Matrix Population Model analysis v12

Created: 2013-08-13 09:18:08 Last updated: 2014-07-09 12:32:27

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The Matrix Population Models Workflow provides an environment to perform several analyses on a stage-matrix with no density dependence:

- Eigen analysis; - Age specific survival; - Generation time (T); - Net reproductive rate (Ro); - Transient Dynamics; - Bootstrap of observed census transitions (Confidence intervals of lambda); - Survival curve; - Keyfitz delta; - Cohen's cumulative distance.

This workflow requires an instance of Rserve on localhost

This workflow has been created by the Biodiversity Virtual e-Laboratory (BioVeL http://www.biovel.eu/) project. BioVeL is funded by the EU’s Seventh Framework Program, grant no. 283359.

This workflow uses R packages ‘popbio’ (Stubben & Milligan 2007; Stubben, Milligan & Nantel 2011) and 'popdemo' (Stott, Hodgson and Townley 2013).

References:

Caswell, H. 1986. Life cycle models for plants. Lectures on Mathematics in the Life Sciences 18: 171-233.

Caswell, H. 2001. Matrix population models: Construction, analysis and interpretation, 2nd Edition. Sinauer Associates, Sunderland, Massachusetts.

de Kroon, H. J., A. Plaiser, J. van Groenendael, and H. Caswell. 1986. Elasticity: The relative contribution of demographic parameters to population growth rate. Ecology 67: 1427-1431.

Horvitz, C., D.W. Schemske, and Hal Caswell. 1997. The relative "importance" of life-history stages to population growth: Prospective and retrospective analyses. In S. Tuljapurkar and H. Caswell. Structured population models in terrestrial and freshwater systems. Chapman and Hall, New York.

Jongejans E. & H. de Kroon. 2012. Matrix models. Chapter in Encyclopaedia of Theoretical Ecology (eds. Hastings A & Gross L) University of California, p415-423

Mesterton-Gibbons, M. 1993. Why demographic elasticities sum to one: A postscript to de Kroon et al. Ecology 74: 2467-2468.

Oostermeijer J.G.B., M.L. Brugman; E.R. de Boer; H.C.M. Den Nijs. 1996. Temporal and Spatial Variation in the Demography of Gentiana pneumonanthe, a Rare Perennial Herb. The Journal of Ecology, Vol. 84(2): 153-166.

Stott, I., S. Townley and D.J. Hodgson 2011. A framework for studying transient dynamics of population projection matrix models. Ecology Letters 14: 959–970

Stubben, C & B. Milligan. 2007. Estimating and Analysing Demographic Models Using the popbio Package in R. Journal of Statistical Software 22 (11): 1-23

Stubben, C., B. Milligan, P. Nantel. 2011. Package ‘popbio’. Construction and analysis of matrix population models. Version 2.3.1

van Groenendael, J., H. de Kroon, S. Kalisz, and S. Tuljapurkar. 1994. Loop analysis: Evaluating life history pathways in population projection matrices. Ecology 75: 2410-2415.

