R.1. Basic anatomy of life cycles and MPMs

Matrix population models are a mathematical tool that integrate population dynamics and population structure to model the dynamics of populations with different ‘types’ of individuals, whether due to age, life stage, sex, genotype or any other attribute that causes differences in demographic rates. Consider a plant species with three different types of individuals: seedlings that have newly recruited into the population from seeds, non-reproductive rosettes, and flowering adults that make seeds and also reproduce asexually by vegetative budding. We can represent the biology of this life cycle graphically (Fig. 1) where each life stage is a node and the life stages are connected with arrows showing the paths (and probabilities) that individuals move between stages.

Figure 1. Life cycle diagram for a hypothetical plant with three life stages (nodes). Arrows are coloured by conventional groupings of demographic processes: growth and survival (green), sexual reproduction via seeds (gold) and asexual reproduction via budding (purple). The numbers along each arrow indicate the transition probability–how many individuals will there be in the stage at the arrow’s end at the next time step for each individual at the arrow’s start?

The transition or projection matrix (Fig. 2; called A in matrix notation) for this life cycle is a square matrix where the rows and columns correspond to the life stages (nodes) and the elements correspond to transition probabilities (arrows). By convention, the columns reflect the current life stage (i.e., the arrow’s start) and the rows reflect the arrow’s end.

Figure 2. Transition (projection) matrix corresponding to the life cycle in Fig. 1.

For basic MPM projection and equilibrium analyses, this single matrix is all that is needed. However, more advanced analyses like many of those in Rage use decompositions of the A matrix into its components of growth and survival (U) and reproduction (R). Reproduction can be further decomposed into offspring produced sexually (F) and asexually/clonally (C; Fig. 3). The submatrices combine additively to create the transition matrix,

\[ \begin{aligned} \mathbf{A} &= \mathbf{U} + \mathbf{R} \\ &= \mathbf{U} + \left(\mathbf{F} + \mathbf{C}\right). \end{aligned} \] In our example, all of the elements of A are found in only one of the submatrices. However, other life cycles may have multiple pathways from one life stage to another.

Figure 3. Decomposition of the transition matrix, A, into U, F and C submatrices. The coloured elements correspond to the coloured arrows in Fig. 1.

R.2. Projecting population change

Simple matrix multiplication allows the full transition matrix, A, to be projected forward in time for any starting population. We initiate the model with a population vector, \(\mathbf{n}_{t=0}\), that contains the number of individuals in each life stage. Right multiplying the initial population vector to the transition matrix, A, yields the population vector at the next time step via the projection equation,

\[ \mathbf{n}_{t+1} = \mathbf{A}\mathbf{n}_{t}. \] Suppose we initialize the example plant population, \(\mathbf{n}_{t=0}\), with five seedlings, 10 rosettes, and 15 flowering individuals. We expect that the population in the next time step will be,

\[ \begin{aligned} \mathbf{n}_{t=1} &= \mathbf{A}\mathbf{n}_{t=0}, \\ \left[ \begin{matrix} 48.0 \\ 17.5 \\ 17.5 \end{matrix} \right] &= \left[ \begin{matrix} 0 & 0 & 3.2 \\ 0.5 & 0.3 & 0.8 \\ 0 & 0.4 & 0.9 \end{matrix} \right] \left[ \begin{matrix} 5 \\ 10 \\ 15 \end{matrix} \right]. \end{aligned} \] Summing the population vectors gives us the total population size (\(N_{t=0} =\sum\mathbf{n}_{t=0} = 30\) and \(N_{t=1} = 83\)). Likewise, we can measure the instantaneous per-capita population growth rate as \(N_{t+1}/N_{t} = 83/30 = 2.7\overline{6}\)). Recursively substituting the new \(\mathbf{n}_{t+1}\) into the projection equation in place of \(\mathbf{n}_{t}\) allows us to continue projecting the population dynamics and life stage composition forward in time.

R.3. Analysis of population equilibrium (eigenanalysis)

Provided that the transition matrix, A, meets certain conditions, iterative projections from any starting population vector \(\mathbf{n}_{t=0}\) will eventually converge to an equilibrium per-capita population growth rate (\(\lambda\)) and stable stage distribution (\(\mathbf{w}\)). We can determine \(\lambda\) and \(\mathbf{w}\) by finding the dominant eigenvalue and its associated (right) eigenvector, respectively. The popdemo package provide functions to do these calculations (the package popbio also provides extensive functionality for matrix model analysis):

library(popdemo)

# define the transition matrix, A
A <- rbind(c(0.0, 0.0, 3.2),
           c(0.5, 0.3, 0.8),
           c(0.0, 0.4, 0.9))

# lambda: equilibrium per-capita population growth rate
popdemo::eigs(A = A, what = "lambda")
#> [1] 1.51273

# w: stable stage distribution (relative frequencies)
popdemo::eigs(A = A, what = "ss")
#> [1] 0.4551941 0.3296228 0.2151831

From \(\lambda\), we infer that the population would eventually grow at an equilibrium rate of 51% per time step. Likewise, from \(\mathbf{w}\) we infer that nearly half of individuals will be seedlings, about a third will be rosettes and a fifth will be flowering at equilibrium.

