In this vignette, we will use the `lathyrus`

dataset to
illustrate the estimation of **raw MPMs**, with the
intention of producing matrices similar to those published in Ehrlén (2000). These matrices will not be the
same, as the dataset included in this package includes data for more
individuals as well as an extra year of monitoring. It also includes
differences in classification due to different assumptions regarding
transitions to and from vegetative dormancy, which is an unobservable
life history stage in which herbaceous perennial plants spend the
growing season belowground, without aboveground structures. To reduce
output size, we have prevented some statements from running if they
produce long stretches of output. Examples include most
`summary()`

calls. In these cases, we include hashtagged
versions of these calls, and our text assumes that the user runs these
statements without hashtags. We have also written this vignette using
the default Ehrlén format as the output format for all historical MPMs.
All examples work with deVries format as well. This vignette is only a
sample analysis. Detailed information and instructions on using
`lefko3`

are available through a free online e-book called
*lefko3: a gentle introduction*, available on
the projects page
of the Shefferson lab website. This website also includes links to
long-format vignettes and other demonstrations of this package.

*Lathyrus vernus* (family Fabaceae) is a long-lived forest herb,
native to Europe and large parts of northern Asia. Individuals increase
slowly in size and usually flower only after 10-15 years of vegetative
growth. Flowering individuals have an average conditional lifespan of
44.3 years (Ehrlén and Lehtila 2002).
*L. vernus* lacks organs for vegetative spread and individuals
are well delimited (Ehrlén 2002). One or
several erect shoots of up to 40 cm height emerge from a subterranean
rhizome in March and April. Flowering occurs about four weeks after
shoot emergence. Shoot growth is determinate, and the number of flowers
is determined in the previous year (Ehrlén and
Van Groenendael 2001). Individuals do not necessarily produce
aboveground structures every year, and instead can remain dormant in one
or more seasons. *L. vernus* is self-compatible but requires
visits from bumble-bees to produce seeds. Individuals produce few, large
seeds and establishment from seeds is relatively frequent (Ehrlén and Eriksson 1996). The pre-dispersal
seed predator *Bruchus atomarius* often consumes a large fraction
of developing seeds, and roe deer (*Capreolus capreolus*)
sometimes consume the shoots (Ehrlén and
Munzbergova 2009).

Data for this study were collected from six permanent plots in a
population of *L. vernus* located in a deciduous forest in the
Tullgarn area, SE Sweden (58.9496 N, 17.6097 E), from 1988 to 1991 (Ehrlén 1995). The six plots were relatively
similar with regard to soil type, elevation, slope, and canopy cover.
Within each plot, all individuals were marked with numbered tags that
remained over the study period, and their locations were carefully
mapped. New individuals were included in the study in each year.
Individuals were recorded at least three times every growing season. At
the time of shoot emergence, we recorded whether individuals were alive
and produced above-ground shoots, and if shoots had been grazed. During
flowering, we recorded flower number and the height and diameter of all
shoots. At fruit maturation, we counted the number of intact and damaged
seeds. To derive a measure of aboveground size for each individual, we
calculated the volume of each shoot as \(\pi ×
(\frac{1}{2} diameter)^2 × height\), and summed the volumes of
all shoots. This measure is strongly correlated with the dry mass of
aboveground tissues (\(R^2 = 0.924\),
\(P < 0.001\), \(n = 50\), log-transformed values; Ehrlén
1995). Size of individuals that had been grazed was estimated based on
measures of shoot diameter in grazed shoots, and the relationship
between shoot diameter and shoot height in non-grazed individuals. Only
individuals with an aboveground volume of more than 230 mm^{3}
flowered and produced fruits during this study. Individuals that lacked
aboveground structures in one season but reappeared in the following
year were considered dormant. Individuals that lacked aboveground
structures in two subsequent seasons were considered dead from the year
in which they first lacked aboveground structures. Probabilities of
seeds surviving to the next year, and of being present as seedlings or
seeds in the soil seed bank, were derived from separate yearly sowing
experiments in separate plots adjacent to each subplot (Ehrlén and Eriksson 1996).

The dataset that we have provided is organized in horizontal format,
meaning that rows correspond to unique individuals and columns
correspond to individual condition in particular observation occasions
(which we also refer to as *years* here). The original
spreadsheet file used to keep the dataset has a repeating pattern to
these columns, with each year having a similarly arranged group of
variables. Let’s take a look at this dataset. First, we clear memory and
load it. We will then look at its dimensions and its summary.

```
rm(list=ls(all=TRUE))
library(lefko3)
data(lathyrus)
dim(lathyrus)
#> [1] 1119 38
#summary(lathyrus)
```

This dataset includes information on 1,119 individuals arranged
horizontally, so there are 1,119 rows with data. There are 38 columns.
The first two columns are variables giving identifying information about
each individual. This is followed by four sets of nine columns, each
named `VolumeXX`

, `lnVolXX`

, `FCODEXX`

,
`FlowXX`

, `IntactseedXX`

, `Dead19XX`

,
`DormantXX`

, `Missing19XX`

, and
`SeedlingXX`

, where `XX`

corresponds to the year
of observation and with years organized consecutively. Thus, columns
3-11 refer to year 1988, columns 12-20 refer to year 1989, etc. This
strictly repeated pattern allows us to manipulate the original dataset
quickly and efficiently via `lefko3`

. We should note the
number of years, which is 4 here (includes all years from and including
1988 to 1991). Ideally, we should also have arranged the columns in the
same order for each year, with years in consecutive order with no extra
columns between them. This order is not required, provided that we are
willing to input all variable names in the correct order when
transforming the dataset later.

We will now create a **stageframe** describing the life
history of the species and linking it to the data. A stageframe is a
data frame that describes all stages in the life history of the
organism, in a way usable by the functions in this package and using
stage names and classifications that completely match those used in the
dataset. In essence, it links the dataset to the *life cycle
graph* used to model the organism’s life history. It should include
complete descriptions of all stages that occur in the dataset, with each
stage defined uniquely. Since this object can be used for automated
classification of individuals, all sizes, reproductive states, and other
characteristics defining each stage in the dataset need to be accounted
for explicitly. The final description of each stage occurring in the
dataset must not completely overlap with any other stage also found in
the dataset, although partial overlap is allowed and expected.

Here, we create a stageframe named `lathframe`

based on the
classification used in Ehrlén (2000). We
build this by creating vectors of the characteristics describing each
stage, with each element always in the same order within the vector. Of
particular note are the vectors input as `sizes`

and
`binhalfwidth`

in the `sf_create()`

function. The
values in the former vector correspond to the midpoint values of each
size class bin, while values in the latter vector correspond to one-half
of the width of the bin. If size values are not to be binned, then
narrow half bin-widths can be used (the default is 0.5, and all entries
must be positive). For example, in this dataset, vegetatively dormant
individuals necessarily have a size of zero, and so we can set the
`halfbinwidth`

for this stage to 0.5 provided that the
resulting size bin does not overlap with any other size bin matching the
other characteristics of vegetative dormancy. Here, we only use a single
size variable, but up to three size variables may be used concurrently.
Additionally, `stagenames`

must include unique names only.
The vectors `repstatus`

, `obsstatus`

,
`matstatus`

, `immstatus`

, `propstatus`

,
and `indataset`

are binomial vectors referring to status as a
reproductive stage, status as an observed stage, status as a mature
stage, status as an immature stage, status as a propagule stage, and
status as a stage occurring within the user-supplied dataset,
respectively. The combination of these characteristics must be
completely unique for each stage. The final vector, called
`comments`

, is a vector of text stage descriptions useful in
later interpretation.

```
<- c(0, 100, 13, 127, 3730, 3800, 0)
sizevector <- c("Sd", "Sdl", "Tm", "Sm", "La", "Flo", "Dorm")
stagevector <- c(0, 0, 0, 0, 0, 1, 0)
repvector <- c(0, 1, 1, 1, 1, 1, 0)
obsvector <- c(0, 0, 1, 1, 1, 1, 1)
matvector <- c(1, 1, 0, 0, 0, 0, 0)
immvector <- c(1, 0, 0, 0, 0, 0, 0)
propvector <- c(0, 1, 1, 1, 1, 1, 1)
indataset <- c(0, 100, 11, 103, 3500, 3800, 0.5)
binvec <- c("Dormant seed", "Seedling", "Tiny vegetative", "Small vegetative",
comments "Large vegetative", "Flowering", "Vegetatively dormant")
<- sf_create(sizes = sizevector, stagenames = stagevector,
lathframe repstatus = repvector, obsstatus = obsvector, propstatus = propvector,
immstatus = immvector, matstatus = matvector, indataset = indataset,
binhalfwidth = binvec, comments = comments)
#lathframe
```

Care should be taken in assigning sizes to stages, particularly when
stages occur with `size = 0`

. In most cases, a size of zero
will mean that the individual is alive but not observable, such as in
the case of vegetative dormancy. However, a size of zero may have other
meanings. For example, if the size metric used is a logarithm of the
measured size, then observable sizes of zero and lower may be possible.
These situations may impact matrix construction and analysis,
particularly when dealing with function-based MPMs, such as IPMs.

