popdemo (version 0.2-3)

truelambda: Calculate simulated asymptotic growth

Description

Calculate the true asymptotic growth of a specified population projection matrix (PPM) model using simulation.

Usage

truelambda(A, vector = "n", accuracy=1e-7, iterations=1e+5)

Arguments

A

a square, non-negative numeric matrix of any dimension

vector

(optional) a specified initial age/stage distribution of class vector or class matrix with which to project the model.

accuracy

the accuracy with which to determine convergence on asymptotic growth, expressed as a proportion (see details).

iterations

the maximum number of iterations of the model before the code breaks. For slowly-converging models and/or high specified convergence accuracy, this may need to be increased.

Value

• If vector is specified, a numeric vector of length 2 giving the range in which asymptotic growth of the model lies. If vector is not specified, a 2-column matrix with each row giving the range in which asymptotic growth lies for its corresponding stage-biased projection. Therefore, the number of rows is equal to the dimension of A.

Details

truelambda works by simulating the given model and manually determining growth when convergence to the given accuracy is reached. Convergence on an asymptotic growth is deemed to have been reached when the growth of the model stays within the window determined by accuracy for 10*s iterations of the model, with s equal to the dimension of A. So for example, projection of an 8 by 8 matrix with convergence accuracy of 1e-2 is deemed to have converged on asymptotic growth when 10*8=80 consecutive iterations of the model have a growth within 1-1e-2=0.99 (i.e. 99%) and 1+1e-2=1.01 (i.e. 101%) of each other. If vector is specified, then the asymptotic growth of the projection of vector through A is returned. If vector="n" then asymptotic growths of the set of stage-biased vectors is calculated. In practise, these projections are achieved using a set of standard basis vectors, each with every element equal to 0, except for a single element that is equal to 1 (i.e. for a 3 by 3 matrix, the set of stage-biased vectors are: c(1,0,0), c(0,1,0) and c(0,0,1).)

References

Stott et al. (2010) Methods Ecol. Evol., 1, 242-252.

Examples

# Create a 3x3 irreducible PPM
    A <- matrix(c(0,0,2,0.5,0.1,0,0,0.6,0.6), byrow=TRUE, ncol=3)
    A

    # Create an initial stage structure
    initial <- c(1,3,2)
    initial

    # Calculate the true asymptotic growth of the stage-biased
    # projections of A
    truelambda(A)

    # Calculate the true asymptotic growth of the projection of
    # A and initial
    truelambda(A, vector=initial)

    # Create a 3x3 reducible, nonergodic PPM
    B<-A
    B[3,2] <- 0
    B

    # Calculate the true asymptotic growth of the 3 stage-biased
    # projections of B
    truelambda(B)