Finally, we are reminded that MPM projections rely on consistency of the demographic vital rates (i.e., transition matrix elements) over time. Projections tell us what would happen if this strong assumption were to be met. Projections differ radically from forecasts, which are designed to predict what will happen. Forecasting models may use MPMs to drive population change, but will also typically require a component that predicts changes in the underlying demographic vital rates. Keyfitz (1972) unpacks these differences further, with special emphasis on the value of projection as a window into the fundamental processes that drive the behaviour of populations. Much of the functionality of Rage rests on this ethos.

1. Standardized vital rates

The elements of an MPM transition matrix (A) generally are composites of two or more vital rates (sometimes called ‘lower-level vital rates’). For example, the transition from stage 1 to stage 2 (element \(a_{21}\)) may colloquially be thought of as ‘growth,’ but importantly this transition is growth conditional on the individual surviving. Assuming a post-breeding census design, we can retroactively partition each transition in MPM submatrices into survival (using the column sums of U) and one of the following: growth (using the lower triangle of U), shrinkage (upper triangle of U), stasis (diagonal of U), dormancy (from U with user-indicated dormant stages), fecundity (from F) or clonality (from C).

The vr_vec_* family of functions provide a means to calculate lower-level vital rates for each life stage in the input matrix or matrices. For example, we can calculate stage-specific survival and stasis from our example U matrix:

vr_vec_survival(matU = mpm1$matU)
#>    seed   small  medium   large dormant 
#>    0.15    0.50    0.66    0.86    0.39
vr_vec_stasis(matU = mpm1$matU)
#>      seed     small    medium     large   dormant 
#> 0.6666667 0.2400000 0.1818182 0.6046512 0.4358974

Multiplying these two together yields the probability of stasis conditional on survival, in other words the diagonal of our example U matrix.

# product of Pr(survival) and Pr(stasis) yields Pr(stasis|survived)
vr_vec_survival(matU = mpm1$matU) * vr_vec_stasis(matU = mpm1$matU)
#>    seed   small  medium   large dormant 
#>    0.10    0.12    0.12    0.52    0.17
diag(mpm1$matU)  # equivalent to the diagonal of U matrix
#>    seed   small  medium   large dormant 
#>    0.10    0.12    0.12    0.52    0.17

Rage also provides functions to summarize these vital rates across stage classes using the vr_* family of functions. These functions return a single value that is the mean of the corresponding stage-specific vital rate vector from vr_vec_*. Life stages may be excluded, allowing the user to tailor these calculations to the life history of the organism and their working definition of each vital rate. Similarly, custom weights are allowed to control the contributions of life stages to the average. This functionality is more fully described in the Vital Rates vignette.

vr_survival(matU = mpm1$matU, exclude_col = 1)  # exclude 'seed' stage
#> [1] 0.6025
mean(vr_vec_survival(mpm1$matU)[-1])     # equivalent to the mean without 'seed'
#> [1] 0.6025

2. Deriving life tables from MPMs for age-from-stage analyses

Classical demographic analyses were based on age-structured populations, in which surviving individuals always progress to the next age class. However, MPMs allow for demographic rates to be structured by any number of biological criteria, whether stage, sex, size, or condition. In these alternatively-structured models, individuals may remain in the same state or even ‘regress’ to states they had previously. Here there is no one-to-one mapping between state and age. Each individual still has an age, and understanding their age-specific demographic trajectories nevertheless offers a useful window into their life history. Rage provides functions that decompose MPMs into life tables, thereby allowing for a wide array of demographic analyses based on age-specific trajectories (described below in section 4).

Following the age-from-stage methods of Caswell (2001, ch. 5.3), we can convert an MPM to a full, age-structured life table or any of its component columns.

