width, and age class width is the same for all age classes (Fig. 1). I discovered that any Leslie matrix can be represented by a smaller one of any size and that the reduced matrix retains important properties of the original matrix.

Figure 2. Effectiveness of aggregation (analogous to r-squared), measures how consistent the aggregated model is with the original model. Here m is the size of the aggregated matrix.

What does aggregating a Leslie matrix model do to its transient dynamics? To answer this question, I selected Leslie matrices from the COMADRE Animal Matrix Database (4.21.8.0) that represented short- to long-lived species with a wide range of population growth rates. I found that aggregated models usually converged more rapidly to the stable age distribution than the original models, and that transient dynamics were lost with high aggregation. For long-lived species, aggregation error was generally small, but could be high with shorter-lived species (Fig. 2). 

I plan to develop aggregators for any population projection matrix, and I seek colleagues interested in collaborating. I also plan to develop aggregation methods robust to error-prone vital rates. I will use the thousands of matrices in COM(P)ADRE to determine what level of aggregation is appropriate for different species.

As for music, I have reduced my band to a duo (www.tangocowboys.com), and I often busk at the Pike Place Market of Seattle—just me and my melodica—performing songs of the Old West. With a solitary instrument, I attempt to reach the hearts of passersby.

R code for aggregating a Leslie matrix model: https://www.github.com/hinrich62.

Literature Cited

Iwasa, Y., Andreasen, V., & Levin, S. (1987). Aggregation in model ecosystems. I. Perfect aggregation. Ecological Modelling, 37(3–4), 287–302.

Rich Hinrichsen on Apr 20, 2023