Interstage flow matrices: Population statistic derived from matrix population models

Many population statistics describe the characteristics of populations within and among species. These are useful for describing population dynamics, understanding how environmental factors alter demographic patterns, testing hypotheses related to the evolution of life history characteristics and informing the effective management of populations. In this study, we propose a population statistic: the interstage flow. The interstage flow is defined as the product of the element in the ith row, the jth column of the population projection matrix and the jth element of the normalized stable stage distribution. The sum of the interstage flow matrix elements is equal to the population growth rate (PGR), which is the dominant eigenvalue of the population projection matrix. The interstage flow matrix elements allow decomposition of PGR into component contributions made by transitions between developmental stages. We demonstrate the utility of interstage flow matrices using matrix population models from the COMPADRE plant matrix database. We compared interstage flows among four life history/functional groups (FGs) (semelparous herbs, iteroparous herbs, shrubs and trees) and described how PGR reflected individual transitions related to stasis, fecundity and growth. We found that the individual flows are different among FGs. Synthesis. The proposed population statistic, the interstage flow matrix, describes the contribution of individual developmental stage transitions to the PGR. The flow of individuals between developmental stages differs in distinctive ways among different life histories and FGs. The interstage flow matrix is a valuable statistic for describing these differences.


| INTRODUC TI ON
Detailed demographic data have been compiled by plant ecologists for populations that span a range of taxonomic, life history and environmental conditions.Researchers have developed a number of statistics to describe these data, and the development of convenient population statistics has been an important aspect of theoretical population ecology.Matrix population models (MPMs) can be used to derive many useful population statistics, such as population growth rate (PGR), generation time, age at maturation, sensitivity and elasticity.These statistics are helpful for describing population dynamics, understanding how environmental factors alter demographic patterns and testing hypotheses related to the evolution of life history characteristics (Salguero-Gómez, 2017;Salguero-Gomez et al., 2016;Silvertown et al., 1993Silvertown et al., , 1996)).
The growing availability of plant demographic data has allowed more robust hypothesis tests and spurred innovative new directions in comparative plant demography.For example, Silvertown et al. (1996) calculated MPM elasticities of about 90 plant species and showed that the relative influence of different developmental stage transitions (e.g.recruitment from seed, stasis, transitions to larger size classes) on population growth varied systematically across plant functional groups (FGs).These differences likely reflect morphological, physiological and environmental constraints that influence the evolution of plant life history.In 2015, the COMPADRE database (Salguero-Gómez et al., 2015;www. compa dredb. org) began compiling published plant MPMs in a standardized and accessible format.Salguero-Gomez et al. (2016) used COMPADRE to quantitatively describe the life history strategies of 418 plant species explicitly in terms of their demographic characteristics.They found that species assorted along two relatively independent strategies: a slowly growing, long-lived strategy and a reproduction-focused strategy.
In this study, we introduce a statistic called the interstage flow, which is expressed in matrix form and describes the flow of individuals between developmental stages or ages.The approach provides information that is not provided by standard population statistics based on the population projection matrix such as elasticity.
Elasticity describes the proportional change in PGR with a proportional change in a population matrix element, while interstage flow describes the contribution of demographic transitions to PGR itself.
Comparing both interstage flow and elasticity can help to better understand the ecological characteristics of species.Furthermore, the elements of a population projection matrix, that is, survival, growth and fecundity, do not explicitly describe the demographic contribution of the individuals transitioning between classes.For example, a particular developmental stage might have high seed production, but if the proportional abundance of individuals in that stage is low, its contribution to PGR may be minimal.In contrast to the population