lt <- mpm_to_table(matU = mpm1$matU, matF = mpm1$matF)  # full life table
lt
#>   x         lx         dx        hx        qx        px        ex       mx
#> 1 0 1.00000000 0.85000000 1.4782609 0.8500000 0.1500000 0.7306918  0.00000
#> 2 1 0.15000000 0.11000000 1.1578947 0.7333333 0.2666667 1.0379453  0.00000
#> 3 2 0.04000000 0.02016000 0.6737968 0.5040000 0.4960000 1.5172950  9.54125
#> 4 3 0.01984000 0.00742535 0.4604204 0.3742616 0.6257384 1.5509980 19.49277
#> 5 4 0.01241465 0.00397750 0.3815018 0.3203876 0.6796124 1.1796124 24.83256
#> 6 5 0.00843715 0.00843715 2.0000000 1.0000000 0.0000000 0.5000000 26.79977
#>        lxmx
#> 1 0.0000000
#> 2 0.0000000
#> 3 0.3816500
#> 4 0.3867365
#> 5 0.3082875
#> 6 0.2261137
lx <- mpm_to_lx(matU = mpm1$matU)  # survivorship to start of each age class
lx
#> [1] 1.00000000 0.15000000 0.04000000 0.01984000 0.01241465 0.00843715

Rage also supports translation between the different ways of expressing age-specific mortality/survival, whether proportional survivorship (lx), survival probability (px) or mortality hazard (hx), following an x_to_y function name pattern.

lx_to_px(lx = lx)  # survivorship to survival probability
#> [1] 0.1500000 0.2666667 0.4960000 0.6257384 0.6796124        NA
lx_to_hx(lx = lx)  # survivorship to mortality hazard
#> [1] 1.8971200 1.3217558 0.7011794 0.4688229 0.3862326        NA

When individuals can remain in the last life stage class, age-specific trajectories of mortality and fertility can appear to plateau as a cohort asymptotically approaches its stable stage distribution (SSD). For these MPMs, we can choose a quasi-stationary distribution (QSD) of stages that is within an arbitrarily close threshold of the SSD. Subsetting age-specific demographic trajectories to time points before a cohort reaches the QSD will minimise these plateau artefacts. In the example MPM, mpm1, a cohort of germinated individuals quickly converges towards the SSD:

# project a germinated cohort through the U matrix
cohort <- popdemo::project(A = mpm1$matU, vector = c(0,1,0,0,0), time = 10)
popStructure <- vec(cohort)/rowSums(vec(cohort))

matplot(popStructure, type = "l", xlab = "time", ylab = "proportion in stage class")

Using qsd_converge, we can find the time at which the cohort reaches the QSD and subset the life table or any component age-specific trajectory accordingly.

# calculate time to QSD from the U matrix of an MPM
(q <- qsd_converge(mat = mpm1$matU, start = "small"))
#> [1] 6

# subset the life table rows to ages prior to the QSD
lt_preQSD <- lt[1:q,]

# plot mortality trajectory from the life table subset (blue),
# showing plateau effect if the trajectory (grey) was allowed to continue to the
# QSD (dashed vertical line) and beyond
plot(qx ~ x, data = lt, type = "l", col = "darkgrey", ylim = c(0, 1), xlab = "age")
lines(qx ~ x, data = lt_preQSD, type = "l", col = "blue", lwd = 4)
abline(v = q, lty = "dashed")

We refer readers to the AgeFromStage vignette for further detail on these methods.

3. Perturbation analyses

Perturbation analyses of MPMs measure the response of demographic statistics, such as the equilibrium per-capita population growth rate (\(\lambda\)), to perturbations typically applied at the level of individual matrix elements and which may be perturbations of a fixed or proportionate amount (sensitivities and elasticities, respectively). By extension, the vital rates (survival, growth, etc.) or transition type (stasis, retrogression, etc.) underlying multiple matrix elements may be similarly perturbed. Rage provides functions for each scope of perturbation, allowing the user to specify the desired demographic statistic and type of perturbation.

# construct the transition matrix A = U + F (+ C when present)
mpm1$matA <- with(mpm1, matU + matF)

# sensitivity of lambda to...
# ...matrix element perturbations
perturb_matrix(matA = mpm1$matA,
               type = "sensitivity", demog_stat = "lambda")
#>               seed      small      medium       large     dormant
#> seed     0.2173031 0.01133203 0.004786307 0.002986834 0.001150702
#> small    4.4374613 0.23140857 0.097739871 0.060993321 0.023498190
#> medium  10.8654599 0.56661979 0.239323187 0.149346517 0.057537004
#> large   21.3053309 1.11104269 0.469270739 0.292842888 0.112820081
#> dormant  3.6111989 0.18831948 0.079540420 0.049636196 0.019122780
# ...vital rate perturbations
perturb_vr(matU = mpm1$matU, matF = mpm1$matF,
           type = "sensitivity", demog_stat = "lambda")
#> $survival
#> [1] 2.986054
#> 
#> $growth
#> [1] 1.077597
#> 
#> $shrinkage
#> [1] -0.1653284
#> 
#> $fecundity
#> [1] 0.00572764
#> 
#> $clonality
#> [1] 0
# ...transition type perturbations
perturb_trans(matU = mpm1$matU, matF = mpm1$matF,
              type = "sensitivity", demog_stat = "lambda")
#> $stasis
#> [1] 1.000001
#> 
#> $retro
#> [1] 0.4174435
#> 
#> $progr
#> [1] 6.713571
#> 
#> $fecundity
#> [1] 0.007773141
#> 
#> $clonality
#> [1] NA

4. Deriving life history traits

What is the life expectancy of an individual? At what age will it begin to reproduce? How likely is it to survive to reproduction? What is the generation time? These high-level questions address the population-level life history traits that emerge from aggregating individual-level demographic rates, and tracing trajectories through the life cycle.

