Age-specific sensitivity analysis of stable, stochastic and transient growth for stage-classified populations

Abstract

Sensitivity analysis in ecology and evolution is a valuable guide to rank demographic parameters depending on their relevance to population growth. Here, we propose a method to make the sensitivity analysis of population growth for matrix models solely classified by stage more fine-grained by considering the effect of age-specific parameters. The method applies to stable population growth, the stochastic growth rate, and transient growth. The method yields expressions for the sensitivity of stable population growth to age-specific survival and fecundity from which general properties are derived about the pattern of age-specific selective forces molding senescence in stage-classified populations.

Keywords: aging; elasticity analysis; population growth; senescence; sensitivity analysis; stage; stochastic growth rate; transient growth.

FIGURE 1

(a) Life cycle graph of Dipsacus sylvestris. Nodes represent stages. Blue arrows indicate survival transitions and correspond to entries of the $\mathbf{U}$ matrix. Magenta arrows indicate fecundities and correspond to entries of the $\mathbf{F}$ matrix. There is a single reproductive stage (magenta), Flowering plant, that can be accessed from any Rosette stage (small, medium and large). There are two Seed stages (dormant year 1 and dormant year 2). Offspring stages (green) are those where new recruits are first censused. (b) Stable age distribution of flowering plants and selection on their age‐specific fecundity in Dipsacus. Theoretical considerations (see main text) suggest that: age distribution and selection are proportional to one another, flowering plants can only be aged $\ge 2$ and the age distribution initially increases and eventually decreases with age. Data for this analysis are from Caswell (, p. 60). For this model, stable population growth is $\lambda =2.33$ and individuals of every stage experience some nonzero mortality. The Supporting Information (Giaimo & Traulsen, 2022a) contains code to generate this panel.

FIGURE 2

Age‐specific selective forces within females stages of Arisaema serratum. These forces are computed using the method proposed here (circles) and a previous method (diagonal crosses) by Caswell (2012) that Caswell and Salguero‐Gómez (, their fig. 3) employed to compute the same quantities. The methods are in perfect agreement. The distinctive analytical power of the proposed method, however, allows us to make a step further and predict (horizontal lines) from Equation 22 the existence of (and not only the theoretical possibility of visualizing) an age‐independent ratio between the force of selection against mortality and the force of selection on fecundity at each age within a single stage. The Supporting Information (Giaimo & Traulsen, 2022a) contains code to generate this figure.

FIGURE 3

Age‐specific elasticities of stochastic growth ${\lambda }_s$. We initiated a population with random stage distribution. We demographically updated the populations for 70,000 time‐steps using a randomly generated sequence of projection matrices. We computed elasticities of ${\lambda }_s$ to different demographic processes: fecundity (F), individual growth (G), retrogression (R), and stasis (St). Age‐specific elasticities were computed for each case using Equation 30. Only age‐specific elasticities up to age 7 are reported here, as elasticities for later ages contributed only minimally to the overall age‐independent elasticity for this dataset. In computing elasticities, we removed the first and last 10,000 steps to minimize transient effects. The analysis is based on demographic data for population E of kidney vetch in Davison et al. (2010). The Supporting Information (Giaimo & Traulsen, 2022a) contains code to generate this figure.

FIGURE 4

Age‐specific elasticities of transient growth ${\lambda }_t$. We initiated a population with uniform stage distribution and random age distribution with maximum initial age of 30. We demographically updated the populations for 30 time‐steps using a randomly generated sequence of projection matrices. We computed elasticities of ${\lambda }_t$ to different demographic processes: fecundity (F), individual growth (G), retrogression (R), and stasis (St). Age‐specific elasticities were computed for each case using Equation 32 with initial vectors $\partial \mathbf{n}\left(0\right)/\partial \theta$ set equal to the zero vector. Only age‐specific elasticities up to age 7 are reported here, as elasticities for later ages contributed only minimally to the overall age‐independent elasticity for $t\ge 6$ in these simulations. Note that elasticities are on different scales. The analysis is based on demographic data for population E of kidney vetch in Davison et al. (2010). The Supporting Information (Giaimo & Traulsen, 2022a) contains code to generate this figure.

FIGURE B1

Convergence to a common age distribution. We used demographic data for population C of kidney vetch in Davison et al. (2010). We initiated ten populations. To each, we assigned a random age and stage distribution with maximum age $30$. We updated the populations for several time steps by subjecting the populations to the same randomly generated sequence of projection matrices. Here, we report for different time points (columns) and within each stage (row), the differences between the age distribution of each population and the arithmetic mean of the ten age distributions. Different lines correspond to different populations. As time increases, differences vanish and lines overlap so that, eventually, only a single horizontal line at level 0 is visible. The Supporting Information (Giaimo & Traulsen, 2022a) contains code to generate this figure.