| Definition of interstage flow
The dynamics of populations over a discrete time interval can be described as follows: where A is the population projection matrix, and n t indicates a vector of population sizes of each stage at time t.We propose a metric (statistic): the interstage flow matrix F IS .It is defined as follows: where a ij is a matrix element of A. w j is the jth element of the stable stage distribution (w) of matrix A. The stable stage distribution is normalized such that the sum of all elements is equal to 1, that is, where d denotes the matrix dimension (MD) of A.
We present how to obtain the interstage flow matrix in Figure 1 using a population projection matrix of a Japanese perennial herb, Trillium apetalon (Ohara et al., 2001), as an example.The population projection matrix A is and the normalized stable stage distribution can be obtained from the matrix as: (1) 0.451 0.643 0 0 0 0.021 0.8 0 0 0 0.08 0.981 Therefore, the (2, 1) element of the interstage flow matrix is 0.451 × 0.402 (=0.181) and the (1, 4) element of the interstage flow matrix is 5.13 × 0.08 (=0.410).These are not the probabilities of transitions between stages, but rather the normalized number of individuals transitioning between stages.Then, the interstage flow matrix is The unique property of f IS ij is that the relationship between the PGR (λ) and the elements of the flow matrix is as follows: Equation (4) implies that the elements of the interstage flow matrix quantify the extent to which each transition between developmental stages contributes to the PGR.
If matrix A is primitive, it is confirmed by the strong ergodic theorem that the stable stage distribution (the vector of population size at time t) is: which is proportional to the normalized stable stage distribution (w):  that is where N t = ∑ d j=1 n j,t .During a single timestep, N t w j individuals, the number of individuals at stage j, flow to several stages, including stage j.These flows reflect the demographic processes of survival and reproduction.During the survival process, N t w j individuals flow to stage i with probability a ij , and the number decreases because a ij is less than 1.The amount of interstage flow from stage j to stage i (the number of survivors) is N t a ij w j .In the reproduction process, N t w j individuals reproduce their offspring with per-capita fecundity a ij .The amount of interstage flow in the reproduction process is similar to that in the survival process, N t a ij w j .The sum of all interstage flows is j=1 a ij w j , which must be equal to N t+1 because the sum of the surviving individuals and new offspring at the next time step is N t+1 .This can be proven by the following mathematical calculations: The number of n j,t increases or decreases as a single time step proceeds.This was calculated using Equation (1) as: Equation ( 6) can be written based on Equation (5) as follows: The sum of the elements of the left-hand side of the equation is N t+1 , and the sum of the elements of the right-hand side is It is supposed that the population dynamics described by Equation (1) grow at a PGR and with a structure proportional to the stable stage distribution after sufficient time has elapsed (p.86 in Caswell, 2001).Hence, N t+1 = N t .Therefore, ∑ d i=1 ∑ d j=1 a ij w j = .This means that all interstage flow matrix elements sum to the PGR.We provide another proof of Equation ( 4) in Appendix S1.

| Relationship between interstage flow and elasticity
Elasticity is frequently used to quantify relative change in the PGR per relative change in matrix elements (Caswell et al., 1984;De Kroon et al., 1986;Kaneko & Takada, 2014;Pfister, 1998;Takada & Kawai, 2020;Yokomizo et al., 2017).Patterns of proportional values of interstage flow and elasticity do not necessarily mirror each other.We illustrate this using the following population matrices that only differ in matrix elements  7). (5)

| COMPADRE plant matrix database
We described how interstage flows (Equation 2) are related to life

| Matrix selection
We used matrices satisfying the following eight criteria: 1.No maximum stage-specific survival in the submatrix U exceeded one.
2. At least one element in submatrices F or C was greater than zero.
3. Matrices were irreducible and primitive.This is because we considered that the population density in a stage does not become zero or oscillate between years if the stage distribution is stable.A matrix is irreducible or primitive if d is the dimension of the population projection matrix of A (Caswell, 2001).
4. Dimensions were equal to or larger than four because small matrices cannot reflect stage-specific information about transitions.
Note, however, that an interstage flow matrix can be obtained even from matrices with small dimensions.
5. Matrices were for unmanipulated populations.We removed matrices for experimentally manipulated populations.
6. Matrices satisfied A = U + F + C. For some populations, this equation does not hold true, for various reasons (e.g.errors in the database).
7. Higher stages in matrices are for larger sizes, active or matured stages.In some matrices, dormant stages are in larger rows than active ones with the same size categories.We did not use those matrices in this study, because we assumed transitions to larger rows are classified as growth.
8. Only for analyses involving elasticity, we used matrices in which all matrix elements can be classified into only one category out of stasis, fecundity and growth (See the next section).
When multiple matrices were available for a species, we calculated the mean matrix and the average of each element.However, we could not calculate a mean matrix if there were multiple population projection matrices with different dimensions.In such cases, we selected the population projection matrix associated with the study with the longest research period.