Next, we will standardize the dataset into *vertical format*.
Vertically formatted datasets are structured such that each row
corresponds to the state of a single individual in two (if ahistorical)
or three (if historical) consecutive monitoring occasions. The
`verticalize3()`

function handles this, creating a
*historical vertical dataset*. We can use one of two approaches
for this task, both using this function. In the most general case, we
may input all column names from the original dataset in each input
option in the same order (corresponding to monitoring occasion). This
allows any format of dataset to be used. However, an easier approach may
be used in datasets with strictly repeating sets of columns. In this
case, most of the inputs to this function’s options constitute the names
of the first variables coding for particular states. For example,
`Seedling1988`

is the first variable in the dataset coding
for status as a juvenile, `Volume88`

is the first variable
coding for the main size metric, and `Intactseed88`

is the
first variable coding for fecundity. The dataset includes a repeating
pattern of such variables organized as blocks of nine variables for each
year (noted as `blocksize`

). There are four monitoring
occasions (noted as `noyears`

). We also have a repeated
censor variable, the first of which is `Missing1988`

, and we
note here that we wish to censor the data and to keep data points with
`NA`

values in the censor term. The `patchidcol`

and `individcol`

terms are variables denoting which
patch/subpopulation and individual each row of data belongs to,
respectively. The setting `NAas0 = TRUE`

tells R that missing
values in size and fecundity should be interpreted as zeros, which
allows us to infer incidents of vegetative dormancy as cases where size
is equal to zero. Finally, `stageassign`

ties the dataset to
the correct stageframe.

```
<- verticalize3(lathyrus, noyears = 4, firstyear = 1988,
lathvert patchidcol = "SUBPLOT", individcol = "GENET", blocksize = 9,
juvcol = "Seedling1988", sizeacol = "Volume88", repstracol = "FCODE88",
fecacol = "Intactseed88", deadacol = "Dead1988", nonobsacol = "Dormant1988",
stageassign = lathframe, stagesize = "sizea", censorcol = "Missing1988",
censorkeep = NA, censor = TRUE, NAas0 = TRUE)
#summary(lathvert)
```

The `verticalize3()`

function includes error-checking
measures designed to find instances where individual characteristics do
not match those assigned to stages in the associated stageframe. In
those instances, warning messages will be displayed and the instances
will be marked `NoMatch`

in `stage1`

,
`stage2`

, or `stage3`

.

It pays to explore the dataset thoroughly, and summaries can help to do
that. Our summary reveals that the `verticalize3()`

function
has automatically subset the data to only those instances in which the
individual is alive in occasion *t* (please look at the variable
`alive2`

). Knowing this can help us interpret other
variables. For example, the mean value for `alive3`

suggests
very high survival to occasion *t*+1 (91.4%). Further, the
minimum values for all size variables are zero, suggesting that
unobservable stages occur within the dataset (these are instances of
vegetative dormancy).

The resulting reorganization has dramatically changed the dimensions of the dataset. The following lines show the difference between the original dataset and the vertically formatted dataset. Data has been neither lost nor added - it has just been restructured for analysis.

```
writeLines("Dimensions of original dataset:")
#> Dimensions of original dataset:
dim(lathyrus)
#> [1] 1119 38
writeLines("Dimensions of verticalized dataset:")
#> Dimensions of verticalized dataset:
dim(lathvert)
#> [1] 2552 42
```

Subset summaries may shed more light on these data. For example, after
subsetting to only cases in which individuals are vegetatively dormant
in occasion *t* and narrowing the summary to only variables
corresponding to occasion *t* (columns 22 to 33), we see that
this particular stage is associated exclusively with a size of zero. We
can also see from the dimensions of this subset that vegetative dormancy
occurs often in this dataset.

```
summary(lathvert[which(lathvert$stage2 == "Dorm"),c(22:27)])
#> repstra2 feca2 juvgiven2 obsstatus2 repstatus2 fecstatus2
#> Min. :0 Min. :0 Min. :0 Min. :0 Min. :0 Min. :0
#> 1st Qu.:0 1st Qu.:0 1st Qu.:0 1st Qu.:0 1st Qu.:0 1st Qu.:0
#> Median :0 Median :0 Median :0 Median :0 Median :0 Median :0
#> Mean :0 Mean :0 Mean :0 Mean :0 Mean :0 Mean :0
#> 3rd Qu.:0 3rd Qu.:0 3rd Qu.:0 3rd Qu.:0 3rd Qu.:0 3rd Qu.:0
#> Max. :0 Max. :0 Max. :0 Max. :0 Max. :0 Max. :0
summary(lathvert[which(lathvert$stage2 == "Dorm"),c(28:33)])
#> matstatus2 alive2 stage2 stage2index sizea3 repstra3
#> Min. :1 Min. :1 Length:138 Min. :7 Min. : 0.00 Min. :0.0000
#> 1st Qu.:1 1st Qu.:1 Class :character 1st Qu.:7 1st Qu.: 16.02 1st Qu.:0.0000
#> Median :1 Median :1 Mode :character Median :7 Median : 88.45 Median :0.0000
#> Mean :1 Mean :1 Mean :7 Mean : 242.33 Mean :0.0942
#> 3rd Qu.:1 3rd Qu.:1 3rd Qu.:7 3rd Qu.: 300.90 3rd Qu.:0.0000
#> Max. :1 Max. :1 Max. :7 Max. :2637.60 Max. :1.0000
writeLines("\nDimensions of data subset corresponding to dormant individuals in occasion t: ")
#>
#> Dimensions of data subset corresponding to dormant individuals in occasion t:
dim(lathvert[which(lathvert$stage2 == "Dorm"),])
#> [1] 138 42
```

The `verticalize3()`

function has made stage assignments for
all individuals at each time. This can be seen in the subset summary in
the `stage2index`

column, which shows all individuals that
are alive and have a size of 0 in occasion *t* to be in the
seventh stage. Type `lathframe`

and enter at the prompt to
check that the seventh stage in the stageframe is really vegetative
dormancy.

A further subset summary can teach us how reproduction is handled here.
The reproductive status of flowering adults is set as reproductive in
`lathframe`

, and a subset summary of just those individuals
flowering in occasion *t* shows all of those individuals set to
reproductive (see the distribution of values for `repstatus2`

in the output below). However, fecundity ranges from 0 to over 60 (see
`feca2`

, which codes for fecundity in occasion *t*),
meaning that some flowering adults do not actually produce any
offspring. In fact, only 44.3% of flowering plants produced seed in
occasion *t*. This has happened because our reproductive status
variable, `FCODE88`

, notes whether these individuals flowered
but not whether they produced seed. Since this plant must be pollinated
by an insect vector, some flowers should yield no seed. This issue does
not cause problems in the creation of raw matrices, but it might cause
difficulties in the creation of function-based matrices under some
conditions. It helps to consider whether the definitions used for stages
are appropriate, and so whether reproductive status must necessarily be
associated with *successful* reproduction or merely the attempt.
Here, we associate it with the latter, but in other vignettes we will
reconsider this assumption.

```
summary(lathvert[which(lathvert$stage2 == "Flo"),c(22:27)])
#> repstra2 feca2 juvgiven2 obsstatus2 repstatus2 fecstatus2
#> Min. :1 Min. : 0.000 Min. :0 Min. :1 Min. :1 Min. :0.0000
#> 1st Qu.:1 1st Qu.: 0.000 1st Qu.:0 1st Qu.:1 1st Qu.:1 1st Qu.:0.0000
#> Median :1 Median : 0.000 Median :0 Median :1 Median :1 Median :0.0000
#> Mean :1 Mean : 4.807 Mean :0 Mean :1 Mean :1 Mean :0.4426
#> 3rd Qu.:1 3rd Qu.: 6.000 3rd Qu.:0 3rd Qu.:1 3rd Qu.:1 3rd Qu.:1.0000
#> Max. :1 Max. :66.000 Max. :0 Max. :1 Max. :1 Max. :1.0000
summary(lathvert[which(lathvert$stage2 == "Flo"),c(28:33)])
#> matstatus2 alive2 stage2 stage2index sizea3 repstra3
#> Min. :1 Min. :1 Length:601 Min. :6 Min. : 0.0 Min. :0.0000
#> 1st Qu.:1 1st Qu.:1 Class :character 1st Qu.:6 1st Qu.: 308.0 1st Qu.:0.0000
#> Median :1 Median :1 Mode :character Median :6 Median : 795.8 Median :1.0000
#> Mean :1 Mean :1 Mean :6 Mean : 959.8 Mean :0.5591
#> 3rd Qu.:1 3rd Qu.:1 3rd Qu.:6 3rd Qu.:1326.8 3rd Qu.:1.0000
#> Max. :1 Max. :1 Max. :6 Max. :6539.4 Max. :1.0000
```

