Constructs projection matrix A given a vec-permutation matrix P, a demographic block diagonal matrix BB, and a movement (dispersal) block diagonal matrix MM as described in Hunter and Caswell (2005) and Lebreton (1996).
Arguments
- P
The vec-permutation matrix (see `vec.perm` function).
- BB
The block diagonal demographics matrix (see `blk.diag` function).
- MM
The block diagonal movement matrix (see `blk.diag` function).
- group_by
The structural form of the population vector `n` as either "patches" or "stages".
- lh_order
The order of demographic and movement (dispersal) processes.
Note
The demographic block matrix BB consists of s × s number of demographic matrices (one for each patch). The demographic block matrix MM consists of p × p number of stage movement matrices (one for each stage). Please refer to Hunter and Caswell (2005) for details.
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 `blk.diag`.
# 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
# Movement 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