In this vignette, we will use the `cypdata`

dataset to
illustrate the estimation of **function-based MPMs**. To
reduce file size, we have prevented some statements from running if they
produce long stretches of output. Examples include most
`summary()`

calls - to run these statements, eliminate the
leading hashtags and then run them. We have also written this vignette
using the default Ehrlén format for all historical MPMs. All examples
can be run in 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.

The white lady’s slipper, *Cypripedium candidum*, is a rare and
long-lived North American perennial in the orchid family. The study
population is located within a state nature preserve located in
northeastern Illinois, USA. The population was monitored annually from
2004 to 2009, with two monitoring sessions per year. Monitoring sessions
included complete censuses of the population divided into sections
referred to as patches. Each monitoring session consisted of searches
for previously recorded individuals, which were located according to
coordinates relative to fixed stakes at the site, followed by a search
for new individuals. Data recorded per individual included: location,
number of non-flowering sprouts (also refered to as non-flowering
stems), number of flowering sprouts, number of flowers per sprout, and
number of fruits per sprout. Location was used to infer individual
identity. More information about this population and its characteristics
is given in Shefferson et al. (2001) and
Shefferson, Mizuta, and Hutchings (2017).

Population matrix projection modeling requires an appropriate life
history model showing how all stages and transitions are related. The
figure below shows a very general life history model detailing these
relationships in *Cypripedium candidum*. The first stage of life
is the dormant seed, although a seed may germinate in the year following
seed production, in which case the first stage of life is the protocorm.
The protocorm is the first germinated stage - an underground,
mycoheterotrophic stage found only in the families Orchidaceae and
Pyrolaceae. There are three years of protocorm stages, followed by a
seedling stage, and finally a set of stages that comprise the
size-classified adult portion of life. The figure shows 49 such stages,
each for a different number of stems (including zero for vegetative
dormancy) and one of two reproductive statuses.

Figure 5.1. Life history model of *Cypripedium candidum*.
Survival and fecundity transitions are indicated with solid and dashed
arrows, respectively.

In this figure, the juvenile stages are limited to simple, progressive
transitions. New recruits may enter the population directly from a seed
produced the previous year, in which case they start in the protocorm 1
stage, or they may begin as dormant seed. Dormant seed may remain
dormant, die, or germinate into the protocorm 1 stage. Protocorms exist
for up to three years, yielding the protocorm 1, 2, and 3 stages.
Protocorm 3 leads to the seedling stage, in which the plant may persist
for many years before becoming mature. The first mature stage is usually
either vegetative dormancy (`Dorm`

), during which time the
plant does not sprout, or a small, non-flowering adult (`1V`

,
which has a single stem). Once in this portion of the life history, the
plant may transition among 49 mature stages, including vegetative
dormancy, 1-24 stems without flowers, or 1-24 stems with at least one
flower.

Let’s load the dataset, which is called `cypdata`

. The
horizontal dataset `cypdata`

and the ahistorical vertical
dataset `cypvert`

are the same data but are structured
differently. Both include only data for the adult stages, and so later
we will need to set juvenile transitions to constants from the
literature.

```
data(cypdata)
#summary(cypdata)
```

We will first describe the life history characterizing the population,
matching it to our dataset with a *stageframe* for our
*Cypripedium candidum* dataset. Since this analysis will be
function-based and all sizes have reasonable representation, we will
include all observed size classes here. Size is count-based, and refers
to all numbers of stems that occurred, representing the life history
diagram shown previously. Note that we are using the midpoint approach
to determining the size bins here, using the default bin halfwidths of
0.5 (since all size bins are using this default, we do not include a
`binhalfwidth`

vector option in the `sf_create()`

input). However, we could have used the `sizemin`

and
`sizemax`

options to more deliberately set the size bin
minima and maxima instead.

```
<- c(0, 0, 0, 0, 0, seq(from = 0, t = 24), seq(from = 1, to = 24))
sizevector <- c("SD", "P1", "P2", "P3", "SL", "D", "V1", "V2", "V3", "V4",
stagevector "V5", "V6", "V7", "V8", "V9", "V10", "V11", "V12", "V13", "V14", "V15", "V16",
"V17", "V18", "V19", "V20", "V21", "V22", "V23", "V24", "F1", "F2", "F3",
"F4", "F5", "F6", "F7", "F8", "F9", "F10", "F11", "F12", "F13", "F14", "F15",
"F16", "F17", "F18", "F19", "F20", "F21", "F22", "F23", "F24")
<- c(0, 0, 0, 0, 0, rep(0, 25), rep(1, 24))
repvector <- c(0, 0, 0, 0, 0, 0, rep(1, 48))
obsvector <- c(0, 0, 0, 0, 0, rep(1, 49))
matvector <- c(0, 1, 1, 1, 1, rep(0, 49))
immvector <- c(1, rep(0, 53))
propvector <- c(0, 0, 0, 0, 0, rep(1, 49))
indataset <- c("Dormant seed", "Yr1 protocorm", "Yr2 protocorm", "Yr3 protocorm",
comments "Seedling", "Veg dorm", "Veg adult 1 stem", "Veg adult 2 stems",
"Veg adult 3 stems", "Veg adult 4 stems", "Veg adult 5 stems",
"Veg adult 6 stems", "Veg adult 7 stems", "Veg adult 8 stems",
"Veg adult 9 stems", "Veg adult 10 stems", "Veg adult 11 stems",
"Veg adult 12 stems", "Veg adult 13 stems", "Veg adult 14 stems",
"Veg adult 15 stems", "Veg adult 16 stems", "Veg adult 17 stems",
"Veg adult 18 stems", "Veg adult 19 stems", "Veg adult 20 stems",
"Veg adult 21 stems", "Veg adult 22 stems", "Veg adult 23 stems",
"Veg adult 24 stems", "Flo adult 1 stem", "Flo adult 2 stems",
"Flo adult 3 stems", "Flo adult 4 stems", "Flo adult 5 stems",
"Flo adult 6 stems", "Flo adult 7 stems", "Flo adult 8 stems",
"Flo adult 9 stems", "Flo adult 10 stems", "Flo adult 11 stems",
"Flo adult 12 stems", "Flo adult 13 stems", "Flo adult 14 stems",
"Flo adult 15 stems", "Flo adult 16 stems", "Flo adult 17 stems",
"Flo adult 18 stems", "Flo adult 19 stems", "Flo adult 20 stems",
"Flo adult 21 stems", "Flo adult 22 stems", "Flo adult 23 stems",
"Flo adult 24 stems")
<- sf_create(sizes = sizevector, stagenames = stagevector,
cypframe repstatus = repvector, obsstatus = obsvector, matstatus = matvector,
propstatus = propvector, immstatus = immvector, indataset = indataset,
comments = comments)
#cypframe
```

The resulting object, `cypframe`

, shows a data frame that
includes the following information in order for each stage: the stage’s
name; primary, secondary, and tertiary size; minimum and maximum ages
associated with the stage; reproductive status; status as observable;
status as a propagule; status as immature; status as mature; whether it
occurs in the dataset; half-width, minimum, maximum, centroid, and full
width of primary, secondary, and tertiary size class bins; stage group
number; and a comments field. We used one size variable, so values for
the secondary and tertiary sizes are set to `NA`

. Stage names
and combinations of characteristics occurring within the dataset must be
unique to prevent estimation errors, and the comments field may be
edited to include any information deemed pertinent.

