Scale a matrix population model so that lambda equals 1

Scales a matrix population model (MPM) so that the dominant eigenvalue (\(\lambda\)) of the full projection matrix equals 1. The function accepts either a full projection matrix (matA) or a set of component matrices consisting of the survival component (matU) plus at least one reproductive component (matF and/or matC).

scale_mpm_to_lambda1(matU = NULL, matF = NULL, matC = NULL, matA = NULL)

Arguments

matU

The survival component of a matrix population model (i.e., a square projection matrix reflecting survival-related transitions; e.g., progression, stasis, and retrogression). Required when using component inputs.

matF

The sexual reproduction component of a matrix population model. Optional if matC is supplied. If omitted while matC is supplied, it is assumed to be a zero matrix.

matC

The clonal reproduction component of a matrix population model. Optional if matF is supplied. If omitted while matF is supplied, it is assumed to be a zero matrix.

matA

The full projection matrix of a matrix population model. Optional if component matrices are supplied. If supplied together with components, it must equal matU + matF + matC.

Value

A list with six elements:

matA

The scaled full projection matrix.

matU

The scaled survival matrix. Returned as NULL if only matA is supplied.

matF

The scaled sexual reproduction matrix. Returned as NULL if only matA is supplied.

matC

The scaled clonal reproduction matrix. Returned as NULL if only matA is supplied.

lambda_original

The dominant eigenvalue of the original projection matrix.

lambda_scaled

The dominant eigenvalue of the scaled projection matrix.

Details

When component matrices are used, matU must be supplied together with at least one of matF or matC. Missing reproductive components are treated as zero matrices of the appropriate dimension.

See also

Other transformation: is_leslie_matrix(), leslie_collapse(), mpm_collapse(), mpm_rearrange(), mpm_split(), mpm_standardize(), name_stages(), repro_stages(), standard_stages()

Author

Owen R. Jones <jones@biology.sdu.dk>

Examples

data(mpm1)

# scale from the full projection matrix
matA <- mpm1$matU + mpm1$matF
scale_mpm_to_lambda1(matA = matA)
#> $matA
#>               seed      small      medium       large    dormant
#> seed    0.08920313 0.00000000 15.96736076 40.67662853 0.00000000
#> small   0.04460157 0.10704376  0.08920313  0.00000000 0.00000000
#> medium  0.00000000 0.31221096  0.10704376  0.20516721 0.10704376
#> large   0.00000000 0.02676094  0.24976877  0.46385629 0.08920313
#> dormant 0.00000000 0.00000000  0.14272501  0.09812345 0.15164533
#> 
#> $matU
#> NULL
#> 
#> $matF
#> NULL
#> 
#> $matC
#> NULL
#> 
#> $lambda_original
#> [1] 1.121037
#> 
#> $lambda_scaled
#> [1] 1
#> 

# scale from component matrices, assuming no clonal reproduction
scale_mpm_to_lambda1(matU = mpm1$matU, matF = mpm1$matF)
#> $matA
#>               seed      small      medium       large    dormant
#> seed    0.08920313 0.00000000 15.96736076 40.67662853 0.00000000
#> small   0.04460157 0.10704376  0.08920313  0.00000000 0.00000000
#> medium  0.00000000 0.31221096  0.10704376  0.20516721 0.10704376
#> large   0.00000000 0.02676094  0.24976877  0.46385629 0.08920313
#> dormant 0.00000000 0.00000000  0.14272501  0.09812345 0.15164533
#> 
#> $matU
#>               seed      small     medium      large    dormant
#> seed    0.08920313 0.00000000 0.00000000 0.00000000 0.00000000
#> small   0.04460157 0.10704376 0.08920313 0.00000000 0.00000000
#> medium  0.00000000 0.31221096 0.10704376 0.20516721 0.10704376
#> large   0.00000000 0.02676094 0.24976877 0.46385629 0.08920313
#> dormant 0.00000000 0.00000000 0.14272501 0.09812345 0.15164533
#> 
#> $matF
#>         seed small   medium    large dormant
#> seed       0     0 15.96736 40.67663       0
#> small      0     0  0.00000  0.00000       0
#> medium     0     0  0.00000  0.00000       0
#> large      0     0  0.00000  0.00000       0
#> dormant    0     0  0.00000  0.00000       0
#> 
#> $matC
#>         seed small medium large dormant
#> seed       0     0      0     0       0
#> small      0     0      0     0       0
#> medium     0     0      0     0       0
#> large      0     0      0     0       0
#> dormant    0     0      0     0       0
#> 
#> $lambda_original
#> [1] 1.121037
#> 
#> $lambda_scaled
#> [1] 1
#>