The scale-variant interstage flow makes biological insights possible: A response to Hinrichsen

1 INTRODUCTION

Hinrichsen (2025) identifies two primary limitations of the interstage flow matrix introduced by Yokomizo et al. (2024). The first limitation is described as biological unrealism due to equal weighting for each individual across all the stages. According to Hinrichsen (2025), Yokomizo et al. (2024) assign the same weight to seeds and reproducing adults, which he argues is biologically unrealistic because it undervalues the differing contributions of life stages to population growth. Abundant stages, such as seed banks, disproportionately influence the entries in the interstage flow matrix.

The second limitation is that interstage flow is scale-variant: Simply rescaling the stages alters its entries. Hinrichsen (2025) asserts that this limits the utility of the matrix for robust comparative analyses, as results can vary with changes in scaling.

However, we assert that these two points are not shortcomings of the interstage flow approach but valuable features of it.

1.1 Limitation 1: Equal weighting among developmental stages

Hinrichsen (2025) claims that ‘seeds receive the same weight or ‘value’ as reproducing adults’ in the interstage flow matrix and proposes reweighting (rescaling) approaches to obtain scale-invariant forms of the interstage flow matrix. However, it is not biologically unreasonable for seeds and reproducing individuals to have equal weight, as population growth rate is fundamentally based on individual-level rates of increase.

Life-history strategies vary among species. Some species produce many offspring with low survivorship, while others produce fewer offspring with higher survivorship (Pearl, 1928). The imbalance that Hinrichsen critiques—such as the concentration of populations in immature stages—often reflects biologically meaningful traits rather than a problem requiring correction. Yokomizo et al. (2024) demonstrated that four functional groups (semelparous herbs, iteroparous herbs, shrubs and trees) exhibit distinct patterns of interstage flow allocation across stasis, fecundity and growth. Semelparous herbs are characterized by flows dominated by growth and fecundity, while trees emphasize stability with flows primarily dominated by stasis. Iteroparous herbs and shrubs display intermediate patterns, balancing flows across stasis, growth and fecundity. These differences align with life-history strategies and ecological adaptations.

While balancing, as proposed by Hinrichsen (2024, 2025), could be applicable to examine species characteristics, it is unclear how the balanced results should be interpreted. How meaningful is balancing that amplifies very small stages for understanding species traits? Balancing could also overly emphasize older individuals which represent a small proportion and contribute less to fecundity. Hinrichsen (2025) merely compares interstage flows from Yokomizo et al. (2024) with those derived from balancing using Keyfitz's delta, without discussing their biological significance.

1.2 Limitation 2: Scale variance

Hinrichsen (2025) also highlights differences between interstage flows derived from pre-breeding and post-breeding census models. Although balancing produces identical results when applied to data from pre- and post-breeding censuses, these identical results do not justify the use of balancing. Balancing could overlook the life-history strategies inherent to developmental stages, potentially leading to an oversimplification of biologically significant traits. Therefore, we should favour either the pre-breeding or post-breeding census model over balancing.

If Hinrichsen (2024, 2025) advocates for a scale-invariant approach, he should clarify its advantages in practical applications—for example, what novel insights or methodologies it facilitates. Converting variables to achieve scale invariance, without considering their biological significance, could be counterproductive. We do not see any inherent issue with interstage flow being scale-variant. At the same time, we do not consider a scale-invariant approach to be inherently wrong either if balancing helps clarify its biological significance.