Now we will standardize our vertical dataset into a
historically-formatted vertical (*hfv*) data frame using the
`verticalize3()`

function. The resulting dataset will have
each individual’s observed life history broken up into states
corresponding to three consecutive monitoring occasions (years) per row,
with plant identity marked in each row.

```
<- verticalize3(data = cypdata, noyears = 6, firstyear = 2004,
vertdata patchidcol = "patch", individcol = "plantid", blocksize = 4,
sizeacol = "Inf2.04", sizebcol = "Inf.04", sizeccol = "Veg.04",
repstracol = "Inf.04", repstrbcol = "Inf2.04", fecacol = "Pod.04",
stageassign = cypframe, stagesize = "sizeadded", NAas0 = TRUE)
summary_hfv(vertdata, full = FALSE)
#>
#> This hfv dataset contains 320 rows, 57 variables, 1 population,
#> 3 patches, 74 individuals, and 5 time steps.
```

In the above code, we described the input dataset in a way that allows R
to reorganize it appropriately. In the input to the function, we
utilized a repeating pattern of variable names arranged in the same
order for each monitoring occasion. This arrangement allows us to enter
only the first variable in each set, as long as `noyears`

and
`blocksize`

are set properly and no gaps or shuffles appear
among columns in the dataset. `lefko3`

’s data management
functions do not require such repeating patterns, but they do make the
required function input much more succinct. Without these patterns,
column names should be input as vectors in the same temporal order for
each input option, and the `blocksize`

argument should be
omitted.

The output dataset includes a number of summary variables (to see the
variables please remove `full = FALSE`

from the
`summary_hfv()`

call), but the data is broken down into
groups of three consecutive monitoring occasions per row (occasions
*t*+1, *t*, and *t*-1, corresponding to
`year3`

, `year2`

, and `year1`

in the
output, respectively), with individuals spread across multiple rows. The
output dataset is further limited to those entries in which the
individual is alive in occasion *t* (`year2`

), meaning
that all rows in which an individual is dead or not yet recruited in
occasion *t* were dropped. We end up with 320 rows of data and 57
variables.

This reorganized dataset includes the `sizeadded`

group of
three variables, which were automatically calculated. These are sums of
the size variables for each occasion, such that `size1added`

is calculated as `sizea1 + sizeb1 + sizec1`

. This term may or
may not make sense depending on the dataset. Here, the full size of the
individual in each occasion equals this sum, because size is determined
as the number of stems per plant, and the three original size columns
give the numbers of three different kinds of stems per plant. Since size
is given as the total number of stems in this example, we will use the
`sizeadded`

group of variables to code individual size.

Size and fecundity are both count variables here, and we need to
determine the appropriate distribution to use in both cases. Let’s look
at histograms of size and fecundity in occasion *t*.

```
par(mfrow=c(1,2))
hist(vertdata$size2added, main = "Size in time t", xlab = "No. of stems in time t")
hist(subset(vertdata, repstatus2 == 1)$feca2, main = "Fecundity in time t",
xlab = "No. of fruit pods in time t")
```

Stem number has a very low mean, and the curve declines fairly smoothly. The Poisson or negative binomial distribution might work for size. Fecundity appears to have many zeros, suggesting that we might need a zero-inflated count distribution. Let’s conduct a formal set of tests to determine the appropriate distributions for size and fecundity, using a function that checks for the appropriate distributions for all possible vital rate functions in our dataset.

```
hfv_qc(data = vertdata, vitalrates = c("surv", "obs", "size", "repst", "fec"),
suite = "full", size = c("size3added", "size2added", "size1added"))
#> Survival:
#>
#> Data subset has 58 variables and 320 transitions.
#>
#> Variable alive3 has 0 missing values.
#> Variable alive3 is a binomial variable.
#>
#>
#> Observation status:
#>
#> Data subset has 58 variables and 303 transitions.
#>
#> Variable obsstatus3 has 0 missing values.
#> Variable obsstatus3 is a binomial variable.
#>
#>
#> Primary size:
#>
#> Data subset has 58 variables and 288 transitions.
#>
#> Variable size3added has 0 missing values.
#> Variable size3added appears to be an integer variable.
#>
#> Variable size3added is fully positive, lacking even 0s.
#>
#> Overdispersion test:
#> Mean size3added is 3.653
#> The variance in size3added is 13.41
#> The probability of this dispersion level by chance assuming that
#> the true mean size3added = variance in size3added,
#> and an alternative hypothesis of overdispersion, is 3.721e-138
#> Variable size3added is significantly overdispersed.
#>
#> Zero-inflation and truncation tests:
#> Mean lambda in size3added is 0.02592
#> The actual number of 0s in size3added is 0
#> The expected number of 0s in size3added under the null hypothesis is 7.465
#> The probability of this deviation in 0s from expectation by chance is 0.9964
#> Variable size3added is not significantly zero-inflated.
#>
#> Variable size3added does not include 0s, suggesting that a zero-truncated distribution may be warranted.
#>
#>
#> Reproductive status:
#>
#> Data subset has 58 variables and 288 transitions.
#>
#> Variable repstatus3 has 0 missing values.
#> Variable repstatus3 is a binomial variable.
#>
#>
#> Fecundity:
#>
#> Data subset has 58 variables and 118 transitions.
#>
#> Variable feca2 has 0 missing values.
#> Variable feca2 appears to be an integer variable.
#>
#> Variable feca2 is fully non-negative.
#>
#> Overdispersion test:
#> Mean feca2 is 0.7881
#> The variance in feca2 is 1.536
#> The probability of this dispersion level by chance assuming that
#> the true mean feca2 = variance in feca2,
#> and an alternative hypothesis of overdispersion, is 0.1193
#> Dispersion level in feca2 matches expectation.
#>
#> Zero-inflation and truncation tests:
#> Mean lambda in feca2 is 0.4547
#> The actual number of 0s in feca2 is 68
#> The expected number of 0s in feca2 under the null hypothesis is 53.65
#> The probability of this deviation in 0s from expectation by chance is 5.904e-06
#> Variable feca2 is significantly zero-inflated.
```

The results of these tests show us that size is overdispersed and zero-truncated (the zero truncation is due to the fact that we observed instances of size equal to zero within the vegetative dormancy stage). We will use the zero-truncated negative binomial distribution for this vital rate. Our tests of fecundity show no significant overdispersion, but the number of zeros exceeds expectation, suggesting that we use a zero-inflated Poisson distribution. All other vital rates appear to match expectation for the binomial distribution.