| Species functional classification
We categorized the selected matrices into the following growth form categories: herbaceous perennials, shrubs and trees based on the 'OrganismType' classification in the metadata of the COMPADRE database.We also categorized herbaceous perennials as either semelparous or iteroparous.Semelparous herbs die immediately after reproduction and typically have high fecundity to compensate for the loss of future reproductive opportunities (Charnov & Schaffer, 1973;Pianka, 1976Pianka, , 1978)).The trade-off between fecundity and adult survival in semelparous herbs affects elasticity (Takada & Kawai, 2020).Semelparous herbs were identified from the elements of population projection matrices by using the method described by Takada et al. (2018).
We obtained interstage flow matrices for 14 species of semelparous herbs, 143 species of iteroparous herbs, 36 species of shrubs, 93 species of trees and 286 species in total.In addition, we obtained elasticity matrices for 14 species of semelparous herbs, 135 species of iteroparous herbs, 36 species of shrubs, 91 species of trees and 276 species in total.

| Classification of matrix elements into stasis, fecundity and growth
We classified each interstage flow matrix element as reflecting the developmental stage transitions of fecundity, growth to larger size classes, or stasis within the same stage using an approach similar to the method used by Silvertown et al. (1996) (see Figure 3a).The diagonal or upper diagonal elements of U (transitions to the same or lower stages) were classified as stasis and lower diagonal elements of U (transitions to larger stages) were classified as growth.All elements of F (number of seeds produced per individual) were classified as fecundity.All elements of C (clonal reproduction) were classified as growth.
An example of the classification process is shown in Figure 3b.In this case, each element is classified into stasis, fecundity or growth.
The flow matrix is derived according to Equation (2) using the population projection matrix A. We summed the elements of the flow matrix for stasis, fecundity and growth, respectively (see Figure 3b).(a)

| Statistical analysis
We calculated interstage flow matrices for the selected plant species based on Equation (2).We used Dirichlet regression to describe how the patterns of interstage flows vary across plant FGs.
Dirichlet regression is used when a set of bounded variables has a constant sum, for example with proportions and probabilities (Adler et al., 2014).In our study, the sum of proportional values of interstage flows related to stasis, fecundity and growth was one, making Dirichlet regression a suitable choice.MD (the number of developmental stage transitions described by the matrix) is not an innate characteristic of plant species, but rather is defined by researchers and reflects the design and constraints of individual studies.MD has been shown to influence other statistics derived from matrices, such as PGR and elasticity (Ramula et al., 2008;Ramula & Lehtila, 2005).We, therefore, included MD as an explanatory variable in our analysis.Categorical variables for FG, PGR, MDs, interaction terms of FG and PGR, interaction terms of FG and MD, and interaction terms of PGR and MD were included in our model as explanatory variables.All analyses used R version 4.1.0software (R Core Team, 2021), and we used the DirichReg() function in the DirichletReg package (Maier, 2021).

| RE SULTS
FGs had distinct patterns of flow allocation among stasis, fecundity and growth (Table 1; Figures 4 and 5).The ternary plots in Figure 4 illustrate these differences.Semelparous herbs had interstage flows that tended to be dominated by growth and fecundity (Figure 4a).
The spatial median, which shows the central tendency of the distribution of interstage flows, of semelparous herbs is located in an area with large values of growth and fecundity, suggesting that the interstage flows of semelparous herbs tended to be dominated by growth and fecundity.At the other end of the spectrum, the interstage flows of trees tended to be dominated by stasis (Figure 4d).Iteroparous herbs and shrubs tended to have intermediate patterns that spanned a range between those two extremes (semelparous herbs and trees) (Figure 4).
PGR and MD also influenced patterns of interstage flow.
Growing populations tended to have large interstage flows related to fecundity and relatively small flows related to both growth and stasis (Figure 5).Overall, PGR had a statistically significant influence on interstage flows related to stasis and growth, but the influence differed among FGs (see interaction terms between FG and PGR in Table 1).Interstage flows related to stasis tended to decrease with MD, but this influence also varied among FGs and PGR (Table 1).
The pattern of elasticities among FGs broadly mirrored those observed for interstage flow.Trees tended to have large elasticities for stasis, and semelparous herbs tended to have large elasticities for growth, whereas iteroparous herbs and shrubs had elasticities that spanned an intermediate range between growth and stasis.But in contrast to interstage flow, elasticities generally ascribed less of an influence on fecundity (Figure 4).Also, in contrast to interstage flow, the largest elasticities in growing populations tended to be related to growth (Figure 6).Overall, PGR, MD, and their interaction had a statistically significant influence on the elasticities related to stasis, fecundity and growth (Table 2).

