project: Project population dynamics

Description

Project dynamics of a specified population matrix projection model.

Usage

project(
  A,
  vector = "n",
  time = 100,
  standard.A = FALSE,
  standard.vec = FALSE,
  return.vec = TRUE,
  Aseq = "unif",
  Astart = NULL,
  draws = 1000,
  alpha.draws = "unif",
  PREcheck = TRUE
)

Arguments

a matrix, or list of matrices. If A is a matrix, then project performs a 'deterministic' projection, where the matrix does not change with each timestep. If A is a list of matrices, then project performs a 'stochastic' projection where the matrix varies with each timestep. The sequence of matrices is determined using Aseq. Matrices must be square, non-negative and numeric. If A is a list, all matrices must have the same dimension. 'Projection' objects inherit names from A: if A is a matrix, stage names (in mat and vec slots) are inherited from its column names..

vector

(optional) a numeric vector or matrix describing the age/stage distribution(s) used to calculate the projection. Single population vectors can be given either as a numeric vector or one-column matrix. Multiple vectors are specified as a matrix, where each column describes a single population vector. Therefore the number of rows of the matrix should be equal to the matrix dimension, whilst the number of columns gives the number of vectors to project. vector may also take either "n" (default) to calculate the set of stage-biased projections (see details), or "diri" to project random population vectors drawn from a dirichlet distribution (see details).

time

the number of projection intervals.

standard.A

(optional) if TRUE, scales each matrix in A by dividing all elements by the dominant eigenvalue. This standardises asymptotic dynamics: the dominant eigenvalue of the scaled matrix is 1. Useful for assessing transient dynamics.

standard.vec

(optional) if TRUE, standardises each vector to sum to 1, by dividing each vector by its sum. Useful for assessing projection relative to initial population size.

return.vec

(optional) if TRUE, returns the time series of demographic (st)age vectors as well as overall population size.

Aseq

(optional, for stochastic projections only) the sequence of matrices in a stochastic projection. Aseq may be either:

"unif" (default), which results in every matrix in A having an equal, random chance of being chosen at each timestep.
a square, nonnegative left-stochastic matrix describing a first-order Markov chain used to choose the matrices. The transitions are defined COLUMNWISE: each column j describes the probability of choosing stage (row) i at time t+1, given that stage (column) j was chosen at time t. Aseq should have the same dimension as the number of matrices in A.
a numeric vector giving a specific sequence which corresponds to the matrices in A.
a character vector giving a specific sequence which corresponds to the names of the matrices in A.

Astart

(optional) in a stochastic projection, the matrix with which to initialise the projection (either numeric, corresponding to the matrices in A, or character, corresponding to the names of matrices in A). When Astart = NULL (the default), a random initial matrix is chosen.

draws

if vector="diri", the number of population vectors drawn from dirichlet.

alpha.draws

if vector="diri", the alpha values passed to rdirichlet: used to bias draws towards or away from a certain population structure.

PREcheck

many functions in popdemo first check Primitivity, Reducibility and/or Ergodicity of matrices, with associated warnings and/or errors if a matrix breaks any assumptions. Set PREcheck=FALSE if you want to bypass these checks.

Value

A Projection-class item. 'Projection' objects inherit from a standard array, and can be treated as such. Therefore, if if vector is specified, the 'Projection' object will behave as:

if a single vector is given, a numeric vector of population sizes of length time+1
if multiple vectors are given, a numeric matrix of population projections where each column represents a single population projection and is of length time+1
if vector="n", a numeric matrix of population projections where each column represents a single stage-biased projection and is of length time+1.
if vector="diri", a numeric matrix of population projections where each column represents projection of a single vector draw and each column is of length time+1

See documentation on Projection-class objects to understand how to access other slots (e.g. (st)age vectors through the population projection) and for S4 methods (e.g. plotting projections). Some examples for understanding the structure of 3D arrays returned when return.vec=TRUE: when projecting a 3 by 3 matrix for >10 time intervals (see examples), element [11,3,2] represents the density of stage 3 at time 10 for either vector 2 (multiple vectors), stage-bias 2 (vector="n") or draw 2 (vector="diri"); note that because element 1 represents t=0, then t=10 is found at element 11. The vector [,3,2] represents the time series of densities of stage 3 in the projection of vector 2 / stage-bias 2 / draw 2. The matrix [,,2] represents the time series of all stages in the projection of vector 2 / stage-bias 2 / draw 2.

Note that the projections inherit the labelling from A and vector, if it exists. Both stage and vector names are taken from the COLUMN names of A and vector respectively. These may be useful for selecting from the projection object, and for labelling graphs when plotting Projection objects.

Details

If vector is specified, project will calculate population dynamics through time by projecting this vector / these vectors through A. If multiple vectors are specified, a separate population projection is calculated for each.

