Generate random Lefkovitch matrix population models (MPMs) based on life history archetypes

Generates a random matrix population model (MPM) with element values based on defined life history archetypes. Survival and transition/growth probabilities from any particular stage are restricted to be less than or equal to 1 by drawing from a Dirichlet distribution. The user can specify archetypes (from Takada et al. 2018) to restrict the MPMs in other ways:

Archetype 1: all elements are positive, although they may be very small. Therefore, transition from/to any stage is possible. This model describes a life history where individuals can progress and retrogress rapidly.
Archetype 2: has the same form as archetype 1 (transition from/to any stage is possible), but the survival probability (column sums of the survival matrix) increases monotonously as the individuals advance to later stages. This model, as the one in the first archetype, also allows for rapid progression and retrogression, but is more realistic in that stage-specific survival probability increases with stage advancement.
Archetype 3: positive non-zero elements for survival are only allowed on the diagonal and lower sub-diagonal of the matrix This model represents the life cycle of a species where retrogression is not allowed, and progression can only happen to the immediately larger/more developed stage (slow progression, e.g., trees).
Archetype 4: This archetype has the same general form as archetype 3, but with the further assumption that stage-specific survival increases as individuals increase in size/developmental stage. In this respect it is similar to archetype 2.

Usage

rand_lefko_mpm(n_stages, fecundity, archetype = 1, split = TRUE)

Arguments

n_stages

An integer defining the number of stages for the MPM.

fecundity

A measure of reproductive output. The average number of offspring produced per projection interval from each stage. Values can be provided in 4 ways:

An numeric vector of length 1 to provide a single fecundity measure to the top right corner of the matrix model only.
A numeric vector of integers of length equal to n_stages to provide fecundity estimates for the whole top row of the matrix model. Use 0 for cases with no fecundity.
A matrix of numeric values of the same dimension as n_stages to provide fecundity estimates for the entire matrix model. Use 0 for cases with no fecundity.
A list of two matrices of numeric values, both with the same dimension as n_stages, to provide lower and upper limits of mean fecundity for the entire matrix model. Use 0 for both lower and upper limits in cases with no fecundity.

In the latter case, a fecundity value will be drawn from a uniform distribution for the defined range. If there is no fecundity in a particular age class, use a value of 0 for both the lower and upper limit.

archetype

Indication of which life history archetype should be used, based on Takada et al. 2018. An integer between 1 and 4.

split

TRUE/FALSE, indicating whether the matrix produced should be split into a survival matrix and a reproductive output matrix. If true, then the output becomes a list with a matrix in each element. Otherwise, the output is a single matrix. Default is TRUE.

Value

Returns a random matrix population model with characteristics determined by the archetype selected and fecundity vector. If split = TRUE, the matrix is split into separate reproductive output and growth/survival matrices, returned as a list.

Details

In all 4 of these Archetypes, reproductive output is placed as a single element on the top right of the matrix, if it is a single value. If it is a vector of length n_stages then the fecundity vector spans the entire top row of the matrix.

The function is constrained to only output ergodic matrices.

References

Caswell, H. (2001). Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer.

Lefkovitch, L. P. (1965). The study of population growth in organisms grouped by stages. Biometrics, 21(1), 1.

Takada, T., Kawai, Y., & Salguero-Gómez, R. (2018). A cautionary note on elasticity analyses in a ternary plot using randomly generated population matrices. Population Ecology, 60(1), 37–47.