We may also wish to see how to proceed if our original dataset is
already in vertical, ahistorical format, or a format in which each row
gives the state of an individual in a single monitoring occasion. This
package also includes dataset `cypvert`

, which is the same
dataset as `cypdata`

but set in ahistorical vertical format.
Here, we use the `historicalize3()`

function to deal with
this dataset.

```
data(cypvert)
dim(cypdata)
#> [1] 77 29
dim(cypvert)
#> [1] 322 14
#summary(cypvert)
```

This dataset has more rows and fewer columns, because we constructed the
dataset by splitting the data for each individual across multiple rows.
After three columns of identifying information (`plantid`

,
`patch`

, and `censor`

), a single column designates
the time of occasion *t*, given as `year2`

. This
dataset then includes columns showing individual state in pairs of
consecutive years corresponding to occasions *t* and
*t*+1. State in occasion *t*-1 is not presented because
this is an ahistorical dataset. This dataset includes the
`plantid`

variable, which is an individual identity term that
allows us to associate rows with their individuals and so allows
conversion. The `historicalize3()`

function uses individual
identity to reorganize datasets into historical vertical format.

```
<- historicalize3(data = cypvert, patchidcol = "patch",
vertdata2 individcol = "plantid", year2col = "year2", sizea2col = "Inf2.2",
sizea3col = "Inf2.3", sizeb2col = "Inf.2", sizeb3col = "Inf.3",
sizec2col = "Veg.2", sizec3col = "Veg.3", repstra2col = "Inf2.2",
repstra3col = "Inf2.3", repstrb2col = "Inf.2", repstrb3col = "Inf.3",
feca2col = "Pod.2", feca3col = "Pod.3", repstrrel = 2, stageassign = cypframe,
stagesize = "sizeadded", censorcol = "censor", censorkeep = 1, censor = FALSE,
NAas0 = TRUE, reduce = TRUE)
summary_hfv(vertdata2, full = FALSE)
#>
#> This hfv dataset contains 320 rows, 57 variables, 1 population,
#> 3 patches, 74 individuals, and 5 time steps.
```

Comparing the dimensions of the new vertical datasets, we find that the lengths of the datasets are the same in terms of rows and columns, and the variables and data are the same although the order of the columns and rows might not match (see the summaries for comparison).

One important consideration is the use of censor variables. Censoring a
demographic dataset refers to the elimination of suspect data points
from analysis. This is typically accomplished by including a binary
variable in the dataset denoting whether an individual datum is to be
kept or excluded. The objects `vertdata`

and
`vertdata2`

were both created without censoring. However,
because the datasets included censor variables (all data were set to be
included, with no suspect data), we wished to incorporate those
variables in the final datasets. Hence, although
`censor = FALSE`

in both the call to
`verticalize3()`

and the call to
`historicalize3()`

, we also noted
`censorcol = "censor"`

and `censorkeep = 1`

in the
call to `historicalize3()`

(the latter option specifies the
value of the censor variable coding for a “good” datum). Failing to add
these options to the call to `historicalize3()`

will produce
approximately the same dataset, but with some zeroes entering variables
`censor1`

, `censor2`

, and `censor3`

in
the historicalized dataset that do not exist in the verticalized
dataset.

The next steps involve adding some external data to parameterize the
matrices properly. The `supplemental()`

function provides a
means of inputting four kinds of data into MPM construction:

- fixed transition values derived from other studies and added as constants to matrices;
- proxy transition values when data for particular transitions do not exist and other, estimable transitions will be used as proxies;
- reproductive multipliers to indicate which stages lead to the production of which stages, and at what level relative to estimated fecundity; and
- survival multipliers, in cases in which proxy survival transitions are used but must be set at elevated or reduced levels relative to the original transition.

We will create two supplement tables, the first covering the historical
analysis, and the second covering the ahistorical analysis. Each row
refers to a specific transition, and in the historical case there are
codes for 23 given transitions (12 for the ahistorical case). The first
nine of the historical transitions are set to specific probabilities
(six for the ahistorical), and the next 12 are transitions that will be
set to other, estimated transitions acting as proxies (four for the
ahistorical; these are the non-NA transitions in `eststage`

set below). The final two terms are fecundity multipliers. Based on the
literature, the proxies for entry into the adult classes are transitions
from dormancy, as below. Where necessary, we also use `rep`

and `mat`

as shorthand to code for all reproductive stages
and all mature stages, respectively.

```
<- supplemental(stage3 = c("SD", "SD", "P1", "P1", "P2", "P3", "SL",
cypsupp3 "SL", "SL", "D", "V1", "V2", "V3", "D", "V1", "V2", "V3", "mat", "mat",
"mat", "mat", "SD", "P1"),
stage2 = c("SD", "SD", "SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL",
"SL", "SL", "SL", "SL", "SL", "SL", "D", "V1", "V2", "V3", "rep", "rep"),
stage1 = c("SD", "rep", "SD", "rep", "SD", "P1", "P2", "P3", "SL", "P3", "P3",
"P3", "P3", "SL", "SL", "SL", "SL", "SL", "SL", "SL", "SL", "mat", "mat"),
eststage3 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "V1", "V2", "V3", "D",
"V1", "V2", "V3", "mat", "mat", "mat", "mat", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "D", "D", "D", "D",
"D", "D", "D", "D", "V1", "V2", "V3", NA, NA),
eststage1 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "D", "D", "D", "D",
"D", "D", "D", "V1", "V1", "V1", "V1", NA, NA),
givenrate = c(0.08, 0.08, 0.1, 0.1, 0.1, 0.1, 0.125, 0.2, 0.2, NA, NA, NA, NA,
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
NA, 1, 1, 1, 1, 0.5, 0.5),
type = c("S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S",
"S", "S", "S", "S", "S", "S", "S", "R", "R"),
type_t12 = c("S", "F", "S", "F", "S", "S", "S", "S", "S", "S", "S", "S", "S",
"S", "S", "S", "S", "S", "S", "S", "S", "S", "S"), stageframe = cypframe)
<- supplemental(stage3 = c("SD", "P1", "P2", "P3", "SL", "SL", "D",
cypsupp2 "V1", "V2", "V3", "SD", "P1"),
stage2 = c("SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL", "SL", "rep",
"rep"),
eststage3 = c(NA, NA, NA, NA, NA, NA, "D", "V1", "V2", "V3", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, "D", "D", "D", "D", NA, NA),
givenrate = c(0.08, 0.1, 0.1, 0.1, 0.125, 0.2, NA, NA, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 0.5, 0.5),
type = c("S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "R", "R"),
stageframe = cypframe, historical = FALSE)
#cypsupp3
#cypsupp2
```

Matrix estimation can proceed either via raw matrix methods, as initially outlined in Ehrlén (2000), or via function-based matrix development, essentially equivalent to the creation of discretized complex integral projection models per Ellner and Rees (2006) and as further described in the non-Gaussian case in Shefferson, Warren II, and Pulliam (2014). In the raw MPM case, no vital rate models are estimated, whereas in function-based MPMs, vital rates are first analyzed via linear or non-linear models of the raw demographic dataset, and then functions are created that estimate these vital rates according the inputs given. Matrices are then estimated using these functions, as opposed to the raw data.

