Author: Richard P. Shefferson

# Community matrix projection

#### Richard P. Shefferson

In this vignette, we will use the `cypa_data` dataset in package `adapt3`, as well as the `cypdata` dataset from package `lefko3`, to illustrate community matrix projection. Package `adapt3` allows both pre-existing MPMs to be utilized, as well as purely function-based MPMs, which must be constructed at each step. Both approaches utilize functions from and structures created by the package `lefko3`, and we encourage users to become familiar with that package to use package `adapt3` properly. Here, we illustrate the community matrix projection with pre-existing MPMs.

To reduce vignette 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 we encourage the user to run these statements without hashtags.

This vignette is only a sample analysis. Detailed information and instructions on using `adapt3` will be made available over time. Please check available resources on the projects page of the `{r}evolutionary demography` website.

### Organisms and populations

In this analysis, we will use data for two species that occur sympatrically. Cypripedium parviflorum and Cypripedium candidum (family Orchidaceae) are long-lived herbaceous perennials, native to North America and particularly common near the Great Lakes. They typically live in wet, calcareous wetlands on the edges of woodlands, with C. parviflorum prefering more wooded habitat than C. candidum (Shefferson et al. 2001). Individuals begin life as dust seeds, which may or may not germinate the year following production. Germination leads to the protocorm stage, a life history stage in which the individual grows underground in a non-photosynthetic, mycoheterotrophic state (Rasmussen 1995). They may live in this state for several years prior to becoming a seedling, at which point they may or may not sprout. Maturity involves the production of flowering sprouts, each with several leaves and typically only a single flower, although an individual may produce mostly non-flowering sprouts. Most individuals produce only a single sprout, and vegetative dormancy is common and can occur for overa decade in a single individual (Shefferson et al. 2018).

Data for this study were collected from a population in northeastern Illinois, from 2004 to 2009 (Shefferson, Mizuta, and Hutchings 2017). Each year at the start of the flowering season, all individuals at the site were located, with previously identified individuals identified based on their previous locations, and new individuals recorded via an exhaustive survey. Plant characteristics including the number of sprouts, which sprouts were flowering vs. non-flowering, and the number of flowers per sprout, were recorded for every individual in every year.

### Dataset preparation

Package `adapt3` is built to take advantage of package `lefko3`, which offers a general language and architecture to build and analyze most kinds of matrix projection models, including even discretized integral projection models (Shefferson, Kurokawa, and Ehrlén 2021). To start, we will load both `lefko3` and `adapt3`, and also load the datasets that we will use for examples.

    rm(list=ls(all=TRUE))

    library(lefko3)
    library(adapt3)

    data(cypa_data)
    data(cypdata)

Let’s now take a look at the C. parviflorum dataset.