Author

Owen Jones jones@biology.sdu.dk

Examples

set.seed(42) # set seed for repeatability

rand_lefko_mpm(n_stages = 2, fecundity = 20, archetype = 1, split = FALSE)
#>            [,1]       [,2]
#> [1,] 0.73066610 20.0402850
#> [2,] 0.06797628  0.7190785
rand_lefko_mpm(n_stages = 2, fecundity = 20, archetype = 2, split = TRUE)
#> $mat_A
#>           [,1]        [,2]
#> [1,] 0.5129277 20.78121699
#> [2,] 0.2895418  0.04959239
#> 
#> $mat_U
#>           [,1]       [,2]
#> [1,] 0.5129277 0.78121699
#> [2,] 0.2895418 0.04959239
#> 
#> $mat_F
#>      [,1] [,2]
#> [1,]    0   20
#> [2,]    0    0
#> 
rand_lefko_mpm(n_stages = 3, fecundity = 20, archetype = 3, split = FALSE)
#>            [,1]      [,2]       [,3]
#> [1,] 0.26019259 0.0000000 20.0000000
#> [2,] 0.09187956 0.8172080  0.0000000
#> [3,] 0.00000000 0.1640643  0.1762325
rand_lefko_mpm(n_stages = 4, fecundity = 20, archetype = 4, split = TRUE)
#> $mat_A
#>            [,1]       [,2]      [,3]       [,4]
#> [1,] 0.24002446 0.00000000 0.0000000 20.0000000
#> [2,] 0.03254107 0.05480185 0.0000000  0.0000000
#> [3,] 0.00000000 0.62726117 0.7851646  0.0000000
#> [4,] 0.00000000 0.00000000 0.0786406  0.9661587
#> 
#> $mat_U
#>            [,1]       [,2]      [,3]      [,4]
#> [1,] 0.24002446 0.00000000 0.0000000 0.0000000
#> [2,] 0.03254107 0.05480185 0.0000000 0.0000000
#> [3,] 0.00000000 0.62726117 0.7851646 0.0000000
#> [4,] 0.00000000 0.00000000 0.0786406 0.9661587
#> 
#> $mat_F
#>      [,1] [,2] [,3] [,4]
#> [1,]    0    0    0   20
#> [2,]    0    0    0    0
#> [3,]    0    0    0    0
#> [4,]    0    0    0    0
#> 
rand_lefko_mpm(
  n_stages = 5, fecundity = c(0, 0, 4, 8, 10), archetype = 4,
  split = TRUE
)
#> $mat_A
#>             [,1]      [,2]      [,3]      [,4]       [,5]
#> [1,] 0.008672344 0.0000000 4.0000000 8.0000000 10.0000000
#> [2,] 0.027810843 0.2560970 0.0000000 0.0000000  0.0000000
#> [3,] 0.000000000 0.5284182 0.1992335 0.0000000  0.0000000
#> [4,] 0.000000000 0.0000000 0.6157474 0.3219698  0.0000000
#> [5,] 0.000000000 0.0000000 0.0000000 0.6398980  0.9915655
#> 
#> $mat_U
#>             [,1]      [,2]      [,3]      [,4]      [,5]
#> [1,] 0.008672344 0.0000000 0.0000000 0.0000000 0.0000000
#> [2,] 0.027810843 0.2560970 0.0000000 0.0000000 0.0000000
#> [3,] 0.000000000 0.5284182 0.1992335 0.0000000 0.0000000
#> [4,] 0.000000000 0.0000000 0.6157474 0.3219698 0.0000000
#> [5,] 0.000000000 0.0000000 0.0000000 0.6398980 0.9915655
#> 
#> $mat_F
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    0    4    8   10
#> [2,]    0    0    0    0    0
#> [3,]    0    0    0    0    0
#> [4,]    0    0    0    0    0
#> [5,]    0    0    0    0    0
#> 
# Using a range of values for fecundity
rand_lefko_mpm(n_stages = 2, fecundity = 20, archetype = 1, split = TRUE)
#> $mat_A
#>           [,1]        [,2]
#> [1,] 0.2558635 20.54431954
#> [2,] 0.0543606  0.07233605
#> 
#> $mat_U
#>           [,1]       [,2]
#> [1,] 0.2558635 0.54431954
#> [2,] 0.0543606 0.07233605
#> 
#> $mat_F
#>      [,1] [,2]
#> [1,]    0   20
#> [2,]    0    0
#>