| Comparison between interstage flow and elasticity
We showed the difference between elasticity and interstage flow in Figure 2a,b using a hypothetical example of two population matrices.The results indicate that even if interstage flows of different demographic processes differ, their elasticities can be similar.
That is, demographic processes can similarly change the PGR proportionally (elasticity represents the proportional change in PGR with a proportional change in a population matrix element), while their contribution to PGR itself (interstage flow) differs.Similarly, the results imply that even if elasticities of two different demographic processes differ, interstage flows of their processes could be similar.

Patterns of interstage flow varied among plant functional types
(Table 1; Figure 4) in ways that are consistent with life history tradeoffs and that are broadly similar to those based on elasticities.
However, interstage flow generally ascribed a larger role for fecundity than did elasticity and that pattern was consistent across FGs (Figure 4).In addition, while growing populations tended to be dominated by interstage flows related to fecundity, they tended to have high elasticities related to growth (Figures 5 and 6).However, these are potentially distinct demographic processes (Salguero-Gómez, 2018).

| Retrospective and prospective approaches
The different patterns we observed for interstage flow and elasticity reflect their different perspectives on population growth.Interstage flow is derived from the steady-state properties of populations (i.e. stable stage distributions and vital rates).Interstage flow, therefore, describes how PGR would be driven by the different stage transitions.Interstage flow is a retrospective analysis that describes how PGR has been influenced by events and conditions in the recent past.In contrast, elasticity is a prospective analysis that describes the relative contribution of vital rates to expected future PGR after a theoretical perturbation (Caswell, 2001).
Retrospective and prospective approaches provide complementary perspectives that are useful in different contexts (Horvitz et al., 1997).For example, elasticity analysis has been commonly used in the management of non-native invasive populations to predict how targeting control efforts at different life stages or vital rates will influence future population growth (Kerr et al., 2016).However,  2. Both the abundance of individuals and the composition functional traits can vary among populations within a species.These intraspecific differences can in turn influence the strength of species interactions and community composition (Start, 2020).
Interstage flow could be a useful tool for better understanding the linkages between abundance, functional composition and community dynamics.For instance, interstage flow could be helpful in developing mechanistic stage structured demographic models that link the often stage dependent expression of functional traits and the density dependent dynamics of competitors or predators.
3. Life history stages can vary in their environmental requirements, and this can be an important consideration for conservation efforts.For example, the elevational distribution of plant species is shifting in response to global climate change, but the ability of species to shift their elevational range is complicated by the fact that seedling and adult stages often have different abiotic requirements (Lenoir et al., 2009).Interstage flow could be a useful metric for planning and monitoring conservation efforts such as population translocations.
4. Interstage flow may also be useful for addressing questions that involve the link between population dynamics and broader ecosystem properties.For example, plants allocate assimilated energy to processes that support fecundity, growth and survival (Harper, 2010).The degree to which relative allocations vary across FGs or across populations experiencing different environmental conditions can affect ecosystem processes.For example, when resource availability is low or environmental stressors such as herbivory and abiotic stress are high, most of the gross plant assimilated energy may be allocated to support functions related to survival (such as herbivore defence), making less primary production available to consumers (Fridley, 2017) through an increase in biomass and is, therefore, a way to link demography and energy flow.