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The Matrix Population Models Workflow provides an environment to perform several analyses on a stage-matrix with no density dependence: - Eigen analysis; - Age specific survival; - Generation time (T); - Net reproductive rate (Ro); - Transient Dynamics; - Bootstrap of observed census transitions (Confidence intervals of lambda); - Survival curve; - Keyfitz delta; - Cohen's cumulative distance. This workflow requires an instance of Rserve on localhost This workflow has been created by the Biodiversity Virtual e-Laboratory (BioVeL http://www.biovel.eu/) project. BioVeL is funded by the EU’s Seventh Framework Program, grant no. 283359. This workflow uses R packages ‘popbio’ (Stubben & Milligan 2007; Stubben, Milligan & Nantel 2011) and 'popdemo' (Stott, Hodgson and Townley 2013). References: Caswell, H. 1986. Life cycle models for plants. Lectures on Mathematics in the Life Sciences 18: 171-233. Caswell, H. 2001. Matrix population models: Construction, analysis and interpretation, 2nd Edition. Sinauer Associates, Sunderland, Massachusetts. de Kroon, H. J., A. Plaiser, J. van Groenendael, and H. Caswell. 1986. Elasticity: The relative contribution of demographic parameters to population growth rate. Ecology 67: 1427-1431. Horvitz, C., D.W. Schemske, and Hal Caswell. 1997. The relative "importance" of life-history stages to population growth: Prospective and retrospective analyses. In S. Tuljapurkar and H. Caswell. Structured population models in terrestrial and freshwater systems. Chapman and Hall, New York. Jongejans E. & H. de Kroon. 2012. Matrix models. Chapter in Encyclopaedia of Theoretical Ecology (eds. Hastings A & Gross L) University of California, p415-423 Mesterton-Gibbons, M. 1993. Why demographic elasticities sum to one: A postscript to de Kroon et al. Ecology 74: 2467-2468. Oostermeijer J.G.B., M.L. Brugman; E.R. de Boer; H.C.M. Den Nijs. 1996. Temporal and Spatial Variation in the Demography of Gentiana pneumonanthe, a Rare Perennial Herb. The Journal of Ecology, Vol. 84(2): 153-166. Stott, I., S. Townley and D.J. Hodgson 2011. A framework for studying transient dynamics of population projection matrix models. Ecology Letters 14: 959–970 Stubben, C & B. Milligan. 2007. Estimating and Analysing Demographic Models Using the popbio Package in R. Journal of Statistical Software 22 (11): 1-23 Stubben, C., B. Milligan, P. Nantel. 2011. Package ‘popbio’. Construction and analysis of matrix population models. Version 2.3.1 van Groenendael, J., H. de Kroon, S. Kalisz, and S. Tuljapurkar. 1994. Loop analysis: Evaluating life history pathways in population projection matrices. Ecology 75: 2410-2415.

Dependencies (0)

Inputs (6)

Name	Description
label	Descriptive title for labelling generated outputs.
shortTermYears	This value will be use to plot a graph that shows a simulation of the number of individuals per stage a few years after the study. This value represents the years of axis X of the output graph: StageVectorPlotShortTerm.
longTermYears	This value contains the maximum number of iterations in the transient dynamic analysis, and hence the total number of years for the long term graphs. The number of years will be used in two output graphs: 1) StageVectorPlotLongTermProportional: the proportion of individuals per stage in the long term. 2) StageVectorPlotLongTermLogarithmic: the number of individuals per stage in the long term.
stageMatrixFile	The stage matrix input file
stages	Stage input port: The names of the stages or categories of the input matrix. It is very important that the stages names are not longer than 8 characters. The name of the stages must be added one by one. The respective name stages must be filled one by one. First press add value, fill a stage name (not longer than 8 characters) and press enter, then press add value and fill once again the next stage name, repeat the action until you have fill all the stages names. In the following example, the matrix has 5 stages or categories: S J V G D S 0.0000 0.0000 0.0000 7.6660 0.0000 J 0.0579 0.0100 0.0000 8.5238 0.0000 V 0.4637 0.8300 0.9009 0.2857 0.8604 G 0.0000 0.0400 0.0090 0.6190 0.1162 D 0.0000 0.0300 0.0180 0.0000 0.0232 The stages of this matrix are called: 1) Seedlings S 2) Juveniles J 3) Vegetative V 4) Reproductive individuals G 5) Dormant plants D
iterations	Number of iterations for calculation of confidence interval

Processors (14)

