Perturbation analysis of a matrix population model

Perturbs elements within a matrix population model and measures the response (sensitivity or elasticity) of the per-capita population growth rate at equilibrium (\(\lambda\)), or, with a user-supplied function, any other demographic statistic.

perturb_matrix(
  matA,
  pert = 1e-06,
  type = "sensitivity",
  demog_stat = "lambda",
  ...
)

Arguments

matA

A matrix population model (i.e., a square projection matrix).

pert

Magnitude of the perturbation. Defaults to 1e-6.

type

Whether to return sensitivity or elasticity values.

demog_stat

The demographic statistic to be used, as in "the sensitivity/elasticity of demog_stat to matrix element perturbations." Defaults to the per-capita population growth rate at equilibrium (\(\lambda\)). Also accepts a user-supplied function that performs a calculation on a projection matrix and returns a single numeric value.

...

Additional arguments passed to the function demog_stat

Value

A sensitivity or elasticity matrix.

References

Caswell, H. 2001. Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer Associates; 2nd edition. ISBN: 978-0878930968

See also

Other perturbation analysis: perturb_stochastic(), perturb_trans(), perturb_vr(), pop_vectors()

Author

Rob Salguero-Gomez <rob.salguero@zoo.ox.ac.uk>

Examples

matA <- rbind(
  c(0.1, 0, 1.5, 4.6),
  c(0.5, 0.2, 0.1, 0),
  c(0, 0.3, 0.3, 0.1),
  c(0, 0, 0.5, 0.6)
)

perturb_matrix(matA)
#>           [,1]      [,2]       [,3]       [,4]
#> [1,] 0.1956454 0.1034597 0.03892421 0.03337485
#> [2,] 0.4238210 0.2241222 0.08432050 0.07229906
#> [3,] 1.3889133 0.7344748 0.27632874 0.23693290
#> [4,] 1.7815032 0.9420805 0.35443500 0.30390419

# use a larger perturbation than the default
perturb_matrix(matA, pert = 0.01)
#>           [,1]      [,2]       [,3]       [,4]
#> [1,] 0.1965109 0.1035815 0.03891965 0.03335242
#> [2,] 0.4210788 0.2251614 0.08436470 0.07224203
#> [3,] 1.3788340 0.7267363 0.27758497 0.23689003
#> [4,] 1.7863694 0.9339710 0.35225173 0.30591837

# calculate the sensitivity/elasticity of the damping ratio to perturbations
damping <- function(matA) { # define function for damping ratio
  eig <- eigen(matA)$values
  dm <- rle(Mod(eig))$values
  return(dm[1] / dm[2])
}

perturb_matrix(matA, demog_stat = "damping")
#>            [,1]         [,2]       [,3]        [,4]
#> [1,] -0.1404735  0.543012141  0.1507766 -0.07610673
#> [2,] -0.2478520 -0.001715736  0.3708966  0.07466982
#> [3,]  3.0820361 -0.289454277  0.1468064  0.71772081
#> [4,]  6.2816365  1.540591979 -0.5566369  0.79082548