Most function-based matrices, whether integral projection models or
otherwise, are developed using either a *generalized linear modeling
(GLM)* strategy, or a *generalized linear mixed modeling (GLMM,
or mixed)* strategy. The former is more common and simpler, but the
latter is more statistically sound because it corrects the lack of
independence in data originating from repeated sampling of the same
individuals. The difference between the two methods with regards to
vital rate modeling is strongly related to assumptions regarding
individual and spatiotemporal variation in vital rates.

In both GLM and GLMM-based MPMs, the underlying dataset utilized is a
vertical dataset. Each row of data gives the state of the individual in
either two consecutive occasions (the ahistorical case), or three
consecutive occasions (the historical case). Under the GLM framework,
time in occasion *t* is a fixed categorical variable, and
individual identity is ignored. Treating time as fixed implies that the
actual monitoring occasions are the only times for which inference is
sought. Thus, it would be inappropriate to infer future population
dynamics after 2009 for a dataset collected between 2004 and 2009.
Ignoring individual identity treats all transitions as independent, even
though data originating from the same sampled individual is not
independent of that individual’s previous or future transitions. This
statistical dependence may inflate Type 1 error in the linear modeling,
yielding more significant terms in the chosen best-fit model and causing
the retention of more terms than is warranted.

Under the GLMM (generalized linear mixed model, or mixed) framework, both time and individual identity are typically treated as random categorical terms. This has two major implications. First, both time and individuals can be assumed to be random samples from a broader population of times and individuals for which inference is sought. Thus, sampled monitoring occasions represent a greater universe of years for which inference can be made. Second, treating individual as a random categorical term eliminates the pseudoreplication that is inherent in the GLM approach when individuals are monitored potentially many times, because each individual is assumed to be randomly drawn and associated with its own response distribution. Subpopulations may also be considered random, in which case they are assumed to have been sampled from all possible subpopulations whether or not other subpopulations actually exist. We will use this approach in this vignette.

The next step is to choose the underlying distributions. The probabilities of survival, observation, reproductive status, and maturity status are set to the binomial distribution, and this cannot be altered. Size transition probability and fecundity rate can be set to the Gaussian, Gamma, Poisson, or negative binomial distributions, with zero-inflated and zero-truncated versions of the Poisson and negative binomial also available. If size or fecundity rate is continuous, then it should be set to the Gaussian distribution if not too right-skewed; otherwise, the Gamma distribution may be better (although our implementation of the Gamma distribution currently requires fully positive data). If size or fecundity is a count variable, then it should be set to the Poisson distribution if the mean equals the variance. The negative binomial distribution is provided in cases where the assumption that the mean equals the variance is clearly broken.

In *lefko3*, function `modelsearch()`

is the workhorse
that conducts vital rate model estimation. Here, we will create a full
suite of vital rate models for the *Cypripedium candidum*
dataset. Before proceeding, we still need to decide on the the correct
vital rates to model, the proper statistical distributions for estimated
vital rates, the proper parameterization of each vital rate, and the
strategy for determination of the best-fit model for each vital rate.

Function `modelsearch()`

estimates up to fourteen vital rate
models:

survival probability from occasion

*t*to occasion*t*+1,observation probability in occasion

*t*+1 assuming survival until that time,primary size in occasion

*t*+1 assuming survival and observation in that time,secondary size in occasion

*t*+1 assuming survival and observation in that time,tertiary size in occasion

*t*+1 assuming survival and observation in that time,reproduction status in occasion

*t*+1 assuming survival and observation until that time,fecundity rate assuming survival until and observation and reproduction in the occasion of production of offspring (occasion

*t*or*t*+1; mature only),juvenile survival probability from occasion

*t*to occasion*t*+1,juvenile observation probability in occasion

*t*+1 assuming survival until that time,juvenile primary size in occasion

*t*+1 assuming survival and observation in that time,juvenile secondary size in occasion

*t*+1 assuming survival and observation in that time,juvenile tertiary size in occasion

*t*+1 assuming survival and observation in that time,reproduction status in occasion

*t*+1 assuming survival and observation until that time of a juvenile in occasion*t*that is becoming mature in occasion*t*+1, andmaturity status in occasion

*t*+1 assuming survival until that time of a juvenile in occasion*t*.

The default settings for `modelsearch`

estimate 1) survival
probability, 3) primary size distribution, and 7) fecundity, which are
the minimum three vital rates required for a full MPM. Observation
probability (option `obs`

in `vitalrates`

) should
only be included when a life history stage or size exists that cannot be
observed. For example, in the case of a plant with vegetative dormancy,
the observation probability can be thought of as the sprouting
probability, which is a biologically meaningful vital rate (Shefferson et al. 2001). Reproduction status
(option `repst`

in `vitalrates`

) should only be
modeled if size classification needs to be stratified by the ability to
reproduce, as when equivalently sized stages exist differing in the
ability to reproduce. Secondary and tertiary size should only be modeled
when two or three size variables are used in stage classification. Since
*Cypripedium candidum* is capable of long bouts of vegetative
dormancy, since we wish to stratify the population into reproductive and
non-reproductive adults of the same size classes, and since we have no
data derived from juvenile individuals, we will set
`vitalrates = c("surv", "obs", "size", "repst", "fec")`

and
leave the juvenile-related options at their defaults.

Next, we need to set the proper statistical distribution for size and fecundity. Size was measured as the number of stems and so is a count variable. Likewise, fecundity is actually estimated as the number of fruits produced per plant, and so is also a count variable. We have already performed tests for overdispersion and zero-inflation, and we are also aware that size in observed stages cannot equal 0, requiring zero truncation in that parameter. So we will set size to the zero-truncated negative binomial distribution, and fecundity to the zero-inflated Poisson distribution.