Name	Type	Description
Abundance_Interaction	interaction	Abundance iteraction: Initial abundance: In this dialogue authomatically appears the fields to fill out the initial abundance per stage observed in the field (see data below). After fill out the abundances, the user confirms the numbers. As a example Gentiana pneumonanthe has 5 stages with its respective abundance: stage abundance 1) S (seedlings) 69 2) J (Juveniles) 100 3) V (vegetative) 111 4) G (reproductive individuals) 21 5) D (dormant plants) 43 Interaction Page http://biovel.googlecode.com/svn/tags/mpm-20130805/set_abundance.html
StageMatrix_ReadFromFile	rshell	Script stage.data <- read.table(stage_matrix_file) stage_matrix <- as.matrix(stage.data) dimnames(stage_matrix) <- list(stages, stages) R Server localhost:6311
CategoriseStages_Interaction	interaction	With this dialogue automatically appears the names of the stages or categories of the census data file. When the dialogue appears, the stages are in disorder, so the user drags and organizes the stages according to the order in the life cycle. Then, the author chooses if the stage belongs to the recruited, reproductive category or it should be excluded. Recruited means that new individuals can be recruited to this (these) stage(s). Reproductive stages are those that reproduce (produce offspring) (in this example the stage G). In the census data file Dt1.txt, x is use to denote when a plant has died in the second year, so the user must selected in the excluded column. Then the user clicks in confirm and you will read stages submitted. In the following example, the life cycle of Gentiana pneumonanthe has 5 stages or categories: 1) Seedlings S 2) Juveniles J 3) Vegetative V 4) Reproductive individuals G 5) Dormant plants D Interaction Page http://biovel.googlecode.com/svn/tags/mpm-20130805/select_stages.html
StageMatrixAnalysis	workflow
Confidence_interval	workflow
plot_histogram	rshell	Script options(digits=16);library(popbio) max_num<-max(y) png(output) hist(y, br=30, col="palegreen", xlab="Lambda", plot=TRUE, xlim=c(0, max_num), main=plottitle) abline(v=ci, lty=3) dev.off() R Server localhost:6311
OutputEigenanalysis	workflow
OutputElasticityMatrix	workflow
OutputSensitivityMatrix	workflow
OutputFundamentalMatrix	workflow
OutputConfidenceInterval	workflow
OutputStageMatrix	workflow
FecundityCols_FromReproductiveStages	rshell	Script indices = match(some_values, all_values) R Server localhost:6311
FecundityRows_FromRecruitedStages	rshell	Script indices = match(some_values, all_values) R Server localhost:6311

Beanshells (0)

Outputs (25)