Now we will create **supplement tables**, which provide
external data for matrix estimation not included in the main demographic
dataset. Specifically, we will provide the seed dormancy probability and
germination rate, which are given as transitions from the dormant seed
stage to another year of seed dormancy or to the germinated seedling
stage, respectively. We assume that the germination rate is the same
regardless of whether the seed was produced in the previous year or has
been in the seedbank for longer. We will incorporate both terms as
constants for specific transitions within our matrices, and as constant
multipliers for fecundity, since fecundity will be estimated as the
product of seed produced and either the seed germination rate or the
seed dormancy/survival rate.

```
<- supplemental(stage3 = c("Sd", "Sdl", "Sd", "Sdl"),
lathsupp2 stage2 = c("Sd", "Sd", "rep", "rep"),
givenrate = c(0.345, 0.054, NA, NA),
multiplier = c(NA, NA, 0.345, 0.054),
type = c(1, 1, 3, 3), stageframe = lathframe, historical = FALSE)
#lathsupp2
```

The supplement table above will only work with ahistorical MPMs, while
the next supplement table will work for historical MPMs. The primary
difference is the incorporation of stage in time *t*-1. Going
back a further step to look at time *t*-1 shows us that there are
two ways that a dormant seed in time *t* can be a dormant seed in
time *t*+1 - it could have been a dormant seed in time
*t*-1, or it could have been produced by a flowering adult in
time *t*-1. Likewise, a seed in occasion *t* could have
become a seedling in occasion *t*+1 after being produced by a
flowering adult in occasion *t*-1, or remaining in the seed bank
in occasion *t*-1. So, we will enter four given transitions in
the historical case, rather than two transitions as in the ahistorical
case. We will also designate that transitions from seed in occasion
*t*-1 and seedling in occasion *t* to mature stages in
occasion *t*+1 should originate from the `NotAlive`

stage in occasion *t*-1, because the seed stage does not actually
occur in the dataset and so function `rlefko3()`

will not
estimate these transitions without knowing the appropriate proxy set.

```
<- supplemental(stage3 = c("Sd", "Sd", "Sdl", "Sdl", "Sd", "Sdl", "mat"),
lathsupp3 stage2 = c("Sd", "Sd", "Sd", "Sd", "rep", "rep", "Sdl"),
stage1 = c("Sd", "rep", "Sd", "rep", "npr", "npr", "Sd"),
eststage3 = c(NA, NA, NA, NA, NA, NA, "mat"),
eststage2 = c(NA, NA, NA, NA, NA, NA, "Sdl"),
eststage1 = c(NA, NA, NA, NA, NA, NA, "NotAlive"),
givenrate = c(0.345, 0.345, 0.054, 0.054, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, 0.345, 0.054, NA),
type = c(1, 1, 1, 1, 3, 3, 1), type_t12 = c(1, 2, 1, 2, 1, 1, 1),
stageframe = lathframe, historical = TRUE)
#lathsupp3
```

These two supplement tables show us that we have survival-transition
probabilities (`type = 1`

, whereas fecundity rates would be
given as `type = 2`

and fecundity multipliers would be
`type = 3`

), that the second and fourth transitions involve
reproduction from occasion *t*-1 to occasion *t* while the
others involve survival (`type_t12 = 1`

for survival between
occasions *t*-1 and *t*, and `type_t12 = 2`

for
fecundity), that the given transitions originate from the dormant seed
stage (`Sd`

) in occasion *t* (and seeds or
reproductive stages in occasion *t*-1 in the historical case),
and the specific values to be used in overwriting: `0.345`

and `0.054`

. If we wished, we could have used the values of
transitions to be estimated within this matrix as proxies for these
values, in which case we would enter the stages corresponding to the
correct transitions in the `eststageX`

columns, and the
`givenrate`

column would be blank. This is precisely what we
did for one set of transitions - namely transitions from the seedling
class to mature stages. Finally, we have also included fecundity
multipliers for newly produced seed in the bottom two rows.

These supplement tables provide the best means of adding external data
to our MPMs because they allow both specific transitions to be isolated,
and because they allow the use of shorthand to identify large groups of
transitions (e.g. using `mat`

, `immat`

,
`rep`

, `nrep`

, `prop`

,
`npr`

, `groupX`

, or `all`

to signify
all mature stages, immature stages, reproductive stages,
non-reproductive stages, propagule stages, non-propagule stages, stages
within group `X`

, or simply all stages, respectively). They
also allow more powerful changes to MPMs, such as block reset of groups
of transitions to zero or other constants.

We have chosen to build both ahistorical and historical MPMs in this vignette. However, in a typical analysis, it is most parsimonious to test whether history influences the demography of the population significantly first, and only use historical MPMs if the test supports the hypothesis that it does. A number of methods exist to conduct these tests, and we recommend Brownie et al. (1993), Pradel, Wintrebert, and Gimenez (2003), Pradel (2005), and Cole et al. (2014) for good discussions and tools to help with this.

In `lefko3`

, we provide a function that can be used to test
for the effects of history directly from the historical vertical
dataset: `modelsearch()`

. Function `modelsearch()`

estimates best-fit linear models of the key vital rates used to
propagate elements in function-based MPMs. There are up to 13 different
vital rates possible to test, of which seven are adult vital rates and
seven are juvenile vital rates. The standard vital rates that we may
wish to test are survival (marked as `surv`

in
`vitalrates`

), primary size (`size`

), and
fecundity (`fec`

), which are the default vital rates assumed
by the function. Here, we also test observation status
(`obs`

), which can serve as a proxy for sprouting probability
in cases where plants do not necessarily sprout, and reproductive status
(`repst`

), which assesses the probability of reproduction in
cases where reproduction is not certain. This dataset also includes
juveniles whose vital rates we wish to estimate. We designate this by
setting `juvestimate`

to the correct juvenile stage in the
dataset. Because size varies in juveniles, we set
`juvsize = TRUE`

(this setting defaults to
`FALSE`

, in which case size is not tested in juvenile vital
rate models). To test history with function `modelsearch`

: 1)
use the historical vertical dataset as input, 2) set
`historical = TRUE`

, 3) input the relevant vital rates to
estimate, 4) set the suite of independent factors to test (size and
reproductive status in occasions *t* and *t*-1 and all
interactions, or some subset thereof), 5) set the name of the juvenile
stage (if juvenile vital rates are to be estimated and such stages occur
in the dataset), 6) set the proper distributions to use for size and
fecundity, and 7) note which variables code for individual identity
(used to treat identity as a random factor in mixed linear models),
patch identity (if multiple patches occur in the dataset and vital rates
should be estimated with patch as a factor), and observation occasion.
We also set `quiet = TRUE`

to limit the amount of text output
while the function runs.

```
<- modelsearch(lathvert, historical = TRUE, suite = "size",
histtest vitalrates = c("surv", "obs", "size", "repst", "fec"), juvestimate = "Sdl",
sizedist = "gaussian", fecdist = "gaussian", indiv = "individ",
year = "year2", juvsize = TRUE, quiet = TRUE)
#histtest
```

The object generated is quite long, and likely resulted in a series of
warnings from the model-building functions utilized. For our purposes,
we need only look at elements corresponding to the best-fit models for
our tested vital rates, which are the elements marked
`survival_model`

, `observation_model`

,
`size_model`

, `repstatus_model`

,
`fecundity_model`

, `juv_survival_model`

,
`juv_observation _model`

, `juv_size_model`

,
`juv_reproduction_model`

, and
`juv_maturity_model`

. The line beginning
`Formula:`

in each of these sections shows the best-fit
model, in standard R formula notation (i.e. \(y = ax + b\) is given as
`y ~ x`

). The independent terms tested include variables
coding for size in occasions *t* and *t*-1, given as
`sizea2`

and `sizea1`

, respectively. Since several
vital rates show `sizea1`

as a term in the best-fit models,
most notably for adult survival, primary size, and reproduction
probability, we see that individual history has a significant impact on
the demography of this population. The quality control information in
element `qc`

is also of interest, showing that some of our
models are quite strong, with accuracy of 0.936 for adult survival and
0.986 for juvenile observation, while others are notably weaker, such as
fecundity and juvenile size.