Now we need the proper model parameterizations for each vital rate,
using the `suite`

option. The default,
`suite = "main"`

, under the mixed model setting
(`approach = "mixed"`

) starts with the estimation of global
models that include size and reproductive status in occasions *t*
and *t*-1 as fixed factors, with individual identity and time in
occasion *t* (year *t*) set as random categorical terms.
Other terms can be specified, including individual covariates, spatial
density, and age. Setting `suite = "full"`

will yield global
models that also include all two-way interactions. The default under the
GLM setting (`approach = "glm"`

) makes time in occasion
*t* a fixed term and drops individual identity from
consideration. The global model under `suite = "full"`

then
includes all fixed factors noted before, plus all two-way interactions
between fixed factors (`"full"`

is the only setting with
interaction terms). If the population is not stratified by reproductive
status, then `suite = "size"`

will eliminate reproductive
status terms and use all others in the global model. If size is not
important, then `suite = "rep"`

will eliminate size but keep
reproductive status and all other terms. Finally,
`suite = "cons"`

will result in a global model in which
neither reproductive status nor size are considered, although optional
terms such as age or spatial density may be still be incorporated. We
will set `suite = "full"`

.

Finally, we need to determine the proper strategy for the determination
of the best-fit model. Model building proceeds through the
`dredge`

function in package `MuMIn`

(Bartoń 2014), and each model has an associated
AICc value. The default setting in `lefko3`

(`bestfit = "AICc&k"`

) will compare all models within 2.0
AICc units of the model with \(\Delta AICc =
0\), and choose the one with the lowest degrees of freedom. This
approach is generally better than the alternative, which simply uses the
model with \(\Delta AICc = 0\)
(`bestfit = "AICc"`

), as all models within 2.0 AICc units of
that model are equally parsimonious and so fewer degrees of freedom
result from fewer parameters estimated (Burnham
and Anderson 2002).

```
<- modelsearch(vertdata, historical = TRUE, approach = "mixed",
cypmodels3 vitalrates = c("surv", "obs", "size", "repst", "fec"), sizedist = "negbin",
size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE, suite = "full",
size = c("size3added", "size2added", "size1added"), quiet = "partial")
#>
#> Developing global model of survival probability...
#>
#> Global model of survival probability developed. Proceeding with model dredge...
#>
#> Developing global model of observation probability...
#>
#> Global model of observation probability developed. Proceeding with model dredge...
#>
#> Developing global model of primary size...
#>
#> Global model of primary size developed. Proceeding with model dredge...
#>
#> Developing global model of reproduction probability...
#>
#> Global model of reproduction probability developed. Proceeding with model dredge...
#>
#> Developing global model of fecundity...
#>
#> Global model of fecundity developed. Proceeding with model dredge...
#>
#> Finished selecting best-fit models.
#summary(cypmodels3)
```

As `modelsearch`

works, it produces text describing its
current processes and any issues encountered. It is quite likely that
many warning messages will appear, and these may be of use in
understanding the dataset and how well it conforms to analytical
assumptions. We have silenced most of this with the
`quiet = "partial"`

option, although some red text still
indicates mileposts reached by the function and a few warnings that have
slipped through from the core estimation functions. The warnings are not
to be terribly concerned about. However, we encourage users to allow the
function to run unsilenced in case of severe modeling problems
(`quiet = FALSE`

). Please read the documentation for
functions `lm()`

(`stats`

package),
`glm()`

(`stats`

package), `glm.nb()`

(`MASS`

package), `lmer()`

(`lme4`

package), `glmer()`

(`lme4`

package),
`glmmTMB()`

(`glmmTMB`

package),
`zeroinfl()`

(`pscl`

package), `vglm()`

(`VGAM`

package), and `dredge()`

(`MuMIn`

package) for further information on the sources of
warnings.

In the output, we see historical terms determining primary size,
suggesting that we cannot ignore history. As such, the models created
above are actually only usable for the construction of historical
models. We can also see that although the accuracy of our some vital
rate models is very high, while it was only fair for others. For
comparison, we may wish to estimate ahistorical models. In that case, we
also need linear models in which the global models tested do not include
state at occasion *t*-1, as below.

```
<- modelsearch(vertdata, historical = FALSE, approach = "mixed",
cypmodels2 vitalrates = c("surv", "obs", "size", "repst", "fec"), sizedist = "negbin",
size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE, suite = "full",
size = c("size3added", "size2added"), quiet = "partial")
#>
#> Developing global model of survival probability...
#>
#> Global model of survival probability developed. Proceeding with model dredge...
#>
#> Developing global model of observation probability...
#>
#> Global model of observation probability developed. Proceeding with model dredge...
#>
#> Developing global model of primary size...
#>
#> Global model of primary size developed. Proceeding with model dredge...
#>
#> Developing global model of reproduction probability...
#>
#> Global model of reproduction probability developed. Proceeding with model dredge...
#>
#> Developing global model of fecundity...
#>
#> Global model of fecundity developed. Proceeding with model dredge...
#>
#> Finished selecting best-fit models.
#summary(cypmodels2)
```

Fewer models were estimated per dredge, since fewer parameters were
tested in the global models (size and reproductive status in occasion
*t*-1 were not included). Since historical terms were not tested,
the best-fit model for primary size looks quite different. A thorough
comparison shows that many of the best-fit models are similar between
historical and ahistorical analysis, and that the only model different
appears to be the primary size model.

Now we will create function-based MPMs. Let’s first create a set of ahistorical matrices.

```
<- flefko2(stageframe = cypframe, supplement = cypsupp2,
cypmatrix2 modelsuite = cypmodels2, data = vertdata, patch = NULL)
#summary(cypmatrix2)
```

A glance at the summary highlights that these are large, dense matrices - here, 2,459 out of 2,916 total elements have been estimated (84.3%). In raw matrices, elements associated with transitions from specific stages are only estimated when individuals actually move through those transitions, and so the numbers of transitions estimated and of stages identified are strongly limited by the size of the dataset. In function-based matrices, in contrast, the linear models estimated allow the estimation of all elements that are theoretically possible (i.e. only structural zeros are not estimated), and so stages may be developed for the full range of sizes experienced by individuals.

The output of the `flefko2()`

call includes several key
elements. One of the most important is a data frame showing the order of
matrices produced. Let’s take a look at this object.

`#cypmatrix2$labels`

Here we see that our five matrices are annual matrices corresponding to
the same population/patch (we did not input vital rate models using
patch identity as a term, so our matrices are for the whole population
rather than for individual patches). Other elements in this object
include summaries of the stages used, in the order that they occur in
the matrix. Since this is an ahistorical MPM, the key element here is
`cypmatrix2$ahstages`

. If this were a historical MPM, then
`cypmatrix2$ahstages`

would give the order of stages, and
`cypmatrix2$hstages`

would include the order of stage-pairs
corresponding to the rows and columns in such MPMs.