    dim(cypa_data)
    #> [1] 1103   37
    summary(cypa_data)
    #>       plant_id        Inf.94            Veg.94            Inf.95           Veg.95          Inf.96           Veg.96
    #>  K NA     :  28   Min.   :0.00000   Min.   :0.00000   Min.   :0.0000   Min.   :0.000   Min.   :0.0000   Min.   :0.000
    #>  A NA     :  13   1st Qu.:0.00000   1st Qu.:0.00000   1st Qu.:0.0000   1st Qu.:0.000   1st Qu.:0.0000   1st Qu.:0.000
    #>  T NA     :  10   Median :0.00000   Median :0.00000   Median :0.0000   Median :0.000   Median :0.0000   Median :0.000
    #>  Willow NA:  10   Mean   :0.07253   Mean   :0.02267   Mean   :0.1732   Mean   :0.155   Mean   :0.2557   Mean   :0.282
    #>  X NA     :   7   3rd Qu.:0.00000   3rd Qu.:0.00000   3rd Qu.:0.0000   3rd Qu.:0.000   3rd Qu.:0.0000   3rd Qu.:0.000
    #>  Aspen NA :   6   Max.   :4.00000   Max.   :4.00000   Max.   :4.0000   Max.   :7.000   Max.   :6.0000   Max.   :6.000
    #>  (Other)  :1029
    #>      Inf.97           Veg.97          Inf.98           Veg.98           Inf.99           Veg.99           Inf.00
    #>  Min.   :0.0000   Min.   :0.000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000
    #>  1st Qu.:0.0000   1st Qu.:0.000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000
    #>  Median :0.0000   Median :0.000   Median :0.0000   Median :0.0000   Median :0.0000   Median :0.0000   Median :0.0000
    #>  Mean   :0.1333   Mean   :0.301   Mean   :0.2593   Mean   :0.2928   Mean   :0.3218   Mean   :0.2838   Mean   :0.2103
    #>  3rd Qu.:0.0000   3rd Qu.:0.000   3rd Qu.:0.0000   3rd Qu.:0.0000   3rd Qu.:0.0000   3rd Qu.:0.0000   3rd Qu.:0.0000
    #>  Max.   :4.0000   Max.   :7.000   Max.   :5.0000   Max.   :5.0000   Max.   :7.0000   Max.   :4.0000   Max.   :6.0000
    #>
    #>      Veg.00           Inf.01           Veg.01           Inf.02           Veg.02           Inf.03           Veg.03
    #>  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000
    #>  1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000
    #>  Median :0.0000   Median :0.0000   Median :0.0000   Median :0.0000   Median :0.0000   Median :0.0000   Median :0.0000
    #>  Mean   :0.3554   Mean   :0.2076   Mean   :0.3772   Mean   :0.2267   Mean   :0.2937   Mean   :0.1958   Mean   :0.3699