Now let’s create some raw Lefkovitch MPMs based on Ehrlén (2000). We have seen that history should
be included in these analyses, which justifies creating only historical
matrices. However, to introduce these functions in greater depth and
detail, we will first create ahistorical MPMs. The functions we will use
to build these MPMs include `rlefko2()`

for raw ahistorical
MPMs, and `rlefko3()`

for raw historical MPMs.

Ehrlén (2000) shows a mean matrix covering
years 1989 and 1990 as occasion *t*. We will utilize the entire
dataset instead, covering 1988 to 1991, as follows. Note that we will
not create matrices for subpopulations in this case (to include them,
add the options `patch = "all", patchcol = "patchid"`

to the
input below).

```
<- rlefko2(data = lathvert, stageframe = lathframe, year = "all",
ehrlen2 stages = c("stage3", "stage2"), supplement = lathsupp2, yearcol = "year2",
indivcol = "individ")
#ehrlen2
```

The output from this analysis is a `lefkoMat`

object, which
is an S3 object (list) with the following elements:

**A**: a list of full population projection matrices, in
order of population, patch, and year (order given in
`labels`

)

**U**: a list of matrices where non-zero entries are
limited to survival-transition elements, in the same order as
`A`

**F**: a list of matrices where non-zero entries are
limited to fecundity elements, in the same order as `A`

**hstages**: a data frame showing the order of paired
stages (given if matrices are historical; otherwise `NA`

)

**agestages**: a data frame showing the order of
age-stage pairs (given if matrices are age-by-stage ahistorical;
otherwise `NA`

)

**ahstages**: the stageframe used in analysis, with
stages potentially reordered and edited as they occur in the matrix

**labels**: a table showing the order of matrices by
population, patch, and year

**matrixqc**: a short vector used in
`summary`

statements to describe the overall quality of each
matrix (used in `summary()`

calls)

**dataqc**: a short vector used in `summary`

statements to describe key sampling aspects of the dataset (used in
`summary()`

calls)

Input options for function `rlefko2()`

include
`year = "all"`

, which can be changed to
`year = c(1989, 1990)`

to focus just on years 1989 and 1990,
as in Ehrlén (2000), or
`year = 1989`

to focus exclusively on the transition from
1989 to 1990 (the year entered is the year in occasion *t*).
Matrix-estimating functions in `lefko3`

have a default
behavior of creating matrices for each year in the dataset except the
final year, rather than lumping all years together to produce a single
matrix. However, patches or subpopulations will only be separated if a
patch ID variable is provided as input. Package `lefko3`

includes a great deal of flexibility here to estimate matrices covering
all of the populations, patches, and years occurring in a dataset.

We can understand `lefkoMat`

objects in greater detail
through the `summary()`

function.

`#summary(ehrlen2)`

We start off learning that three full `A`

matrices were
estimated (as well as their `U`

and `F`

decompositions), and that they are ahistorical. This is expected given
that there are four consecutive years of data, yielding three time
steps, and an ahistorical matrix requires two consecutive years to
estimate transitions. The following line notes the dimensions of those
matrices. The third, fourth, fifth, and sixth lines of the summary show
how many survival transition and fecundity elements were actually
estimated, both overall and per matrix, the number of populations,
patches, and time steps covered by the MPM, and the number of
individuals and transitions the matrices are based on (these last
numbers can be used to understand matrix quality). Finally, the last
section shows us summaries of the column sums from the
survival-transition (`U`

) matrices in this
`lefkoMat`

object. These column sums correspond to the
survival probabilities of the different life stages, and so the
summaries must show numbers ranging from 0.0 to 1.0. All of these values
are within the realm of possibility for survival probabilities. If we
look back at the stageframe for this analysis to get the order of
stages, we find that they are also reasonably similar to the values
published in Ehrlén (2000). Additional
error-checking can be accomplished by checking specific elements within
each matrix to see if they match expectation.

One rather important point to bear in mind about matrix estimating
functions in `lefko3`

is that they sort the stages provided
in the stageframe according to a standardized rubric, and so the order
of stages in the `ahstages`

element of a
`lefkoMat`

object may differ from the original input.
Particularly, they sort propagules first, followed by immature stages,
followed by non-reproductive adult stages, followed by reproductive
adult stages. Let’s see the order in our matrices. We see that the key
difference from the original input is that the order of the flowering
stage and the vegetative dormancy stage have been flipped.

`#ehrlen2$ahstages`

Now we’ll estimate historical matrices. Because of the size of these
matrices, we will only show the `lefkoMat`

summary. Note that
we will only create Ehrlén-format matrices - users wishing to create
deVries-format hMPMs should add `format = "deVries"`

to the
input options (the resulting matrices will be bigger, but contain the
same number of estimated elements).

```
<- rlefko3(data = lathvert, stageframe = lathframe, year = "all",
ehrlen3 stages = c("stage3", "stage2", "stage1"), supplement = lathsupp3,
yearcol = "year2", indivcol = "individ")
#summary(ehrlen3)
```

The summary output shows several differences from the ahistorical case. First, there is one less matrix estimated in the historical case than in the ahistorical case, because raw historical matrices require three consecutive occasions to estimate each transition instead of two. Second, these matrices are larger than ahistorical matrices, with the numbers of rows and columns generally equaling the number of ahistorical rows and columns squared (although deVries-format increases the dimensions through the addition of a prior stage for newborns, and in both Ehrlén and deVries formats the numbers of rows and columns may be reduced under some circumstances as shown in the next block). Finally, a much greater proportion of each matrix is composed of zeros in the historical case than in the ahistorical case, although there are certainly more non-zero elements as well. This sparseness results from historical matrices being composed primarily of structural zeros. As a result, the historical matrices in this example have non-zero entries in 86 / 2401 = 3.6% of matrix elements, while the equivalent ahistorical matrices have non-zero entries in 26.667 / 49 = 54.45% of matrix elements.

We can see the impact of structural zeros by eliminating some of them in
the process of matrix estimation. This can be done by setting
`reduce = TRUE`

, which tells `rlefko3()`

to
eliminate stage pairs in which both column and row are zero vectors.
Here, we now have matrices with 19 fewer rows and columns, and 86 / 900
= 9.6% of elements as potentially non-zero.

```
<- rlefko3(data = lathvert, stageframe = lathframe,
ehrlen3red year = c(1989, 1990), stages = c("stage3", "stage2", "stage1"),
supplement = lathsupp3, yearcol = "year2", indivcol = "individ",
reduce = TRUE)
#summary(ehrlen3red)
```

Next we will create the element-wise mean ahistorical matrix.

```
<- lmean(ehrlen2)
ehrlen2mean #ehrlen2mean
```

Function `lmean()`

creates a `lefkoMat`

object
retaining all descriptive information from the original
`lefkoMat`

object. The full output includes the main
composite mean matrices (shown in element `A`

), as well as
the mean survival-transition matrix (`U`

) and the mean
fecundity matrix (`F`

), followed by a data frame outlining
the definitions and order of historical paired stages
(`hstages`

, shown as `NA`

in this case because the
matrices are ahistorical), a data frame outlining the age-by-stage
combinations used (`agestages`

, shown as `NA`

because these are not age-by-stage matrices), a data frame outlining the
actual stages as outlined in the `stageframe`

object used to
create these matrices (`ahstages`

), a data frame outlining
the definitions and order of the matrices (`labels`

), and two
quality control elements used in output for the `summary()`

function (`matrixqc`

and `dataqc`

).

Users will note in the output that we have a single mean matrix. The
default for `lmean()`

outputs both patch and population
means. However, we did not separate patches in the dataset, so the patch
mean is equal to the population mean.