Next, we will create a set of historical Lefkovitch matrices. To make
the resulting lefkoMat object reasonably small, we will use sparse
matrices (`sparse_output = TRUE`

).

```
<- flefko3(stageframe = cypframe, supplement = cypsupp3,
cypmatrix3 modelsuite = cypmodels3, data = vertdata, sparse_output = TRUE)
#> Warning: Supplement table contains multiple entries for the same transition(s).
#summary(cypmatrix3)
```

We see many more total elements per matrix (over 8.5 million, in
contrast to 2,916 in the ahistorical case), and many more rows and
columns (54 rows and columns in the ahistorical case, and 54^{2}
= 2,916 rows and columns in the historical case). However, historical
matrices are composed primarily of structural zeros. Thus, with only
117,908 + 2,352 = 120,260 elements estimated per matrix, only 1.4% of
elements are non-zero. Note also that we used sparse matrix format to
hold these matrices. The current size of object `cypmatrix3`

should be around 14 MB, whereas the equivalent object using standard
matrices would be over 1 GB in size (to compare, repeat the code above
without the `sparse_output`

argument, or set it to
`FALSE`

).

Now let’s estimate the historical and ahistorical element-wise mean matrices.

```
<- lmean(cypmatrix2)
tmeans2r <- lmean(cypmatrix3)
tmeans3r #summary(tmeans2r)
#summary(tmeans3r)
```

It might help to do a more holistic comparison of the ahistorical
vs. historical MPMs. Let’s create matrix plots of the ahistorical and
historical means using the `image3()`

function. Let’s first
view the ahistorical matrix image.

```
image3(tmeans2r, used = 1)
#> [[1]]
```

This is a fairly large matrix, and rather dense though with obviously large clusters of zero elements particularly at the edges. Now let’s see the historical matrix.

```
image3(tmeans3r, used = 1)
#> [[1]]
```

Historical matrices are often huge and always sparse, just as our historical matrix is in the plot above. Although there are roughly 8.5 million elements, very few are non-zero and those are identified as the streaks in the matrix image above.

For further comparison, let’s decompose the mean historical matrix into
conditional historical matrices. We’ll take a look at the matrix
conditional on vegetative dormancy in occasion *t*-1.

```
<- cond_hmpm(tmeans3r)
condm3r #print(condm3r$Mcond[[1]]$D, digits = 3)
```

This conditional matrix takes the theoretically possible elements in
which stage in time *t*-1 is `D`

(vegetative
dormancy), and creates a matrix with the dimensions and row and column
order of an ahistorical matrix with them. Since vegetative dormancy is
an adult stage, we do not see survival transitions from juvenile stages
- only adult portions of the matrix are shown, including both survival
and fecundity transitions.

Let’s now conduct some projection analyses. First, we will estimate the
deterministic and stochastic population growth rates for these means.
The `set.seed()`

function makes our stochastic results
reproducible.

```
set.seed(42)
<- slambda3(cypmatrix2)
cypann2s set.seed(42)
<- slambda3(cypmatrix3)
cypann3s
lambda3(tmeans2r)
#> pop patch lambda
#> 1 1 1 0.9603461
lambda3(tmeans3r)
#> pop patch lambda
#> 1 1 <NA> 0.9629342
cypann2s#> pop patch a var sd se
#> 1 1 1 -0.04043205 0.002352739 0.04850504 0.0004850504
cypann3s#> pop patch a var sd se
#> 1 1 <NA> -0.03771166 0.001646792 0.04058069 0.0004058069
```

These \(\lambda\) values are similar to one another, as are estimates of \(a = \text{log} \lambda _{S}\). Let’s take a look at a plot of annual deterministic growth rates (stochastic growth rate is calculated as a single value across time, and so is not included). We find that annual \(\lambda\) estimates differ only slightly between ahistorical and historical analyses.

```
<- lambda3(cypmatrix3)
cypann3 <- lambda3(cypmatrix2)
cypann2
plot(lambda ~ year2, data = cypann2, xlab = "Year", ylab = "Lambda",
ylim = c(0.93, 0.98), type = "l", lty= 1, lwd = 2, bty = "n")
lines(lambda ~ year2, data = cypann3, lty = 2, lwd= 2, col = "red")
legend("bottomright", c("ahistorical", "historical"), lty = c(1, 2),
col = c("black", "red"), lwd = 2, bty = "n")
```

Now let’s take a look at the stable stage distributions. We will look at deterministic and stochastic versions of the ahistorical and historical MPMs.

```
<- stablestage3(tmeans2r)
tm2ss <- stablestage3(tmeans3r)
tm3ss <- stablestage3(cypmatrix2, stochastic = TRUE, seed = 42)
tm2ss_s <- stablestage3(cypmatrix3, stochastic = TRUE, seed = 42)
tm3ss_s
<- cbind.data.frame(tm2ss$ss_prop, tm3ss$ahist$ss_prop,
ss_put_together $ss_prop, tm3ss_s$ahist$ss_prop)
tm2ss_snames(ss_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(ss_put_together) <- tm2ss$stage_id
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage",
ylim = c(0, 0.15), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
```

Overall, these are very similar patterns. Whether ahistorical or historical, deterministic or stochastic, our analyses suggest that small non-reproductive adults take up the greatest share of the stable stage structure, with small reproductive adults coming next, and then dormant seed and 1st-year protocorms.

Next let’s look at the reproductive values associated with both ahistorical and historical approaches. We will standardize each reproductive value vector to sum to 1.0 to make them comparable in a single plot (normally we would not standardize across analyses like this, using the first or minimum reproductive value in each case for standardization instead).

```
<- repvalue3(tmeans2r)
tm2rv <- repvalue3(tmeans3r)
tm3rv <- repvalue3(cypmatrix2, stochastic = TRUE, seed = 42)
tm2rv_s <- repvalue3(cypmatrix3, stochastic = TRUE, seed = 42)
tm3rv_s
<- cbind.data.frame((tm2rv$rep_value / sum(tm2rv$rep_value)),
rv_put_together $ahist$rep_value / sum(tm3rv$ahist$rep_value)),
(tm3rv$rep_value / sum(tm2rv_s$rep_value)),
(tm2rv_s$ahist$rep_value / sum(tm3rv_s$ahist$rep_value)))
(tm3rv_snames(rv_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(rv_put_together) <- tm2rv$stage_id
barplot(t(rv_put_together), beside=T, ylab = "Relative rep value", xlab = "Stage",
ylim = c(0, 0.03), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topleft", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
```

We see roughly equivalent patterns here. All analyses show the smallest reproductive value in the juvenile stages, while adults have the greatest value, particularly medium- and large-sized adults of both flowering and non-flowering kinds. The key difference appears to be that historical analyses predict decreasing reproductive value for adults above a certain size.

