Scale‐invariant interstage flow matrices: A comment on Yokomizo et al. (2024)

Yokomizo et al. (2024) recently introduced a novel statistic for matrix population models known as the interstage flow matrix. To calculate the interstage flow matrix, one multiplies the population projection matrix on the right by a diagonal matrix whose diagonal entries are the stable stage distribution. Because the sum of the interstage flow matrix entries equals the stable population growth rate, the flow matrix decomposes stable population growth rate into contributions made by transitions between stages. There are two limitations of the interstage flow matrix. First, naturally abundant stages with individuals of relatively low value, such as seed bank, have undue influence on its entries. In the calculation of interstage flow, a seed gets the same weight as a reproducing adult, which is biologically unrealistic. Second, it is scale‐dependent, so a simple rescaling of the stages changes the interstage flow matrix. To overcome these limitations, I use balancing, which rescales stages with the stable stage distribution or, alternatively, the reproductive value distribution. I illustrate how balancing alters the interstage flow matrix using a population projection matrix for Arizona cliffrose (Purshia subintegra). Balancing profoundly changes the conclusions about which interstage flows constitute the greatest share of the stable population growth rate. In a broad application, I use Keyfitz's Δ to compare the original and scale invariant versions of the normalized interstage flow matrix for the 6363 primitive matrices in the COMPADRE plant matrix database. Synthesis. The elasticity matrix has a new interpretation as the normalized matrix of interstage flows of total reproductive value. The scale‐invariant form of the normalized interstage flow matrix that uses the stable stage distribution to rescale is a new statistic which may prove useful as an alternative to the elasticity matrix. Comparative analyses benefit by including a scale‐invariant version, such as elasticities, as Yokomizo et al. (2024) have done. Including scale‐invariant forms derived from balancing gives a more complete and robust picture of interstage flows for comparative plant demography.