Now we will estimate the historical element-wise mean matrix. We will use the reduced matrix to simplify later analyses, and show only the top-left corner of the rather large matrix (a section comprised of the first twenty rows and eight columns of the 30 x 30 matrix).

```
<- lmean(ehrlen3red)
ehrlen3mean #print(ehrlen3mean$A[[1]][1:20,1:8], digits = 3)
```

The prevalence of zeros in this matrix is normal because most elements
are structural zeros and so cannot equal anything else. This matrix is
also a raw matrix, meaning that transitions that do not actually occur
in the dataset cannot equal anything other than 0. To understand the
dominance of structural zeros in the historical case, let’s take a look
at the `hstages`

object associated with this mean matrix.

```
$hstages
ehrlen3mean#> stage_id_2 stage_id_1 stage_2 stage_1
#> 1 1 1 Sd Sd
#> 2 2 1 Sdl Sd
#> 3 3 2 Tm Sdl
#> 4 4 2 Sm Sdl
#> 5 6 2 Dorm Sdl
#> 6 3 3 Tm Tm
#> 7 4 3 Sm Tm
#> 8 5 3 La Tm
#> 9 6 3 Dorm Tm
#> 10 3 4 Tm Sm
#> 11 4 4 Sm Sm
#> 12 5 4 La Sm
#> 13 6 4 Dorm Sm
#> 14 7 4 Flo Sm
#> 15 3 5 Tm La
#> 16 4 5 Sm La
#> 17 5 5 La La
#> 18 6 5 Dorm La
#> 19 7 5 Flo La
#> 20 3 6 Tm Dorm
#> 21 4 6 Sm Dorm
#> 22 5 6 La Dorm
#> 23 6 6 Dorm Dorm
#> 24 7 6 Flo Dorm
#> 25 1 7 Sd Flo
#> 26 2 7 Sdl Flo
#> 27 4 7 Sm Flo
#> 28 5 7 La Flo
#> 29 6 7 Dorm Flo
#> 30 7 7 Flo Flo
```

There are 30 pairs of life history stages, corresponding to the rows and
columns of the historical matrices. The pairs are interpreted so that
matrix columns represent stages in occasions *t*-1 and
*t*, and the rows represent stages in occasions *t* and
*t*+1. For an element in the matrix to contain a number other
than zero, it must first represent the same stage at occasion *t*
in both the column stage pairs and the row stage pairs. The element [1,
1], for example, represents the transition probability from dormant seed
at occasions *t*-1 and *t* (column pair), to dormant seed
at occasions *t* and *t*+1 (row pair) - the occasion
*t* stages match, and so this element is possible. However,
element [1, 2] represents the transition probability from seedling in
occasion *t*-1 and very small adult in occasion *t*
(column pair), to dormant seed in occasion *t* and in occasion
*t*+1 (row pair). Clearly [1, 2] is a structural zero because it
is impossible for an individual to be both a dormant seed and a very
small adult at the same time.

Error-checking is more difficult with historical matrices because they
are typically one or two orders of magnitude bigger than their
ahistorical counterparts, but the same basic strategy can be used here
as with ahistorical matrices. In these cases we can use
`summary()`

function to assess the key quality control
characteristics of the mean hMPM, such as the distribution of survival
probability estimates for historical stages.

`#summary(ehrlen3mean)`

Let’s try another approach, looking at some conditional historical
matrices. Conditional matrices are matrices showing transitions from
stage at occasion *t* to occasion *t*+1, conditional on
all individuals having been in the same stage in occasion *t*-1.
They are calculated from historical MPMs, and the output below shows all
conditional matrices developed from the first `A`

matrix of
`ehrlen3mean`

. The first matrix, for example, shows all
transitions involving individuals that had been in the dormant seed
stage `Sd`

in occasion *t*-1, while the last matrix
shows transitions involving individuals that had been vegetatively
dormant in occasion *t*-1.

```
<- cond_hmpm(ehrlen3mean)
ehrlen3condmn #ehrlen3condmn$Mcond[[1]]
```

Quick scans will show many transitions missing, because each stage has only certain stages that can transition from it, and to which it can transition. Further transitions are missing because the MPM is raw, and some transitions are not parameterized because no individuals made them.

One last technique before we move on to analyze our MPMs: matrix
visualization plots. These plots provide a relatively easy way to
understand the “spatial spread” of values throughout a matrix. Package
`lefko3`

includes function `image3()`

, which
provides an easy way to make these images. Let’s start off by looking at
the ahistorical mean matrix.

`image3(ehrlen2mean, used = 1)`

```
#> [[1]]
#> NULL
```

The resulting image shows non-zero elements as red spaces, and zero elements as white spaces. Rows and columns are numbered, and we can see that this matrix is reasonably dense.

Now let’s take a look at the mean historical matrix. The historical mean matrix has many more rows and columns, and has more of both zero and non-zero elements. However, it has become a sparse matrix, and it turns out that increasing the numbers of life stages will increase the sparsity of historical projection matrices.

`image3(ehrlen3mean, used = 1)`

```
#> [[1]]
#> NULL
```

Package `lefko3`

includes functions to conduct some analyses
of population dynamics. We will start by estimating the asymptotic
population growth rate (\(\lambda\))
and the stochastic population growth rate (\(a
= \text{log} \lambda _{S}\)) from the ahistorical MPMs, including
both the annual MPM and the mean. For the stochastic case, we will set
the seed for R’s random number generation to make our output
reproducible. Note that each \(\lambda\) estimate includes a data frame
describing the matrices in order (given as the `labels`

object within the output list). Here is the set of ahistorical annual
\(\lambda\) estimates, followed by
\(\lambda\) for the mean matrix, and
the stochastic population growth rate (\(a =
\text{log} \lambda _{S}\)).

```
writeLines("Deterministic ahistorical: ")
#> Deterministic ahistorical:
lambda3(ehrlen2)
#> pop patch year2 lambda
#> 1 1 1 1988 0.8801910
#> 2 1 1 1989 0.9201885
#> 3 1 1 1990 1.0061563
writeLines("Deterministic mean ahistorical: ")
#> Deterministic mean ahistorical:
lambda3(ehrlen2mean)
#> pop patch lambda
#> 1 1 1 0.9491146
writeLines("Stochastic ahistorical: ")
#> Stochastic ahistorical:
set.seed(42)
slambda3(ehrlen2)
#> pop patch a var sd se
#> 1 1 1 -0.05385279 0.03349303 0.183011 0.00183011
```

\(\lambda\) for the mean ahistorical matrix may seem high relative to the annual matrices. However, elements in raw annual matrices may differ dramatically due to a lack of individuals transitioning through them in a particular year, which is a situation more likely to happen with smaller datasets and larger numbers of recognized stages. Since different elements within the matrix have different amounts of influence on the population growth rate itself, \(\lambda\) may differ in ways difficult to predict without conducting a perturbation analysis.

We will now look at the same numbers for the historical analyses. There
are several differences in the output in addition to the lower growth
rate estimates. First, because there are four years of data, there are
three ahistorical transitions possible for estimation: year 1 to 2, year
2 to 3, and year 3 to 4. However, in the historical case, only two are
possible: from years 1 and 2 to 3 (technically, from years 1
[*t*-1] and 2 [*t*] to years 2 [*t*] and 3
[*t*+1]), and from years 2 and 3 to 4 (technically, from years 2
[*t*-1] and 3 [*t*] to years 3 [*t*] and 4
[*t*+1]). Second, historical matrices cover more of the
individual heterogeneity in a population by splitting ahistorical
transitions by stage in occasion *t*-1. This heterogeneity may
reflect the impacts of trade-offs operating across years (Shefferson and Roach 2010). One particularly
common trade-off is the cost of growth: an individual that grows a great
deal in one time step due to great environmental conditions in that year
might pay a large cost of survival, growth, or reproduction in the next
if those environmental conditions deteriorate (Shefferson, Warren II, and Pulliam 2014; Shefferson et
al. 2018). Alternatively, these patterns may reflect greater
year-to-year variability in allocation to aboveground relative to
belowground tissues, the latter of which includes this species’
perennating structure. While we do not argue that the drop in \(\lambda\) must be due to these issues, and
it is certainly possible for historical matrix analysis to lead to
higher \(\lambda\) estimates, we do
believe that these estimates of \(\lambda\) and \(a
= \text{log} \lambda _{S}\) are likely to be more realistic than
the higher estimates in the ahistorical case.

```
writeLines("Deterministic historical: ")
#> Deterministic historical:
lambda3(ehrlen3red)
#> pop patch lambda
#> 1 1 1 0.8863458
#> 2 1 1 0.9842366
writeLines("Deterministic mean historical: ")
#> Deterministic mean historical:
lambda3(ehrlen3mean)
#> pop patch lambda
#> 1 1 1 0.9138706
writeLines("Stochastic historical: ")
#> Stochastic historical:
set.seed(42)
slambda3(ehrlen3red)
#> pop patch a var sd se
#> 1 1 1 -0.09382616 0.01900087 0.1378437 0.001378437
```