Let’s now conduct a sensitivity analysis, again using both deterministic and stochastic approaches. First we will look at the ahistorical MPM.

```
<- sensitivity3(tmeans2r)
tm2sens set.seed(42)
<- sensitivity3(cypmatrix2, stochastic = TRUE)
tm2sens_s
# The highest deterministic sensitivity value:
max(tm2sens$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)])
#> [1] 0.2002353
# This value is associated with element:
which(tm2sens$ah_sensmats[[1]] ==
max(tm2sens$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)]))
#> [1] 378
# The highest stochastic sensitivity value:
max(tm2sens_s$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)])
#> [1] 0.1771649
# This value is associated with element:
which(tm2sens_s$ah_sensmats[[1]] ==
max(tm2sens_s$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)]))
#> [1] 1674
```

The highest sensitivity value in the deterministic case appears to be associated with the transition from 1-sprouted non-flowering adult to the largest flowering adult (element 378, in the 7th column and 54th row), and the highest value in the stochastic case is the transition from 1-sprouted flowering adult to the largest flowering adult (element 1674, in the 31st column and 54th row). Inspecting each sensitivity matrix also shows that transitions near that element in both matrices are also associated with rather high sensitivities.

Now let’s compare with the historical case. We will compare both the full historical sensitivity matrices, and also look at the historically-corrected deterministic sensitivities. Note that we need to convert to standard matrix format to get the correct element numbers, since the input matrices are in sparse format. We will also use a small number of time steps for the historical sensitivity analysis to make the time required more manageable.

```
<- sensitivity3(tmeans3r)
tm3sens set.seed(42)
<- sensitivity3(cypmatrix3, stochastic = TRUE, steps = 500)
tm3sens_s
# The highest deterministic sensitivity value:
max(as.matrix(tm3sens$h_sensmats[[1]])[which(as.matrix(tmeans3r$A[[1]]) > 0)])
#> [1] 0.04427405
# This value is associated with element:
which(as.matrix(tm3sens$h_sensmats[[1]]) ==
max(as.matrix(tm3sens$h_sensmats[[1]])[which(as.matrix(tmeans3r$A[[1]]) > 0)]))
#> [1] 962644
# The highest stochastic sensitivity value:
max(as.matrix(tm3sens_s$h_sensmats[[1]])[which(as.matrix(tmeans3r$A[[1]]) > 0)])
#> [1] 0.02567237
# This value is associated with element:
which(as.matrix(tm3sens_s$h_sensmats[[1]]) ==
max(as.matrix(tm3sens_s$h_sensmats[[1]])[which(as.matrix(tmeans3r$A[[1]]) > 0)]))
#> [1] 962641
# The highest deterministic sensitivity value (historically-corrected):
max(as.matrix(tm3sens$ah_sensmats[[1]])[which(as.matrix(tmeans2r$A[[1]]) > 0)])
#> [1] 0.1744897
# This value is associated with element:
which(as.matrix(tm3sens$ah_sensmats[[1]]) ==
max(as.matrix(tm3sens$ah_sensmats[[1]])[which(as.matrix(tmeans2r$A[[1]]) > 0)]))
#> [1] 364
# The highest stochastic sensitivity value (historically-corrected):
max(as.matrix(tm3sens_s$ah_sensmats[[1]])[which(as.matrix(tmeans2r$A[[1]]) > 0)])
#> [1] 0.1357679
# This value is associated with element:
which(as.matrix(tm3sens_s$ah_sensmats[[1]]) ==
max(as.matrix(tm3sens_s$ah_sensmats[[1]])[which(as.matrix(tmeans2r$A[[1]]) > 0)]))
#> [1] 1657
```

Deterministic sensitivity analysis supported transitions from small
non-flowering stages to larger flowering stages, such as 1-sprouted
non-flowering adult in occasions *t*-1 and *t* to
10-sprouted flowering adult in occasion *t*+1, as the transitions
that \(\lambda\) is most sensitive to
(as, an example, we can see this by dividing 962,644 by the number of
stage pairs, which is 2916 - rounding up the answer gives us the matrix
column [331], and the remainder from this division problem gives us the
row [364]; the row and column numbers can then be checked against the
`hstages`

portion of the historical `lefkoMat`

object used as input). The stochastic sensitivity analysis gives a
slightly different result, supporting the transition from 1-sprouted
non-flowering adult in occasions *t*-1 and *t* to
7-sprouted flowering adult in occasion *t*+1. The highest
historically-corrected deterministic sensitivity value is associated
with the transition from 1-sprouted non-flowering to 10-sprouted
flowering (element 364), while the highest historically-corrected
stochastic sensitivity value is associated with the transition from
1-sprouted flowering adult to 7-sprouted flowering adult stage (element
1657).

Let’s now assess the elasticity of \(\lambda\) to matrix elements, comparing the ahistorical to the historically-corrected case in both deterministic and stochastic analyses.

```
<- elasticity3(tmeans2r)
tm2elas <- elasticity3(tmeans3r)
tm3elas set.seed(42)
<- elasticity3(cypmatrix2, stochastic = TRUE)
tm2elas_s set.seed(42)
<- elasticity3(cypmatrix3, stochastic = TRUE, steps = 500)
tm3elas_s
# The largest ahistorical deterministic elasticity is associated with element:
which(tm2elas$ah_elasmats[[1]] == max(tm2elas$ah_elasmats[[1]]))
#> [1] 331
# The largest historically-corrected deterministic elasticity is associated with element:
which(as.matrix(tm3elas$ah_elasmats[[1]]) == max(tm3elas$ah_elasmats[[1]]))
#> [1] 331
# The largest historical deterministic elasticity is associated with element:
which(as.matrix(tm3elas$h_elasmats[[1]]) == max(tm3elas$h_elasmats[[1]]))
#> [1] 962611
# The largest ahistorical stochastic elasticity is associated with element:
which(tm2elas_s$ah_elasmats[[1]] == max(tm2elas_s$ah_elasmats[[1]]))
#> [1] 331
# The largest historically-corrected stochastic elasticity is associated with element:
which(as.matrix(tm3elas_s$ah_elasmats[[1]]) ==
max(as.matrix(tm3elas_s$ah_elasmats[[1]])))
#> [1] 331
# The largest historical stochastic elasticity is associated with element:
which(as.matrix(tm3elas_s$h_elasmats[[1]]) ==
max(as.matrix(tm3elas_s$h_elasmats[[1]])))
#> [1] 962611
```

Ahistorical and historical, deterministic and stochastic analyses
suggest that population growth rate is most elastic in response to
stasis as 1-sprouted non-flowering adult (element 331). Indeed, element
962,611 in the historical case is actually stasis in this stage across
times *t*-1, *t*, and *t*+1 (column:
`round(962611/2916) = 331`

; row:
`962611 %% 2916 = 331`

). This is a remarkably consistent
result, reinforcing the importance of this stage as seen in sensitivity
analyses.

