Calculates the sensitivity or elasticity of lambda to changes in the demographic and movement (dispersal) black matrices as described in Hunter and Caswell (2005) and Lebreton (1996).
Arguments
- BB
The block diagonal demographics matrix (see `blk.diag` function).
- A
The spatial population projection matrix constructed from the vec-permutation matrix P, block diagonal demographic matrix BB, and block diagonal movement matrix MM (see `spmm.project.matrix` for more details).
- P
The vec-permutation matrix (see `vec.perm` function).
- MM
The block diagonal movement matrix (see `blk.diag` function).
Value
A matrix containing sensitivity values for the projection matrix A. According to Morris and Doak (2003) sensitivity values od lambda for a particular matrix element is "directly proportional to the fraction of individuals in the population on which the element will act times the future value of each individual that the element 'creates'" (p. 226).
Note
Ensure that the structural type of population vector `n` and projection matrix `A` are the same. Otherwise, projections may produce incorrect values!
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988). The New S Language. Wadsworth & Brooks/Cole.
Wootton, J.T., and Bell, D.A. (1992). A metapopulation model of the peregrine falcon in California: viability and management strategies. Ecological Applications 2:307--321.
Lebreton, J. D. (1996). Demographic models for subdivided populations: the renewal equation approach. Theoretical Population Biology 49:291--313.
Caswell, H. (2001). Matrix Population Models: Construction, analysis, and interpretation (2nd ed.). Sinauer Associates.
Morris, W. F., and Doak, D. F. (2003). Quantitative Conservation Biology: Theory and practice of population viability analysis. Sinauer Associates.
Hunter, C. M. and Caswell, H. (2005). The use of vec-permutation matrix in spatial matrix population models. Ecological Modelling 188:15--21.
Examples
# Peregrine falcon example from Hunter and Caswell (2005), data from Wootton
# and Bell (1992). Continues example from `spmm.project.matrix`.
# Define the number of patches and stages
n_patches <- 2 # northern = 1x; southern = 2x
n_stages <- 2 # juvenile = x1; adult = x2
group_by <- "patches"
# Construct vec-permutation matrix
P <- vec.perm(n_stages, n_patches, group_by)
# Demographic parameter values
# Northern
f11 <- 0.00 # only adults reproduce
f12 <- 0.26
s11 <- 0.72
s12 <- 0.77
# Southern
f21 <- 0.00
f22 <- 0.19
s21 <- 0.72
s22 <- 0.77
# Demography matrices for patches
B1x <-
matrix(c(f11, f12, s11, s12),
nrow = 2,
byrow = TRUE)
B2x <-
matrix(c(f21, f22, s21, s22),
nrow = 2,
byrow = TRUE)
# Demography block matrix construction
BB <- blk.diag(list(B1x, B2x))
# Movement parameter values
dx1 <- 0.27 # only juveniles disperse
dx2 <- 1 - dx1
# Movement matrices for stages
Mx1 <- matrix(c(dx2, dx1, dx1, dx2), nrow = n_patches, byrow = TRUE)
Mx2 <- diag(x = 1, nrow = n_patches, ncol = n_patches) # no movement by adults
# Movement block matrix construction
MM <- blk.diag(list(Mx1, Mx2))
# Arrangement by patches
group_by <- "patches"
# Assumed movement before demography
lh_order <- "move"
# Projection matrix construction
A <- spmm.project.matrix(P, BB, MM, group_by, lh_order) # BB %*% t(P) %*% MM %*% P
# Calculate sensitivity of lambda to elements of block deomgraphic matrix BB
BB_sens <- spmm.demo.sens(BB, A, P, MM)
BB_elas <- spmm.demo.elas(BB, A, P, MM)
# Calculate sensitivity of lambda to elements of block movement matrix MM
MM_sens <- spmm.move.sens(MM, A, P, BB)
MM_elas <- spmm.move.elas(MM, A, P, BB)
# Calculate sensitivity of lambda to specific movement probability
sens_d <- MM_sens[1, 2] + MM_sens[2, 1] - MM_sens[1, 1] - MM_sens[2, 2]