    #>  3rd Qu.:0.5000   3rd Qu.:0.0000   3rd Qu.:1.0000   3rd Qu.:0.0000   3rd Qu.:0.0000   3rd Qu.:0.0000   3rd Qu.:1.0000
    #>  Max.   :6.0000   Max.   :5.0000   Max.   :5.0000   Max.   :6.0000   Max.   :4.0000   Max.   :6.0000   Max.   :6.0000
    #>
    #>      Inf.04            Veg.04           Inf.05           Veg.05           Inf.06            Veg.06           Inf.07
    #>  Min.   :0.00000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.00000   Min.   :0.0000   Min.   :0.00000
    #>  1st Qu.:0.00000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.00000   1st Qu.:0.0000   1st Qu.:0.00000
    #>  Median :0.00000   Median :0.0000   Median :0.0000   Median :0.0000   Median :0.00000   Median :0.0000   Median :0.00000
    #>  Mean   :0.09791   Mean   :0.2684   Mean   :0.1233   Mean   :0.3073   Mean   :0.03626   Mean   :0.2013   Mean   :0.03264
    #>  3rd Qu.:0.00000   3rd Qu.:0.0000   3rd Qu.:0.0000   3rd Qu.:0.0000   3rd Qu.:0.00000   3rd Qu.:0.0000   3rd Qu.:0.00000
    #>  Max.   :4.00000   Max.   :4.0000   Max.   :6.0000   Max.   :4.0000   Max.   :3.00000   Max.   :4.0000   Max.   :3.00000
    #>
    #>      Veg.07           Inf.08            Veg.08           Inf.09            Veg.09           Inf.10            Veg.10
    #>  Min.   :0.0000   Min.   :0.00000   Min.   :0.0000   Min.   :0.00000   Min.   :0.0000   Min.   :0.00000   Min.   :0.0000
    #>  1st Qu.:0.0000   1st Qu.:0.00000   1st Qu.:0.0000   1st Qu.:0.00000   1st Qu.:0.0000   1st Qu.:0.00000   1st Qu.:0.0000
    #>  Median :0.0000   Median :0.00000   Median :0.0000   Median :0.00000   Median :0.0000   Median :0.00000   Median :0.0000
    #>  Mean   :0.1578   Mean   :0.05712   Mean   :0.1242   Mean   :0.08885   Mean   :0.1868   Mean   :0.08613   Mean   :0.1668
    #>  3rd Qu.:0.0000   3rd Qu.:0.00000   3rd Qu.:0.0000   3rd Qu.:0.00000   3rd Qu.:0.0000   3rd Qu.:0.00000   3rd Qu.:0.0000
    #>  Max.   :4.0000   Max.   :3.00000   Max.   :4.0000   Max.   :5.00000   Max.   :6.0000   Max.   :4.00000   Max.   :4.0000
    #>
    #>      Inf.11           Veg.11
    #>  Min.   :0.0000   Min.   :0.0000
    #>  1st Qu.:0.0000   1st Qu.:0.0000
    #>  Median :0.0000   Median :0.0000
    #>  Mean   :0.1342   Mean   :0.1886
    #>  3rd Qu.:0.0000   3rd Qu.:0.0000
    #>  Max.   :4.0000   Max.   :4.0000
    #>