Now let’s compare the elasticity of population growth rate in relation to the core life history stages, via a barplot comparison.

```
<- cbind.data.frame(colSums(tm2elas$ah_elasmats[[1]]),
elas_put_together ::colSums(tm3elas$ah_elasmats[[1]]), colSums(tm2elas_s$ah_elasmats[[1]]),
Matrix::colSums(tm3elas_s$ah_elasmats[[1]]))
Matrixnames(elas_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_put_together) <- tm2elas$ah_stages$stage_id
barplot(t(elas_put_together), beside=T, ylab = "Elasticity", xlab = "Stage",
col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
```

Elasticity analyses in these plots look very similar. Population growth rates are most elastic in response to changes in stage 7, which is the 1-sprouted non-flowering adult stage. Small adult stages have higher elasticity than larger adults and juveniles. Most of the differences between analyses here appear to lie in elasticity of small non-flowering vs. flowering adults in the ahistorical vs. historical comparison.

Finally, let’s take a look at how the importance of different kinds of transitions changes, by looking at elasticity sums. The plot will show that population growth rates are most elastic in response to changes in growth and shrinkage, with virtually no influence of fecundity.

```
<- summary(tm2elas)
tm2elas_sums <- summary(tm3elas)
tm3elas_sums <- summary(tm2elas_s)
tm2elas_s_sums <- summary(tm3elas_s)
tm3elas_s_sums
<- cbind.data.frame(tm2elas_sums$ahist[,2],
elas_sums_together $ahist[,2], tm2elas_s_sums$ahist[,2], tm3elas_s_sums$ahist[,2])
tm3elas_sumsnames(elas_sums_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_sums_together) <- tm2elas_sums$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 ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
```

Some users of `lefko3`

may wish to use conduct custom
projections, perhaps involving density dependence or projected
individual covariate inputs. Beginning with version 5.0,
`lefko3`

includes function `f_projection3()`

,
which allows users to conduct projection simulations of function-based
MPMs. This function takes the vital rate models that constitute the
`lefkoMod`

object input into matrix generating functions
`flefko3()`

, `flefko2()`

, `aflefko2()`

,
and `afleslie()`

, and builds matrices projecting the
population forward according to whatever inputs the user has in mind.
Function `f_projection3()`

is different from function
`projection3()`

in that the latter requires a
`lefkoMat`

object as input, and uses the matrices in whatever
projection the user has in mind. The former actually creates these
matrices from scratch at each time step and so does not take matrices as
inputs.

To illustrate how we might use this function, let’s run a
conservation-focused analysis using the ahistorical vital rate models
created above. We will ask what happens if we artificially pollinate
these plants enough to double their fecundity. Let’s first create a
`lefkoDens`

object, which holds the density dependence
information. Let’s also create an alternative supplement table with
doubled fecundity.

```
<- density_input(cypmatrix2, stage3 = c("SD", "P1", "SD", "P1"),
c2d stage2 = c("SD", "SD", "rep", "rep"), style = 1,
time_delay = 1, alpha = 1, beta = 0, type = c(1, 1, 2, 2))
#c2d
<- supplemental(stage3 = c("SD", "P1", "P2", "P3", "SL", "SL", "D",
cypsupp2alt "V1", "V2", "V3", "SD", "P1"),
stage2 = c("SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL", "SL", "rep",
"rep"),
eststage3 = c(NA, NA, NA, NA, NA, NA, "D", "V1", "V2", "V3", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, "D", "D", "D", "D", NA, NA),
givenrate = c(0.08, 0.1, 0.1, 0.1, 0.125, 0.2, NA, NA, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 0.5*2, 0.5*2),
type = c("S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "R", "R"),
stageframe = cypframe, historical = FALSE)
#cypsupp2alt
```

Now let’s run two projections, the first using the original ahistorical information, and the second using the altered supplement table. We will run both as stochastic and density dependent, with ten replicates of each and 100 time steps.

```
set.seed(42)
<- f_projection3(format = 3, data = vertdata, supplement = cypsupp2,
basetrial stageframe = cypframe, modelsuite = cypmodels2, nreps = 10, times = 100,
stochastic = TRUE, growthonly = TRUE, substoch = 2, density = c2d)
#> Warning: Option patch not set, so will set to first patch/population.
<- f_projection3(format = 3, data = vertdata, supplement = cypsupp2alt,
alttrial stageframe = cypframe, modelsuite = cypmodels2, nreps = 10, times = 100,
stochastic = TRUE, growthonly = TRUE, substoch = 2, density = c2d)
#> Warning: Option patch not set, so will set to first patch/population.
#summary(basetrial)
#summary(alttrial)
```

Let’s compare these two projections with plots.

```
par(mfrow = c(1,2))
plot(basetrial, ylim = c(0, 100))
plot(alttrial, ylim = c(0, 100))
```

Each plot shows 10 lines, one for each replicate. Both projections show decline, but the population appears to reach a higher high size early on under the boosted fecundity scenario. So, while boosting fecundity does not prevent extinction, it does seem to boost the population and probably lowers the overall chances of extinction within a short time window.

We have also created other literature to help users familiarize
themselves with population analyses in general and using the package.
Users should see our free e-manual called ** lefko3: a gentle
introduction**, as well as other vignettes including
long-format and video vignettes, on
the projects
page of the Shefferson lab website.

We are grateful to Joyce and George Proper, Ken Klick, and the many
other volunteers who have helped amass the *Cypripedium dataset*,
as well as 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.

Bartoń, Kamil A. 2014. “MuMIn: Multi-Model
Inference.” https://CRAN.R-project.org/package=MuMIn.

Burnham, Kenneth P., and David R. Anderson. 2002. *Model Selection
and Multimodel Inference: A Practical Information-Theoretic
Approach*. New York, New York, USA: Springer-Verlag New York, Inc.

Ehrlén, Johan. 2000. “The Dynamics of Plant Populations: Does the
History of Individuals Matter?” *Ecology* 81 (6): 1675–84.
https://doi.org/10.1890/0012-9658(2000)081[1675:TDOPPD]2.0.CO;2.

Ellner, Stephen P., and Mark Rees. 2006. “Integral Projection
Models for Species with Complex Demography.” *American
Naturalist* 167 (3): 410–28. https://doi.org/10.1086/499438.

Shefferson, Richard P., Ryo Mizuta, and Michael J. Hutchings. 2017.
“Predicting Evolution in Response to Climate Change: The Example
of Sprouting Probability in Three Dormancy-Prone Orchid Species.”
*Royal Society Open Science* 4 (1): 160647. https://doi.org/10.1098/rsos.160647.

Shefferson, Richard P., Brett K. Sandercock, Joyce Proper, and Steven R.
Beissinger. 2001. “Estimating Dormancy and Survival of a Rare
Herbaceous Perennial Using Mark-Recapture Models.”
*Ecology* 82 (1): 145–56. https://doi.org/10.1890/0012-9658(2001)082[0145:EDASOA]2.0.CO;2.

Shefferson, Richard P., Robert J. Warren II, and H. Ronald Pulliam.
2014. “Life History Costs Make Perfect Sprouting Maladaptive in
Two Herbaceous Perennials.” *Journal of Ecology* 102 (5):
1318–28. https://doi.org/10.1111/1365-2745.12281.