We can also examine the stable stage distributions, as follows for the ahistorical case.

```
<- stablestage3(ehrlen2mean)
ehrlen2mss
ehrlen2mss#> matrix stage_id stage ss_prop
#> 1 1 1 Sd 0.29878730
#> 2 1 2 Sdl 0.04676671
#> 3 1 3 Tm 0.22350242
#> 4 1 4 Sm 0.18386817
#> 5 1 5 La 0.07717874
#> 6 1 6 Dorm 0.05581804
#> 7 1 7 Flo 0.11407862
```

The data frame output shows us the stages themselves
(`stage`

, and associated number in `stage_id`

),
which matrix they refer to (`matrix`

), and the stable stage
distribution (`ss_prop`

). Interpreting these values, we find
that the mean matrix suggests that, if we project the population forward
indefinitely assuming the population dynamics are static and represented
by this matrix, we will find that approximately 30% of individuals
should be dormant seeds (suggesting a large seedbank). A further 22% and
18% should be very small and small adults, respectively, and 11% should
be flowering adults. Almost 6% of the population should eventually be
composed of vegetatively dormant adults.

We can estimate the stable stage distribution for the historical case,
as well. Because the historical output for the
`stablestage3()`

function is a list with two data frames,
let’s take a look at each of these data frames in turn. The first will
be the stage-pair output.

```
<- stablestage3(ehrlen3mean)
ehrlen3mss #ehrlen3mss$hist
```

This data frame is structured in historical format, and so shows the
stable stage distribution of stage pairs. We may wish to see which stage
pair dominates, in which case we might look at the row with the maximum
`ss_prop`

value.

```
$hist[which(ehrlen3mss$hist$ss_prop == max(ehrlen3mss$hist$ss_prop)),]
ehrlen3mss#> matrix stage_id_2 stage_id_1 stage_2 stage_1 ss_prop
#> 6 1 3 3 Tm Tm 0.2263377
```

Here we see that about 23% of the population is expected to be composed of tiny adults maintaining themselves as tiny adults.

The longer format of the historical stable stage output makes it a bit
hard to read. However, these historical values can also be combined by
stage at occasion *t* (`stage_2`

) to estimate the
**historically-corrected stable stage distribution**, which
allows comparison to a stable stage distribution estimated from a purely
ahistorical MPM. We can do this by examining the `ahist`

element of the historical stable stage distribution object. Notice below
that the `ss_prop`

column shows values that are a bit
different from the ahistorical case, suggesting the influence of
individual history, particularly showing that tiny adults are a larger
share of the stage stage structure.

```
$ahist
ehrlen3mss#> matrix stage_id stage ss_prop
#> 1 1 1 Sd 0.27129927
#> 2 1 2 Sdl 0.04246423
#> 3 1 3 Tm 0.28036166
#> 4 1 4 Sm 0.19642812
#> 5 1 5 La 0.06707576
#> 6 1 6 Dorm 0.04847825
#> 7 1 7 Flo 0.09389270
```

To see the impact of history on the stable stage distribution, let’s
plot the ahistorical and historically-corrected stable stage
distributions together. We will also include the stochastic long-run
stage distribution in our output, which is estimated with the same
function but using the `stochastic = TRUE`

option. This will
allow us to see the impact of random temporal variation. We will also
set the random seed to make our output reproducible.

```
<- stablestage3(ehrlen2, stochastic = TRUE, seed = 42)
ehrlen2mss_s <- stablestage3(ehrlen3red, stochastic = TRUE, seed = 42)
ehrlen3mss_s <- cbind.data.frame(ehrlen2mss$ss_prop, ehrlen3mss$ahist$ss_prop,
ss_put_together $ss_prop, ehrlen3mss_s$ahist$ss_prop)
ehrlen2mss_snames(ss_put_together) <- c("d_ahist", "d_hist", "s_ahist", "s_hist")
rownames(ss_put_together) <- ehrlen2mss$stage
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage",
ylim = c(0, 0.35), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahistorical", "det historical", "sto ahistorical",
"sto historical"), pch = 22, col = "black",
pt.bg = c("black", "orangered", "grey", "darkred"), bty = "n")
```

Let’s take the deterministic portion first. Accounting for individual history increased the prevalence of tiny and small adults, but decreased the prevalence of dormant seeds, and dormant, large, and flowering adults. Now when we also take in the impact of temporal stochasticity, we can see differences in the proportion of dormant seeds, seedlings, and all adult stages, with the greatest differences in dormant seeds and tiny adults.

Let’s take a look at the reproductive values now, in similar order to the stable stage distribution case. Initially, we will create all sets of reproductive value objects, and then we will plot them. The structure of these objects is the same as that of the stable stage structure outputs. Because the four vectors are all standardized such that the first non-zero reproductive value is set to 1.0, they are on different scales, and so we will make them comparable for plotting purposes by standardizing them relative to their respective maxima.

```
<- repvalue3(ehrlen2mean)
ehrlen2mrv <- repvalue3(ehrlen3mean)
ehrlen3mrv <- repvalue3(ehrlen2, stochastic = TRUE, seed = 42)
ehrlen2mrv_s <- repvalue3(ehrlen3red, stochastic = TRUE, seed = 42)
ehrlen3mrv_s <- cbind.data.frame((ehrlen2mrv$rep_value / max(ehrlen2mrv$rep_value)),
rv_put_together $ahist$rep_value / max(ehrlen3mrv$ahist$rep_value)),
(ehrlen3mrv$rep_value / max(ehrlen2mrv_s$rep_value)),
(ehrlen2mrv_s$ahist$rep_value / max(ehrlen3mrv_s$ahist$rep_value)))
(ehrlen3mrv_snames(rv_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(rv_put_together) <- ehrlen2mrv$stage
barplot(t(rv_put_together), beside=T, ylab = "Relative reproductive value",
ylim = c(0, 1.2), xlab = "Stage", col = c("black", "orangered", "grey", "darkred"),
bty = "n")
legend("topleft", c("det ahistorical", "det historical", "sto ahistorical",
"sto historical"), pch = 22, col = "black",
pt.bg = c("black", "orangered", "grey", "darkred"), bty = "n")
```

Both deterministic and stochastic analyses show that flowering adults have the greatest reproductive value in both ahistorical and historical analysis, while dormant seeds have the least. However, the historical MPM suggests lower contributions of dormant seeds, seedlings, and vegetative dormancy.

Next we will plot the reproductive values of paired stages from the historical MPM. Here is the deterministic case.

```
<- apply(as.matrix(ehrlen3mrv$hist[, c("stage_2", "stage_1")]), 1, function(X) {
labels paste(X[1], X[2])
})barplot(ehrlen3mrv$hist$rep_value, main = "Historical", xaxt = "n",
ylab = "Reproductive value", xlab = "Stage")
text(cex=0.5, y = -2.5, x = seq(from = 0,
to = 1.19*length(ehrlen3mrv$hist$stage_2), by = 1.2), labels, xpd=TRUE, srt=45)
```

In the bar plot, the labels show the stages in the order of stage in
occasion *t* followed by stage in occasion *t*-1.
Examining these values reveals that the largest reproductive value is
associated with flowering adults that were previously flowering, while
the next largest reproductive value is associated with flowering adults
that were previously small, and flowering adults that were previously
very large. Interestingly, historical transitions to very large adult
have lower reproductive value here than we might expect given the
ahistorical analysis, in which they were predicted to have the second
highest reproductive value.