This dataset includes information on 1103 individuals arranged horizontally, by row. There are 37 variables, by column. The first column gives identifying information for each individual. This is followed by 18 sets of two columns, each named `Inf.XX` and `Veg.XX`, where `XX` corresponds to the year of observation and with years organized consecutively. Thus, columns 2-3 refer to year 1994, columns 4-5 refer to year 1995, etc. We can see that we have data for 18 years here. This strictly repeating pattern allows us to manipulate the original dataset quickly and efficiently via `lefko3` (see our free online e-book called lefko3: a gentle introduction for more).

Next let’s look at a summary of the C. candidum dataset.

    dim(cypdata)
    #> [1] 77 29
    summary(cypdata)
    #>     plantid       patch        X               Y              censor     Inf2.04       Inf.04           Veg.04           Pod.04
    #>  Min.   : 164.0   A:23   Min.   : 46.5   Min.   :-28.00   Min.   :1   Min.   :0    Min.   :0.0000   Min.   : 0.000   Min.   :0.0
    #>  1st Qu.: 265.0   B:35   1st Qu.: 60.1   1st Qu.: 22.50   1st Qu.:1   1st Qu.:0    1st Qu.:0.0000   1st Qu.: 1.000   1st Qu.:0.0
    #>  Median : 455.0   C:19   Median : 91.4   Median : 75.60   Median :1   Median :0    Median :0.0000   Median : 2.000   Median :0.0
    #>  Mean   : 669.1          Mean   : 92.9   Mean   : 55.54   Mean   :1   Mean   :0    Mean   :0.6923   Mean   : 2.923   Mean   :0.2
    #>  3rd Qu.: 829.0          3rd Qu.:142.2   3rd Qu.: 80.10   3rd Qu.:1   3rd Qu.:0    3rd Qu.:1.0000   3rd Qu.: 4.000   3rd Qu.:0.0
    #>  Max.   :1560.0          Max.   :173.0   Max.   :142.40   Max.   :1   Max.   :0    Max.   :8.0000   Max.   :12.000   Max.   :3.0
    #>                                                                       NA's   :12   NA's   :12       NA's   :12       NA's   :12
    #>     Inf2.05            Inf.05           Veg.05          Pod.05          Inf2.06       Inf.06            Veg.06           Pod.06
    #>  Min.   :0.00000   Min.   : 0.000   Min.   :0.000   Min.   :0.0000   Min.   :0    Min.   : 0.0000   Min.   : 0.000   Min.   :0.0000
    #>  1st Qu.:0.00000   1st Qu.: 0.000   1st Qu.:1.000   1st Qu.:0.0000   1st Qu.:0    1st Qu.: 0.0000   1st Qu.: 1.000   1st Qu.:0.0000
    #>  Median :0.00000   Median : 0.000   Median :1.000   Median :0.0000   Median :0    Median : 0.0000   Median : 2.000   Median :0.0000
    #>  Mean   :0.04478   Mean   : 1.537   Mean   :2.134   Mean   :0.6567   Mean   :0    Mean   : 0.9016   Mean   : 2.213   Mean   :0.3934
    #>  3rd Qu.:0.00000   3rd Qu.: 2.000   3rd Qu.:3.000   3rd Qu.:1.0000   3rd Qu.:0    3rd Qu.: 1.0000   3rd Qu.: 3.000   3rd Qu.:0.0000
    #>  Max.   :1.00000   Max.   :18.000   Max.   :9.000   Max.   :7.0000   Max.   :0    Max.   :18.0000   Max.   :13.000   Max.   :4.0000
    #>  NA's   :10        NA's   :10       NA's   :10      NA's   :10       NA's   :16   NA's   :16        NA's   :16       NA's   :16
    #>  Inf2.07            Inf.07           Veg.07           Pod.07          Inf2.08       Inf.08            Veg.08          Pod.08
    #>  Mode:logical   Min.   :0.0000   Min.   : 0.000   Min.   :0.0000   Min.   :0    Min.   : 0.0000   Min.   : 0.00   Min.   :0.0000
    #>  NA's:77        1st Qu.:0.0000   1st Qu.: 1.000   1st Qu.:0.0000   1st Qu.:0    1st Qu.: 0.0000   1st Qu.: 1.00   1st Qu.:0.0000
    #>                 Median :0.0000   Median : 2.000   Median :0.0000   Median :0    Median : 0.0000   Median : 2.00   Median :0.0000
    #>                 Mean   :0.6271   Mean   : 2.627   Mean   :0.0678   Mean   :0    Mean   : 0.8868   Mean   : 2.83   Mean   :0.1509
    #>                 3rd Qu.:1.0000   3rd Qu.: 4.000   3rd Qu.:0.0000   3rd Qu.:0    3rd Qu.: 1.0000   3rd Qu.: 4.00   3rd Qu.:0.0000
    #>                 Max.   :7.0000   Max.   :13.000   Max.   :1.0000   Max.   :0    Max.   :11.0000   Max.   :13.00   Max.   :2.0000
    #>                 NA's   :18       NA's   :18       NA's   :18       NA's   :24   NA's   :24        NA's   :24      NA's   :24
    #>     Inf2.09       Inf.09           Veg.09           Pod.09
    #>  Min.   :0    Min.   : 0.000   Min.   : 0.000   Min.   :0.000
    #>  1st Qu.:0    1st Qu.: 0.000   1st Qu.: 1.000   1st Qu.:0.000
    #>  Median :0    Median : 1.000   Median : 1.000   Median :1.000
    #>  Mean   :0    Mean   : 1.833   Mean   : 2.233   Mean   :1.133
    #>  3rd Qu.:0    3rd Qu.: 2.000   3rd Qu.: 3.000   3rd Qu.:1.000
    #>  Max.   :0    Max.   :11.000   Max.   :10.000   Max.   :8.000
    #>  NA's   :17   NA's   :17       NA's   :17       NA's   :17