| Interstage flow in integral projection models
Integral projection models (IPMs) were developed as an alternative to MPMs about 20 years ago and are in the form of a dynamical equation with time-discrete and stage-continuous scheme (Easterling et al., 2000).The mathematical expression is as follows: where n(x, t) and K(y, x) represent a size distribution at time t (continuous function of size x) and the contribution of individuals with size x to size y called as kernel, respectively.The concept of interstage flow proposed in this paper can be applied to IPMs.The interstage flow is equivalent to the product of the kernel and size distribution, K(y, x)n(x, t) in IPM.The unique property of interstage flow related to the PGR λ holds also in IPMs.
Suppose that the dynamics of Equation ( 8) converges to a stable size distribution with growing at a PGR ( ) and the stable dis- projection matrix, the matrix elements of interstage flow explicitly decompose population growth into the contributions made by individuals transitioning between different developmental stages.The idea of interstage flow was introduced by Kawano et al. (1987), but they neither formally interpreted its ecological meaning nor comprehensively analysed its properties.No work has demonstrated how the proportional change of PGR with a proportional change in a population matrix element (elasticity) can be decomposed into interstage flows.Also, no work has been done to test how interstage flow matrices vary across taxa or FGs.In this study we formally define the interstage flow matrix, describe some of its useful properties and demonstrate the relationship between interstage flow and elasticity.We then compare how patterns of elasticity and interstage flow vary across plant functional types and interpret the ecological meaning using 286 plant species from the COMPADRE database.We also explore some potential applications of interstage flow to ecological research.

E 1
The transition probabilities between stages and interstage flow of the Japanese perennial herb, Trillium apetalon.Four stages (seedling, one-leaf, three-leaf and flowering) were set to construct the population projection matrix of the species.The population projection matrix and the interstage flow matrix are shown on the top of the figure.The matrix is fromOhara et al. (2001) and was constructed using long-term census data of the Japanese perennial herb, Trillium apetalon.The flow chart between stages is shown on the bottom of the figure.The numbers attached to each arrow in the flow chart are the transition probabilities between stages.The normalized stable stage distribution (w) in the centre was obtained from the population projection matrix.Using the stable stage distribution and the population projection matrix elements, the interstage flows between stages were calculated as shown in the right part of the figure (see Equation2).Interstage flows are shown by each arrow on the right and in the interstage flow matrix.

a
13 and a 33 .The calculated proportional interstage flows and elasticities are shown in Figure 2a,b, respectively.Elasticities derived from M A and M B are similar, even for elements a 13 and a 33 .In contrast, there are more marked differences in proportional interstage flow especially in a 13 .This indicates that their interstage flows can be different even when population matrices have similar elasticities.Interstage flow and elasticity are related as follows (see Appendix S2 for details):where e kl and f IS ij are the elasticity and flow matrix elements, respectively.Equation (7) indicates that the sensitivity ∕ a kl is the sum of changes in interstage flows when the matrix element changes.In other words, we can decompose elasticity into changes in interstage flows ( f IS ij ∕ a kl ) based on Equation (7) (Figure2c).Elasticity provides information on the predicted proportional influence of matrix elements on PGR, but it does not provide information on how changes in growth rate are driven by changes in transitions of individuals between life stages.The decomposition of elasticity into changes in interstage flows explicitly describes these changes to stage transitions and the sum of change is proportion to elasticity, e kl ∝ Figure 3a).The submatrix U contains the transition and survival rates of individuals in each age or stage.Submatrix F contains the number of seeds produced per individual, and submatrix C contains the clonal reproduction rates.
Interstage flows and elasticities.(a) proportions of interstage flows of matrix M A and M B , (b) elasticities of matrix M A and M B , (c) changes in interstage flows, f IS ij ∕ a 33 .a ij is matrix elements of M A and M B .Elasticity to a 33 is e 33 = that, in the (a), the interstage flows are normalized by dividing the population growth rate to compare the interstage flows and elasticities.
To allow comparison of interstage flows among plant species with different PGRs, we divided the stasis, fecundity and growth by its PGR-defined as proportional values of stasis, fecundity and growth, respectively.The sum of the proportional values of stasis, fecundity and growth equals 1.0.F I G U R E 3 Classification of matrix elements into stasis, fecundity and growth.(a) Population projection matrix A is partitioned into three submatrices: U contains only transitions and survival of existing individuals; F sexual reproduction; C clonal reproduction.The diagonal or upper diagonal elements of U were classified as 'stasis', and lower diagonal elements of U were classified as 'growth'.The elements of F and C were classified as 'fecundity' and 'growth', respectively.(b) A hypothetical example of matrix elements partitioned into stasis, fecundity or growth.The classification for each element is identical between population projection matrix and flow matrix.Flow matrix elements are summed for each class (i.e.stasis, fecundity and growth).The sum of the flow matrix elements is the population growth rate.
The interstage flow matrix, like other population statistics, potentially depends on how the population matrix is constructed.For instance, interstage flows and elasticity are dependent on MD.In addition, categorizing matrix elements to different demographic processes involves a degree of ambiguity and discretion.For example, we classified interstage flow matrix elements relating to asexual reproduction (i.e.clonal growth) into growth, the same as transitions into larger size classes (i.e.growth of smaller plants into larger ones).