Now we will look at the sensitivities of \(\lambda\) to the matrix elements. Here we conduct a deterministic sensitivity analysis on the ahistorical mean matrix.

```
<- sensitivity3(ehrlen2mean)
ehrlen2sens print(ehrlen2sens$ah_sensmats[[1]], digits = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.0184 0.00288 0.0137 0.0113 0.00475 0.00343 0.00701
#> [2,] 0.2055 0.03217 0.1537 0.1265 0.05309 0.03839 0.07847
#> [3,] 0.2566 0.04017 0.1920 0.1579 0.06629 0.04794 0.09798
#> [4,] 0.4243 0.06641 0.3174 0.2611 0.10960 0.07927 0.16200
#> [5,] 0.5808 0.09091 0.4345 0.3574 0.15003 0.10851 0.22177
#> [6,] 0.3191 0.04995 0.2387 0.1964 0.08243 0.05962 0.12184
#> [7,] 0.7510 0.11755 0.5618 0.4622 0.19400 0.14030 0.28675
writeLines(paste0("\nThe highest sensitivity value is: ",
max(ehrlen2sens$ah_sensmats[[1]])))
#>
#> The highest sensitivity value is: 0.751027236713977
writeLines(paste0("This occurs in element ",
which(ehrlen2sens$ah_sensmats[[1]] == max(ehrlen2sens$ah_sensmats[[1]]))))
#> This occurs in element 7
writeLines(paste0("The highest sensitivity value among biologically plausible elements is: ",
max(ehrlen2sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])))
#> The highest sensitivity value among biologically plausible elements is: 0.46216824824681
writeLines(paste0("This occurs in element ", which(ehrlen2sens$ah_sensmats[[1]] ==
max(ehrlen2sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)]))))
#> This occurs in element 28
```

The highest sensitivity value is associated with a biologically impossible transition - dormant seeds (stage/column 1) cannot transition to flowering (stage/row 7), but the highest sensitivity value is associated with that element. This is an unfortunate issue in sensitivity analysis, and requires great care to prevent sloppy inference. Within the biologically plausible elements, the highest sensitivity appears to be associated with element 28, which is the transition from small adult (stage/column 4) to flowering (stage/row 7).

We will now look at the sensitivity of \(\lambda\) to elements in the historical mean MPM.

```
<- sensitivity3(ehrlen3mean)
ehrlen3sens #ehrlen3sens
```

The first element produced in this analysis is `h_sensmats`

,
which is a list composed of sensitivity matrices of the historical
matrices, in order (we have only one in our mean matrix object). These
matrices are the same dimensions as the historical matrices used as
input, and so can be quite huge. This is followed by
`ah_sensmats`

, which is a list composed of
historically-corrected sensitivity matrices of corresponding ahistorical
matrix elements (calculated using the historically-corrected stable
stage distribution and reproductive value vector produced in
`ehrlen3mss`

and `ehrlen3mrv`

, respectively). So,
these are different than the sensitivities estimated from the
ahistorical matrices themselves, but have the same dimensions. Next,
`h_stages`

and `ah_stages`

give the order of
paired stages and life history stages used in the historical and
historically-corrected sensitivity matrices, respectively. Finally, we
have the original `A`

, `U`

, and `F`

matrices used as input.

Our historical matrices are large and full of zeros. So, we will look for the highest sensitivity associated with a biologically plausible element (i.e. non-zero matrix elements). Then, we will assess the highest biologically plausible sensitivity in the historically-corrected sensitivity matrices, to compare against the ahistorical sensitivity analysis.

```
writeLines(paste0("The highest sensitivity value among biologically plausible elements: ",
max(ehrlen3sens$h_sensmats[[1]][which(ehrlen3mean$A[[1]] > 0)])))
#> The highest sensitivity value among biologically plausible elements: 0.309420233310141
writeLines(paste0("This value is associated with element: ",
which(ehrlen3sens$h_sensmats[[1]] ==
max(ehrlen3sens$h_sensmats[[1]][which(ehrlen3mean$A[[1]] > 0)]))))
#> This value is associated with element: 314
writeLines(paste0("The highest historically-corrected sensitivity value among biologically
plausible elements is: ",
max(ehrlen3sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])))
#> The highest historically-corrected sensitivity value among biologically
#> plausible elements is: 0.597664284105433
writeLines(paste0("This occurs in element ", which(ehrlen3sens$ah_sensmats[[1]] ==
max(ehrlen3sens$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)]))))
#> This occurs in element 19
```

The maximum biologically plausible sensitivity value in the historical
matrix appears to be element 314, which is associated with column 11
(small adult in occasions *t*-1 and *t*) and row 14 (small
adult in occasion *t* to flowering adult in occasion
*t*+1). This transition is from small adult in occasions
*t*-1 and *t* to flowering adult in *t*+1. The
historically-corrected sensitivity analysis finds that \(\lambda\) is most sensitive to element 19,
which is the transition from tiny adult to large adult. This is a little
different from the ahistorical MPM, which suggested element 28 (small
adult to flowering adult).

Before moving on, let’s repeat the above exercise with stochastic sensitivities. We will compare ahistorical vs. historical/historically-corrected analyses.

```
<- sensitivity3(ehrlen2, stochastic = TRUE)
ehrlen2sens_s <- sensitivity3(ehrlen3red, stochastic = TRUE)
ehrlen3sens_s
writeLines(paste0("Highest stochastic sensitivity among biologically plausible
elements (ahistorical): ",
max(ehrlen2sens_s$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])))
#> Highest stochastic sensitivity among biologically plausible
#> elements (ahistorical): 0.50818027011384
writeLines(paste0("This value is associated with element: ",
which(ehrlen2sens_s$ah_sensmats[[1]] ==
max(ehrlen2sens_s$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)]))))
#> This value is associated with element: 28
writeLines(paste0("Highest stochastic sensitivity among biologically plausible
elements (historical): ",
max(ehrlen3sens_s$h_sensmats[[1]][which(ehrlen3red$A[[1]] > 0)])))
#> Highest stochastic sensitivity among biologically plausible
#> elements (historical): 0.392692392246312
writeLines(paste0("This value is associated with element: ",
which(ehrlen3sens_s$h_sensmats[[1]] ==
max(ehrlen3sens_s$h_sensmats[[1]][which(ehrlen3red$A[[1]] > 0)]))))
#> This value is associated with element: 157
writeLines(paste0("Highest stochastic sensitivity among biologically plausible
elements (historically-corrected): ",
max(ehrlen3sens_s$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)])))
#> Highest stochastic sensitivity among biologically plausible
#> elements (historically-corrected): 0.777074184389309
writeLines(paste0("This value is associated with element: ",
which(ehrlen3sens_s$ah_sensmats[[1]] ==
max(ehrlen3sens_s$ah_sensmats[[1]][which(ehrlen2mean$A[[1]] > 0)]))))
#> This value is associated with element: 19
```

We find strong similarities in these baseline inferences between
deterministic vs. stochastic analyses, with major disagreement coming
primarily from differences between ahistorical vs. historical analyses.
Element 157 in the historical case is associated with the transition
from tiny adult in times *t*-1 and *t* to small adult in
time *t*+1, and so this is another difference from the
deterministic case.

An alternative, or complementary, approach to perturbation analysis is
elasticity analysis. Elasticities are easier to interpret because zero
elements produce zero elasticity values, thus eliminating biologically
impossible transitions from consideration, and because they are scaled
to sum to 1.0. This scaling makes elasticities of survival transitions
easier to compare to those of fecundity. However, they are also
interpreted differently, because while sensitivity analysis shows the
impact of a tiny but *absolute* change to a matrix element on
\(\lambda\), elasticity analysis shows
the impact of a tiny but *proportional* change to a matrix
element on \(\lambda\). In fact, both
sensitivities and elasticities are essentially local slopes, and so are
not unit free. It is therefore not unusual for sensitivity and
elasticity analysis to yield different inferences (of course, the
reality is that they answer different questions).