This is a much smaller dataset, which makes sense given that this population is much smaller than the C. parviflorum population. There are data on 77 individuals across six years, and 29 total variables, each named `Inf2.XX`, `Inf.XX` and `Veg.XX`, where `XX` corresponds to the year of observation and with years organized consecutively.

### Community projection with pre-existing MPMs

Our example will focus on developing MPMs for these populations, and using these MPMs to run community projections in which the two species exhibit density dependent responses in their vital rates. Because we use existing MPMs, the process generally proceeds quite quickly, and only slows when using giant matrices such as might occur in historical MPM projection. To make this process easier, we will develop a relatively small, stage-based empirical MPM

Per `lefko3` terminology, we will develop a stageframe that encapsulates our life history model by noting all of the relevant characteristics of each stage, as below. Because these species share very similar life histories, we will use the same life history for both. Note, however, that we do not need to do this, and can use completely different life histories and associated stageframes.

    sizevector <- c(0, 0, 0, 0, 1, 2.5, 4.5, 8, 17.5)
    stagevector <- c("SD", "P1", "SL", "D", "XSm", "Sm", "Md", "Lg", "XLg")
    repvector <- c(0, 0, 0, 0, 1, 1, 1, 1, 1)
    obsvector <- c(0, 0, 0, 0, 1, 1, 1, 1, 1)
    matvector <- c(0, 0, 0, 1, 1, 1, 1, 1, 1)
    immvector <- c(0, 1, 1, 0, 0, 0, 0, 0, 0)
    propvector <- c(1, 0, 0, 0, 0, 0, 0, 0, 0)
    indataset <- c(0, 0, 0, 1, 1, 1, 1, 1, 1)
    binvec <- c(0, 0, 0, 0.5, 0.5, 1, 1, 2.5, 7)

    cypframe_raw <- sf_create(sizes = sizevector, stagenames = stagevector,
      repstatus = repvector, obsstatus = obsvector, matstatus = matvector,
      propstatus = propvector, immstatus = immvector, indataset = indataset,
      binhalfwidth = binvec)
    cypframe_raw
    #>   stage size size_b size_c min_age max_age repstatus obsstatus propstatus immstatus matstatus indataset binhalfwidth_raw sizebin_min
    #> 1    SD  0.0     NA     NA      NA      NA         0         0          1         0         0         0              0.0         0.0
    #> 2    P1  0.0     NA     NA      NA      NA         0         0          0         1         0         0              0.0         0.0
    #> 3    SL  0.0     NA     NA      NA      NA         0         0          0         1         0         0              0.0         0.0
    #> 4     D  0.0     NA     NA      NA      NA         0         0          0         0         1         1              0.5        -0.5
    #> 5   XSm  1.0     NA     NA      NA      NA         1         1          0         0         1         1              0.5         0.5
    #> 6    Sm  2.5     NA     NA      NA      NA         1         1          0         0         1         1              1.0         1.5
    #> 7    Md  4.5     NA     NA      NA      NA         1         1          0         0         1         1              1.0         3.5
    #> 8    Lg  8.0     NA     NA      NA      NA         1         1          0         0         1         1              2.5         5.5
    #> 9   XLg 17.5     NA     NA      NA      NA         1         1          0         0         1         1              7.0        10.5
    #>   sizebin_max sizebin_center sizebin_width binhalfwidthb_raw sizebinb_min sizebinb_max sizebinb_center sizebinb_width
    #> 1         0.0            0.0             0                NA           NA           NA              NA             NA
    #> 2         0.0            0.0             0                NA           NA           NA              NA             NA
    #> 3         0.0            0.0             0                NA           NA           NA              NA             NA
    #> 4         0.5            0.0             1                NA           NA           NA              NA             NA
    #> 5         1.5            1.0             1                NA           NA           NA              NA             NA
    #> 6         3.5            2.5             2                NA           NA           NA              NA             NA
    #> 7         5.5            4.5             2                NA           NA           NA              NA             NA
    #> 8        10.5            8.0             5                NA           NA           NA              NA             NA
    #> 9        24.5           17.5            14                NA           NA           NA              NA             NA
    #>   binhalfwidthc_raw sizebinc_min sizebinc_max sizebinc_center sizebinc_width group       comments
    #> 1                NA           NA           NA              NA             NA     0 No description
    #> 2                NA           NA           NA              NA             NA     0 No description
    #> 3                NA           NA           NA              NA             NA     0 No description
    #> 4                NA           NA           NA              NA             NA     0 No description
    #> 5                NA           NA           NA              NA             NA     0 No description
    #> 6                NA           NA           NA              NA             NA     0 No description
    #> 7                NA           NA           NA              NA             NA     0 No description
    #> 8                NA           NA           NA              NA             NA     0 No description
    #> 9                NA           NA           NA              NA             NA     0 No description

Next we will create our verticalized datasets, in which our horizontal datasets are restructured to have all data for each individual split into blocks of three consecutive monitoring occasions, as below. Note that, because we are using the `stageassign` argument, we can have `R` assign stages to each individual in each occasion. Note also our use of the `summary_hfv()` function, which provides specialized, useful output for this style of data frame. First, the C.candidum dataset.

    cycaraw_v1 <- verticalize3(data = cypdata, noyears = 6, firstyear = 2004,
      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_raw, stagesize = "sizeadded", NAas0 = TRUE,
      NRasRep = TRUE)
    summary_hfv(cycaraw_v1)
    #>
    #> This hfv dataset contains 320 rows, 57 variables, 1 population,
    #> 3 patches, 74 individuals, and 5 time steps.