If vector="n", project will automatically project the set of 'stage-biased' vectors of A. Effectively, each vector is a population consisting of all individuals in one stage. These projections are achieved using a set of standard basis vectors equal in number to the dimension of A. The vectors have every element equal to 0, except for a single element equal to 1, i.e. for a matrix of dimension 3, the set of stage-biased vectors are: c(1,0,0), c(0,1,0) and c(0,0,1). Stage-biased projections are useful for seeing how extreme transient dynamics can be.

If vector="diri", project draws random population vectors from the dirichlet distribution. draws gives the number of population vectors to draw. alpha.draws gives the parameters for the dirichlet and can be used to bias the draws towards or away from certain population structures. The default is alpha.draws="unif", which passes rep(1,dim) (where dim is the dimension of the matrix), resulting in an equal probability of any random population vector. Relative values in the vector give the population structure to focus the distribution on, and the absolute value of the vector entries (and their sum) gives the strength of the distribution: values greater than 1 make it more likely to draw from nearby that population structure, whilst values less than 1 make it less likely to draw from nearby that population structure.

Projections returned are of length time+1, as the first element represents the population at t=0.

Projections have their own plotting method (see Projection-plots) to enable easy graphing.

In addition to the examples below, see the "Deterministic population dynamics" and "Stochastic population dynamics" vignettes for worked examples that use the project function.

Examples

Run this code

# NOT RUN {
  ### USING PROJECTION OBJECTS

  # Create a 3x3 PPM
  ( A <- matrix(c(0,1,2,0.5,0.1,0,0,0.6,0.6), byrow=TRUE, ncol=3) )

  # Project stage-biased dynamics of A over 70 intervals
  ( pr <- project(A, vector="n", time=70) )
  plot(pr)

  # Access other slots
  vec(pr)  #time sequence of population vectors
  bounds(pr)  #bounds on population dynamics
  mat(pr)  #matrix used to create projection
  Aseq(pr)  #sequence of matrices (more useful for stochastic projections)
  projtype(pr)  #type of projection
  vectype(pr)  #type of vector(s) initiating projection

  # Extra information on the projection
  nproj(pr)  #number of projections
  nmat(pr)  #number of matrices (more usefulk for stochastic projections)
  ntime(pr)  #number of time intervals
  
  # Select the projection of stage 2 bias
  pr[,2]

  # Project stage-biased dynamics of standardised A over 30 intervals
  ( pr2 <- project(A, vector="n", time=30, standard.A=TRUE) )
  plot(pr2)

  #Select the projection of stage 2 bias
  pr2[,2]

  # Select the density of stage 3 in bias 2 at time 10
  vec(pr2)[11,3,2]

  # Select the time series of densities of stage 2 in bias 1
  vec(pr2)[,2,1]

  #Select the matrix of population vectors for bias 2
  vec(pr2)[,,2]

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

  # Project A over 50 intervals using a specified population structure
  ( pr3 <- project(A, vector=initial, time=50) )
  plot(pr3)

  # Project standardised dynamics of A over 10 intervals using 
  # standardised initial structure and return demographic vectors
  ( pr4 <- project(A, vector=initial, time=10, standard.vec=TRUE, 
                   standard.A=TRUE, return.vec=TRUE) )
  plot(pr4)

  # Select the time series for stage 1
  vec(pr4)[,1]

  ### DETERMINISTIC PROJECTIONS

  # Load the desert Tortoise matrix
  data(Tort)

  # Create an initial stage structure
  Tortvec1 <- c(8, 7, 6, 5, 4, 3, 2, 1)
  
  # Create a projection over 30 time intervals
  ( Tortp1 <- project(Tort, vector = Tortvec1, time = 10) )

  # plot p1
  plot(Tortp1)
  plot(Tortp1, bounds = TRUE) #with bounds
 
  # new display parameters
  plot(Tortp1, bounds = TRUE, col = "red", bty = "n", log = "y", 
       ylab = "Number of individuals (log scale)",
       bounds.args = list(lty = 2, lwd = 2) )

  # multiple vectors
  Tortvec2 <- cbind(Tortvec1, c(1, 2, 3, 4, 5, 6, 7, 8))
  plot(project(Tort, vector = Tortvec2), log = "y")
  plot(project(Tort, vector = Tortvec2), log = "y", labs = FALSE) #no labels

  # dirichlet distribution 
  # darker shading indicates more likely population size
  Tortshade <- project(Tort, time = 30, vector = "diri", standard.A = TRUE,
               draws = 500, alpha.draws = "unif")
  plot(Tortshade, plottype = "shady", bounds = TRUE)
  
  ### STOCHASTIC PROJECTIONS
  # load polar bear data
  data(Pbear)
  
  # project over 50 years with uniform matrix selection
  Pbearvec <- c(0.106, 0.068, 0.106, 0.461, 0.151, 0.108)
  p2 <- project(Pbear, Pbearvec, time = 50, Aseq = "unif")

  # stochastic projection information
  Aseq(p2)
  projtype(p2)
  nmat(p2)
  
  # plot
  plot(p2, log = "y")
  
# }