Let’s look at the elasticity of \(\lambda\) to matrix elements in the ahistorical mean matrix.

```
<- elasticity3(ehrlen2mean)
ehrlen2elas print(ehrlen2elas$ah_elasmats, digits = 3)
#> [[1]]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.00668 0.00000 0.000000 0.0000 0.000000 0.00000 0.0117
#> [2,] 0.01169 0.00000 0.000000 0.0000 0.000000 0.00000 0.0205
#> [3,] 0.00000 0.02760 0.143899 0.0159 0.000499 0.00409 0.0000
#> [4,] 0.00000 0.00131 0.040093 0.1583 0.022065 0.02225 0.0171
#> [5,] 0.00000 0.00000 0.000587 0.0289 0.048624 0.01773 0.0542
#> [6,] 0.00000 0.00326 0.007377 0.0225 0.009070 0.00645 0.0110
#> [7,] 0.00000 0.00000 0.000000 0.0355 0.069775 0.00911 0.1723
writeLines(paste0("The maximum elasticity value: ",
max(ehrlen2elas$ah_elasmats[[1]])))
#> The maximum elasticity value: 0.172331468681622
writeLines(paste0("This value is associated with element: ",
which(ehrlen2elas$ah_elasmats[[1]] == max(ehrlen2elas$ah_elasmats[[1]]))))
#> This value is associated with element: 49
```

Elasticity analysis exhibits strong differences from sensitivity analysis. In particular, we find that \(\lambda\) is most strongly elastic in response to changes in stasis transitions in flowering adults (stage 7). We can sum the columns of the elasticity matrix to see which stages \(\lambda\) is most and least elastic in response to, as below.

```
print(colSums(ehrlen2elas$ah_elasmats[[1]]), digits = 3)
#> [1] 0.0184 0.0322 0.1920 0.2611 0.1500 0.0596 0.2867
```

Here we see that \(\lambda\) is most strongly elastic in response to changes in transitions associated with flowering adults, followed by transitions involving small adults. Dormant seeds and seedlings have the smallest impact on \(\lambda\), and the impacts of fecundity (shown in the top-right corner of the elasticity matrix) appear quite small.

Now on to elasticity analysis of the historical MPMs. Once again, we
will not output the matrices. Type `ehrlen3elas`

at the
prompt to see these matrices.

```
<- elasticity3(ehrlen3mean)
ehrlen3elas
writeLines(paste0("The highest deterministic elasticity value: ",
max(ehrlen3elas$h_elasmats[[1]])))
#> The highest deterministic elasticity value: 0.156485128488333
writeLines(paste0("This value is associated with element: ",
which(ehrlen3elas$h_elasmats[[1]] == max(ehrlen3elas$h_elasmats[[1]]))))
#> This value is associated with element: 156
```

The highest elasticity appears to be associated with the element 156,
which is at row 6, column 6. This corresponds to the stasis transition
of tiny adults (very small in *t*-1 to very small in *t*
to very small in *t*+1). So, accounting for history has changed
our perspective on what contributes most to changes in \(\lambda\).

Elasticities are often treated as additive, making the calculation of
historically-corrected elasticity matrices easy. These are stored in the
`ah_elasmats`

element of `elasticity3()`

output
originating from a historical MPM. Eyeballing this matrix, it appears
that the historically-corrected elasticity matrix supports stasis in the
tiny adult stage as the transition that \(\lambda\) is most elastic to, with stasis
as flowering adult a close second. This is a little different from the
ahistorical case, which flipped the importance of these two.

```
print(ehrlen3elas$ah_elasmats, digits = 3)
#> [[1]]
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.00716 0.000000 0.00000 0.0000 0.00e+00 0.00000 0.0118
#> [2,] 0.01181 0.000000 0.00000 0.0000 0.00e+00 0.00000 0.0000
#> [3,] 0.00000 0.010962 0.18270 0.0333 8.33e-05 0.00204 0.0000
#> [4,] 0.00000 0.000268 0.04306 0.1614 2.60e-02 0.02041 0.0290
#> [5,] 0.00000 0.000000 0.00000 0.0313 4.62e-02 0.00963 0.0559
#> [6,] 0.00000 0.000577 0.00331 0.0164 7.69e-03 0.00334 0.0093
#> [7,] 0.00000 0.000000 0.00000 0.0378 6.30e-02 0.00520 0.1704
```

The historical matrices we used as input are reduced, but the historically-corrected output is in the original dimensions corresponding to the number of stages listed in the stageframe. This makes it easy to compare against the actual ahistorical elasticity matrix, for example by subtracting one matrix from the other to see the differences. We can also subtract the column sums in this exercise to assess overall shifts in the importance of stages.

```
print((ehrlen3elas$ah_elasmats[[1]] - ehrlen2elas$ah_elasmats[[1]]), digits = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.000483 0.00000 0.000000 0.00000 0.000000 0.00000 0.000114
#> [2,] 0.000114 0.00000 0.000000 0.00000 0.000000 0.00000 -0.020475
#> [3,] 0.000000 -0.01663 0.038801 0.01741 -0.000416 -0.00205 0.000000
#> [4,] 0.000000 -0.00105 0.002964 0.00302 0.003947 -0.00184 0.011976
#> [5,] 0.000000 0.00000 -0.000587 0.00239 -0.002461 -0.00810 0.001711
#> [6,] 0.000000 -0.00268 -0.004070 -0.00607 -0.001385 -0.00311 -0.001685
#> [7,] 0.000000 0.00000 0.000000 0.00228 -0.006729 -0.00391 -0.001969
writeLines("\nColumn sums of differences:")
#>
#> Column sums of differences:
colSums((ehrlen3elas$ah_elasmats[[1]] - ehrlen2elas$ah_elasmats[[1]]))
#> [1] 0.0005969316 -0.0203610545 0.0371074740 0.0190229198 -0.0070433508 -0.0189963512 -0.0103265687
```

Accounting for history shows \(\lambda\) to be less elastic in response to early-life transitions, transitions from vegetative dormancy, and flowering than suggested by ahistorical matrices, but more elastic in response to changes in survival-transitions in tiny and small adults.

Next, we will create a barplot of the elasticities of life history stages from ahistorical vs. historically-corrected analyses. We will also incorporate stochastic elasticity analysis here to assess the importance of temporal environmental stochasticity on population growth.

```
<- elasticity3(ehrlen2, stochastic = TRUE)
ehrlen2elas_s <- elasticity3(ehrlen3red, stochastic = TRUE)
ehrlen3elas_s <- cbind.data.frame(colSums(ehrlen2elas$ah_elasmats[[1]]),
elas_put_together colSums(ehrlen3elas$ah_elasmats[[1]]), colSums(ehrlen2elas_s$ah_elasmats[[1]]),
colSums(ehrlen3elas_s$ah_elasmats[[1]]))
names(elas_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_put_together) <- ehrlen2elas$ah_stages$stage
barplot(t(elas_put_together), beside=T, ylab = "Elasticity", xlab = "Stage",
col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topleft", c("det ahistorical", "det historical", "sto ahistorical",
"sto historical"), pch = 22, col = "black",
pt.bg = c("black", "orangered", "grey", "darkred"), bty = "n")
```

Historical analyses generally find that population growth rate is less elastic in response to seedlings, and large and dormant adults, and more elastic to tiny and small adults than in ahistorical analyses. Stochastic analyses suggest that population growth rate is barely elastic in response to dormant seeds and seedlings, and more elastic in response to tiny and small adults.

Let’s now look at the elasticities of different kinds of transitions. We
will use the `summary()`

function, which outputs data frames
summarizing elasticity sums by the kind of transition. First, we will
compare ahistorical against historically-corrected transitions.

```
<- summary(ehrlen2elas)
ehrlen2elas_sums <- summary(ehrlen3elas)
ehrlen3elas_sums <- summary(ehrlen2elas_s)
ehrlen2elas_s_sums <- summary(ehrlen3elas_s)
ehrlen3elas_s_sums
<- cbind.data.frame(ehrlen2elas_sums$ahist[,2],
elas_sums_together $ahist[,2], ehrlen2elas_s_sums$ahist[,2],
ehrlen3elas_sums$ahist[,2])
ehrlen3elas_s_sumsnames(elas_sums_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_sums_together) <- ehrlen2elas_sums$ahist$category
barplot(t(elas_sums_together), beside=T, ylab = "Elasticity",
xlab = "Transition", col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahistorical", "det historical", "stoc ahistorical",
"stoc historical"), pch = 22, col = "black",
pt.bg = c("black", "orangered", "grey", "darkred"), bty = "n")
```

We see similar patterns across types of elasticity analysis. Particularly, population growth rate is most elastic in response to changes in stasis transitions, and least elastic to changes in fecundity.

Package `lefko3`

also includes functions to conduct
deterministic and stochastic life table response experiments, and two
general projection functions that can be used for analyses such as
quasi-extinction analysis as well as for density dependent analysis.
Users wishing to conduct these analyses should see our free e-manual
called ** lefko3: a gentle introduction** and other
vignettes, including long-format and video vignettes, on
the projects page
of the Shefferson lab website.

We are grateful to two anonymous reviewers whose scrutiny improved the quality of this vignette. The project resulting in this package and this tutorial was funded by Grant-In-Aid 19H03298 from the Japan Society for the Promotion of Science.

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