Here we find that our new dataset includes 320 rows, 57 variables, taking blocks of 3 consecutive years at a shot and showing the fate of each individual across that time. Let’s do the same for C. parviflorum.

    cyparaw_v1 <- verticalize3(data = cypa_data, noyears = 18, firstyear = 1994,
      individcol = "plant_id", blocksize = 2, sizeacol = "Inf.94",
      sizebcol = "Veg.94", repstracol = "Inf.94", fecacol = "Inf.94",
      stageassign = cypframe_raw, stagesize = "sizeadded", NAas0 = TRUE,
      NRasRep = TRUE)
    summary_hfv(cyparaw_v1)
    #>
    #> This hfv dataset contains 6871 rows, 51 variables, 1 population,
    #> 1 patch, 1025 individuals, and 17 time steps.

This is a much bigger dataset, as we can see, in terms of rows, variables, individuals, and years. Now let’s create our MPMs. See our online book, lefko3: a gentle introduction, for further material on how to construct MPMs properly.

    cypsupp2r <- supplemental(stage3 = c("SD", "P1", "SL", "D", "XSm", "Sm", "SD",
      "P1"),
      stage2 = c("SD", "SD", "P1", "SL", "SL", "SL", "rep", "rep"),
      eststage3 = c(NA, NA, NA, "D", "XSm", "Sm", NA, NA),
      eststage2 = c(NA, NA, NA, "XSm", "XSm", "XSm", NA, NA),
      givenrate = c(0.10, 0.20, 0.25, NA, NA, NA, NA, NA),
      multiplier = c(NA, NA, NA, NA, NA, NA, 5000, 5000),
      type =c(1, 1, 1, 1, 1, 1, 3, 3),
      stageframe = cypframe_raw, historical = FALSE)
    #> Warning: NA values in argument multiplier will be treated as 1 values.

    cycamatrix2r <- rlefko2(data = cycaraw_v1, stageframe = cypframe_raw,
      year = "all", stages = c("stage3", "stage2", "stage1"),
      size = c("size3added", "size2added"), supplement = cypsupp2r,
      yearcol = "year2", indivcol = "individ")

    cypamatrix2r <- rlefko2(data = cyparaw_v1, stageframe = cypframe_raw,
      year = "all", stages = c("stage3", "stage2", "stage1"),
      size = c("size3added", "size2added"), supplement = cypsupp2r,
      yearcol = "year2", patchcol = "patchid", indivcol = "individ")

    writeLines("Cypripedium candidum")
    #> Cypripedium candidum
    summary(cycamatrix2r)
    #>
    #> This ahistorical lefkoMat object contains 5 matrices.
    #>
    #> Each matrix is square with 9 rows and columns, and a total of 81 elements.
    #> A total of 115 survival transitions were estimated, with 23 per matrix.
    #> A total of 42 fecundity transitions were estimated, with 8.4 per matrix.
    #> This lefkoMat object covers 1 population, 1 patch, and 5 time steps.
    #>
    #> The dataset contains a total of 74 unique individuals and 320 unique transitions.
    #>
    #> Survival probability sum check (each matrix represented by column in order):
    #>          [,1]  [,2]  [,3]  [,4]  [,5]
    #> Min.    0.000 0.250 0.250 0.250 0.250
    #> 1st Qu. 0.300 0.870 0.818 0.800 0.857
    #> Median  0.957 0.941 1.000 0.900 0.900
    #> Mean    0.708 0.803 0.804 0.789 0.801
    #> 3rd Qu. 1.000 1.000 1.000 1.000 1.000
    #> Max.    1.000 1.000 1.000 1.000 1.000
    #>
    #>
    #> Stages without connections leading to the rest of the life cycle found in matrices: 1
    writeLines("\nCypripedium parviflorum")
    #>
    #> Cypripedium parviflorum
    summary(cypamatrix2r)
    #>
    #> This ahistorical lefkoMat object contains 17 matrices.
    #>
    #> Each matrix is square with 9 rows and columns, and a total of 81 elements.
    #> A total of 371 survival transitions were estimated, with 21.824 per matrix.
    #> A total of 120 fecundity transitions were estimated, with 7.059 per matrix.
    #> This lefkoMat object covers 1 population, 1 patch, and 17 time steps.
    #>
    #> The dataset contains a total of 1025 unique individuals and 6871 unique transitions.
    #>
    #> Survival probability sum check (each matrix represented by column in order):
    #>          [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16] [,17]
    #> Min.    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    #> 1st Qu. 0.250 0.300 0.300 0.300 0.250 0.300 0.300 0.300 0.300 0.300 0.250 0.300 0.250 0.250 0.250 0.300 0.250
    #> Median  0.500 0.841 0.782 0.862 0.855 0.773 0.771 0.833 0.803 0.792 0.758 0.760 0.686 0.763 0.750 0.728 0.667
    #> Mean    0.483 0.667 0.623 0.662 0.574 0.610 0.637 0.678 0.644 0.641 0.558 0.642 0.514 0.543 0.542 0.562 0.516
    #> 3rd Qu. 0.733 0.909 0.826 0.880 0.904 0.787 0.875 1.000 0.835 0.883 0.952 0.875 0.829 0.806 0.825 0.755 0.788
    #> Max.    1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
    #>
    #>
    #> Stages without connections leading to the rest of the life cycle found in matrices: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