Rage provides functions to calculate a wide range of life history traits from both MPMs and life tables. We illustrate the general pattern of these functions below, referring the reader to the LifeHistoryTraits vignette and the documentation webpage for a complete list of functionality.

MPM-based analyses take the required submatrix (or submatrices), optionally allowing the user to specify the life stage at which to start the calculation, choose among alternative methods and set cut-off thresholds.

# post-germination time steps until post-germination survivorship falls below 5%
longevity(matU = mpm1$matU, start = "small", lx_crit = 0.05)
#> [1] 7
# expected lifetime production of 'small' offspring by a 'small' individual
net_repro_rate(matU = mpm1$matU, matR = mpm1$matF, start = "seed",
               method = "start")
#> [1] 1.852091

When calculating life table-based traits, MPMs can be transformed to the necessary age-based trajectory prior to analysis.

# derive post-germination survivorship trajectory from U matrix
lx <- mpm_to_lx(matU = mpm1$matU, start = "small")
entropy_k(lx = lx)       # calculate Keyfitz' entropy
#> [1] 0.9077186

5. Transforming MPMs

Rage includes a variety of functions that can be used to manipulate or transform MPMs. For example, we can collapse an MPM to a smaller number of stage classes using mpm_collapse().

# collapse 'small', 'medium', and 'large' stages into single stage class
col1 <- mpm_collapse(matU = mpm1$matU, matF = mpm1$matF,
                     collapse = list(1, 2:4, 5))
col1$matA
#>      [,1]        [,2] [,3]
#> [1,] 0.10 11.61331815 0.00
#> [2,] 0.05  0.53908409 0.22
#> [3,] 0.00  0.05728085 0.17

Notice that the stage names are lost during this operation. We can re-add new stage names using the function name_stages, electing to use generic or custom names.

(col1_auto <- name_stages(mat = col1, prefix = "class_"))  # automated stage naming
#> $matA
#>         class_1     class_2 class_3
#> class_1    0.10 11.61331815    0.00
#> class_2    0.05  0.53908409    0.22
#> class_3    0.00  0.05728085    0.17
#> 
#> $matU
#>         class_1    class_2 class_3
#> class_1    0.10 0.00000000    0.00
#> class_2    0.05 0.53908409    0.22
#> class_3    0.00 0.05728085    0.17
#> 
#> $matF
#>         class_1  class_2 class_3
#> class_1       0 11.61332       0
#> class_2       0  0.00000       0
#> class_3       0  0.00000       0
#> 
#> $matC
#>         class_1 class_2 class_3
#> class_1       0       0       0
#> class_2       0       0       0
#> class_3       0       0       0
(col1_cust <- name_stages(mat = col1, names = c("seed", "active", "dormant"),
                         prefix = NULL))  # overwrite with custom stages
#> $matA
#>         seed      active dormant
#> seed    0.10 11.61331815    0.00
#> active  0.05  0.53908409    0.22
#> dormant 0.00  0.05728085    0.17
#> 
#> $matU
#>         seed     active dormant
#> seed    0.10 0.00000000    0.00
#> active  0.05 0.53908409    0.22
#> dormant 0.00 0.05728085    0.17
#> 
#> $matF
#>         seed   active dormant
#> seed       0 11.61332       0
#> active     0  0.00000       0
#> dormant    0  0.00000       0
#> 
#> $matC
#>         seed active dormant
#> seed       0      0       0
#> active     0      0       0
#> dormant    0      0       0

The transition rates in the collapsed matrix are a weighted average of the transition rates from the relevant stages of the original matrix, weighted by the stable distribution at equilibrium. This process guarantees that the collapsed MPM will retain the same population growth rate as the original. However, other demographic and life history characteristics will not necessarily be preserved.

# compare population growth rate of original and collapsed MPM (preserved)
popdemo::eigs(A = mpm1$matA,what = "lambda")
#> [1] 1.121037
popdemo::eigs(A = col1_cust$matA, what = "lambda")
#> [1] 1.121037

# compare net reproductive rate of original and collapsed MPM (not preserved)
net_repro_rate(matU = mpm1$matU, matR = mpm1$matF)
#> [1] 1.852091
net_repro_rate(matU = col1_cust$matU, matR = col1_cust$matF)
#> [1] 1.447468