TA B L E 1
Summary of the results of Dirichlet regression for summed interstage flows for (1) stasis, (2) fecundity and (3) growth.FG_Semelparous, FG_Shrub and FG_Tree are categorical variables for semelparous herbs, shrubs and trees, respectively.F I G U R E 4 Ternary plot of interstage flow (a-d) and elasticity (e-h).Semelparous herbs, are plotted in (a) and (e).Iteroparous herbs are plotted in (b) and (f).Shrubs are plotted in (c) and (g).Trees are plotted in (d) and (h).Colour shading indicates population growth rates.Dashed lines indicate a value of 0.5 for each axis.Filled squares indicate the spatial median(Vardi & Zhang, 2000).F I G U R E 5 Dependence of interstage flow related to stasis, fecundity and growth on the logarithmic population growth rate.(a) semelparous herbs, (b) iteroparous herbs, (c) shrubs and (d) trees.The matrix dimensions used in the regression were the mean values of each functional group.is needed to understand how spatial variation in environmental conditions across the introduced range or periodic variation in conditions (such as associated with disturbance) affect the demography and growth of invasive populations.4.1.2| When are interstage flows useful?Elasticity describes the proportional capacity of different life history processes to influence population growth.Interstage flow describes how that capacity is realized in the supply of individuals transitioning between life stages.Patterns of elasticity and flow do not necessarily align across matrix elements.For instance, the contribution to population growth made by the supply of individuals transitioning between life stages (interstage flow) is distinct from the relative capacity of different life history processes to influence population growth (elasticity).For instance, a population could have a large supply of individuals transitioning between a specific life stage even if the relative capacity of the life history process to drive population growth is low.Conversely, there can be stage transitions that have a large capacity to influence population growth even if the supply of individuals making the life stage transition is low.We think that this distinction could be important in several circumstances.1. Patterns of interstage flow likely vary among populations within and between species in ways that reflect demographic responses to differing environmental conditions or different demographic contexts, such as those associated with the introduction of non-native populations.Interstage flow may, therefore, be a useful descriptor of demographic variation across populations and as a metric for testing the association between environmental variation and components of population growth.
tribution, n * (y), is proportional to a normalized eigen-function, u(y): n * (y) = u(y), where ∫ u(y)dy = 1.Then, Integrating both sides of Equation (9) and using ∫ u(y)dy = 1, we obtain The total sum of interstage flows in IPM is exactly same as PGR, as we proved in Equation (4) in MPMs.This proposed population statistic, the interstage flow matrix, describes the contribution of individual transitions between developmental stages to population growth.We suggest that future research uses the interstage flow matrix to decompose PGR into population projection matrix elements in order to specify contributions to PGRs., x)u(x)dxdy.
Dependence of elasticity related to stasis, fecundity and growth on the logarithmic population growth rate.(a) semelparous herbs, (b) iteroparous herbs, (c) shrubs and (d) trees.The matrix dimensions used in the regression were the mean values of each functional group.Summary of the results of Dirichlet regression for summed elasticities for (1) stasis, (2) fecundity and (3) growth.FG_Semelparous, FG_Shrub and FG_Tree are categorical variables for semelparous herbs, shrubs and trees, respectively.