The summary information that we see is full of interesting information. For example, we see that there are incomplete life cycles in some years in the two populations. This suggests that there are years in which certain stages do not appear. It is not the end of the world, though. Let’s keep going and see what our analyses suggest.

Our next step will be to create a list with our MPMs, where each element of the list is a single set of MPMs. We can accomplish this as follows.

    cyp_mpm_list <- list(cycamatrix2r, cypamatrix2r)

Now that we have our core MPM list, we will need to clarify some of the assumptions of the projection. First, let’s set up starting vectors for each population. We will start our populations with specific numbers of dormant seeds, protocorms, and seedlings, but with more of them in the case of C. parviflorum. We will also use the `start_input()` function in package `lefko3` for this purpose, and combine the results in a list with the same order as our MP{M list. Users should consider the starting vector carefully, as the starting condition of the community impacts its trajectory in projection.

    cyca2_start <- start_input(cycamatrix2r, stage2 = c("SD", "P1", "SL"),
      value = c(500, 100, 200))
    cypa2_start <- start_input(cypamatrix2r, stage2 = c("SD", "P1", "SL"),
      value = c(5000, 1000, 2000))
    cyp_start_list <- list(cyca2_start, cypa2_start)

Next we need to clarify how density dependence will operate on these populations. In herbaceous perennial plants, the germination stage is most commonly negatively density dependent. In orchids, germination may be from a fresh seed, or from a previously dormant seed, and we will assume density dependence in both. Determining the exact nature and proper parameterization of density dependence is too great a topic for this vignette, and so we utilize the following two-parameter Ricker function-based approach, which we know works in this case based on our own previous studies. Note that we will also toss this into a list, but essentially replicate it into the list to make sure that both populations use it.

    cyp2_dv <- density_input(cypamatrix2r, stage3 = c("SD", "P1"),
      stage2 = c("rep", "rep"), style = c(1, 1), alpha = c(0.5, 1.2),
      beta = c(1.0, 2.0), type = c(2, 1))
    cyp_dv_list <- list(cyp2_dv, cyp2_dv)

Our next step will be to build an equivalence data frame. This data frame tells `R` how different stages and different species relate to one another in terms of density dependence. If this structure is left out of the projection, then `R` will assume that each individual of any stage of any species is essentially equal to any other individual of any stage of any species. In our case, we wish to specify that dormant seeds do not affect the density dependence experienced by the populations. For this purpose, we will use the `equiv_input` function.

    cyp_eq <- equiv_input(cycamatrix2r,
      stage2 = c("SD", "P1", "SL", "D", "XSm", "Sm", "Md", "Lg", "XLg"),
      value = c(0, 1, 1, 1, 1, 1, 1, 1, 1))

    eq_list <- list(cyp_eq, cyp_eq)

Finally, we can run our community projection. We do so as follows, projecting the population forward 100 years.

    cyp_comm_proj <- project3(mpms = cyp_mpm_list, starts = cyp_start_list,
      density = cyp_dv_list, equivalence = eq_list, times = 100, integeronly = TRUE)
    summary(cyp_comm_proj)
    #>
    #> The input adaptProj object covers 2 populations, 101 projected steps and 1 projected replicate.
    #> The number of replicates with population size above the threshold size of 1 is as in
    #> the following matrix, with populations given by row and milepost times given by column:
    #> $milepost_sums
    #>      1 25 50 75 101
    #> [1,] 1  1  0  0   0
    #> [2,] 1  1  1  0   0
    #>
    #> $extinction_times
    #> [1] NA

We can see that our projection did not predict extinction in C. candidum, but it did in C. parviflorum. Let’s take a look at a graph of the results, showing the full population sizes over time, as well as the aggregate community density adjusted for stage equivalence (in bold).

    plot(cyp_comm_proj)

Figure C1. Projected population sizes for Cypripedium candidum (solid black) and C. parviflorum (dashed red)

The C. candidum population stayed relatively low but did not go extinct over the 100 years, while the C. parviflorum population exploded in size early on and then dropped. Let’s view the actual population sizes over the course of the projection.

    cyp_comm_proj$N_out
    #> [[1]]
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23]
    #> [1,]  800  347  213  180  157  143  138  128  120   108   100    95    87    81    72    64    60    54    49    42    38    35    30
    #> [2,] 8000 3215 1987 1608 1464 1344 1158 1028  928   815   723   635   542   463   408   377   312   267   200   185   159   144   130
    #>      [,24] [,25] [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38] [,39] [,40] [,41] [,42] [,43] [,44]
    #> [1,]    26    21    17    13     9     5     2     0     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> [2,]   110    95    84    70    60    50    40    33    27    24    18    14     9     6    30    69    19     4     1   556   153
    #>      [,45] [,46] [,47] [,48] [,49] [,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57] [,58] [,59] [,60] [,61] [,62] [,63] [,64] [,65]
    #> [1,]     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> [2,]    72    53    43    36    31    26    20    16    10     9     7    13    96    27     5     2   160    45    14     8     5
    #>      [,66] [,67] [,68] [,69] [,70] [,71] [,72] [,73] [,74] [,75] [,76] [,77] [,78] [,79] [,80] [,81] [,82] [,83] [,84] [,85] [,86]
    #> [1,]     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> [2,]     8    36    11     1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
    #>      [,87] [,88] [,89] [,90] [,91] [,92] [,93] [,94] [,95] [,96] [,97] [,98] [,99] [,100] [,101]
    #> [1,]     0     0     0     0     0     0     0     0     0     0     0     0     0      0      0
    #> [2,]     0     0     0     0     0     0     0     0     0     0     0     0     0      0      0

Although C. parviflorum started off with an order of magnitude more individuals, it went extinct. And, unfortunately, C. candidum appears to be dropping to extinction, though has not done so at the end of the 100 year projection. No doubt further research is required to understand the results.

## Acknowledgements

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 23H02552 from the Japan Society for the Promotion of Science.

## Literature cited

Rasmussen, Hanne. 1995. Terrestrial Orchids: From Seed to Mycotrophic Plant. Cambridge, England, United Kingdom: Cambridge University Press.
Shefferson, Richard P., Tiiu Kull, Michael J. Hutchings, Marc-André Selosse, Hans Jacquemyn, Kimberly M. Kellett, Eric S. Menges, et al. 2018. “Drivers of Vegetative Dormancy Across Herbaceous Perennial Plant Species.” Ecology Letters 21 (5): 724–33. https://doi.org/10.1111/ele.12940.
Shefferson, Richard P., Shun Kurokawa, and Johan Ehrlén. 2021. “Lefko3: Analysing Individual History Through Size-Classified Matrix Population Models.” Methods in Ecology and Evolution 12 (2): 378–82. https://doi.org/10.1111/2041-210X.13526.
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.