Name	Description
stable_stage_distribution	A bar plot which shows the stable stage distribution (w) of the analysed matrix. In other words, the proportion of individuals per stage.
projection_matrix	The stage matrix, displayed with coloured squares to highlight the magnitude of the values.
eigenanalysis_elasticity_matrix	The elasticity matrix, displayed with coloured squares to highlight the magnitude of the values.
eigenanalysis_sensitivity_matrix_1	A sensitivity matrix showing the sensitivities of the actual transitions (i.e. the sensitivity values of non-zero elements), displayed with coloured squares to highlight the magnitude of the values.
eigenanalysis_sensitivity_matrix_2	A sensitivity matrix showing the sensitivities of all possible transitions (i.e. the sensitivity values of zero and non-zero elements), displayed with coloured squares to highlight the magnitude of the values.
stage_matrix	A stage matrix contains transitions probabilities from each stage to the next. In the example, the selected species, Gentiana pneumonanthe, has a matrix of 5 x 5 stages (Oostermeijer et al., 1996).
eigenanalysis	Eigen analysis The Eigen analysis results are a set of demographic statistics: Lambda or dominant eigenvalue: This value describes the population growth rate of a stage matrix. The population will be stable, grow or decrease at a rate given by lambda: e.g.: λ = 1 (population is stable), λ > 1 (population is growing) and finally λ < 1 (population is decreasing). The stable stage distribution (w): It is the proportion of the number of individuals per stage. It is given analytically by the right eigenvector (another property of the transition matrix) that corresponds to the dominant eigenvalue Elasticity and Sensitivity: Sensitivity and elasticity analyses are prospective analyses. The sensitivity matrix: The sensitivity gives the effect on λ of changes in any entry of the matrix, including those that may, at a given context, be regarded as fixed at zero or some other value. The derivative tells what would happened to λ if aij was to change, not whether, or in what direction, or how much, aij actually change. The hypothetical results of such impossible perturbations may or may not be of interest, but they are not zero. It is up to you to decide whether they are useful (Caswell 2001). When comparing the λ-sensitivity values for all matrix elements one can find out in what element a certain increase has the biggest impact on λ. However, a 0.01 increase in a survival matrix element is hard to compare to a 0.01 increase in a reproduction matrix element, because the latter is not bound between 0 and 1 and can sometimes take high values. Increasing matrix element a14 (number of S (seedlings) the next year produced by a G (Reproductive individuals)) with 0.01 from 7.666 to 7.676 does not have a noticeable effect on λ. For comparison between matrix elements it can therefore be more insightful to look at the impact of proportional changes in elements: by what percentage does λ change if a matrix element is changed by a certain percentage? This proportional sensitivity is termed elasticity (Description based on Oostermeijer data, based on Jongejans & de Kroon 2012). The Elasticity matrix: The elasticities of λ with respect to the stage are often interpreted as the “contributions” of each of the stages to λ. This interpretation relies on the demonstration, by de Kroon et al (1986) that the elasticitis of the λ with respect to the stage, always sum to 1. For further information see: de Kroon, et al., 1986. and Caswell 2001. Reproductive value (v): scaled so v[1]=1. To what extent will a plant or animal of a determinate category or stage , contribute to the ancestry of future generation. The damping ratio: it can be considered as a measure of the intrinsic resilience of the population, describing how quickly transient dynamics decay following disturbance or perturbation regardless of population structure, the larger the p, the quicker the population converges.
short_term_stage_vector_plot	A plot that charts the number of individuals per stage vs. years in the short-term (e.g. 5 or 10 years). The number of years is related to the short-term years input value.
long_term_logarithmic_stage_vector_plot	The number of individuals per stage in the long term
population_projection	Population projection: Lambda or dominant eigenvalue: The population will be stable, grow or decrease at a rate given by lambda: e.g.: λ = 1 (population is stable), λ > 1 (population is growing) and finally λ < 1 (population is decreasing). e.g. The projected population growth rate (λ) is 1.237, meaning that the population is projected to increase with 23% per year if these model parameters remain unchanged. The stable stage distribution (stable.stage): It is the proportion of the number of individuals per stage and it is given by (w). Stage vector (stage.vectors): it is the projection of the number of individuals per category per year in the long term, the long term was stipulated in the input port (long term years) (e.g. 50 years). Population sizes (pop.sizes): it is the total population size per year in the long-term (e.g. 50 years). Population change (pop.changes): are the lambda values per year in the long-term (e.g. 50 years).
long_term_proportional_stage_vector_plot	Stage vector plot long term proportional: Is the proportion of individuals per stage in the long term (e.g.: 50 years)
damping_ratio	Damping ratio: The ratio between the dominant eigenvalue and the second highest eigenvalue of a transition matrix is called the damping ratio, and it can be considered as a measure of the intrinsic resilience of the population, describing how quickly transient dynamics decay following disturbance or perturbation regardless of population structure, (the larger the damping ratio, the quicker the population converges). High damping ratios tell you that the dominant stable stage distribution is reached fairly soon.
fundamental_matrix	Age specific survival The fundamental matrix workflow gives the basic information on age-specific survival, this includes the mean, variance and coefficient of variation (cv) of the time spent in each stage class and the mean and variance of the time to death. Fundamental matrix (N): is the mean of the time spent in each stage class. e.g.: For our Gentiana example means a J plants will spends, on average, about 1 year as a Juvenile, 11 as vegetative plant, less than a year a reproductive and dormant plant. Variance (var): is the variance in the amount of time spent in each stage class. Coefficient of variation (cv): is the coefficient of variation of the time spent in each class (sd/mean- the ratio of the standard deviation to the mean). Meaneta: is the mean of time to death, of life expectancy of each stage. e.g. The mean age at death is the life expectancy; the life expectancy of a new individual seedling is 8 years. Vareta: is the variance of time to death.
generation_time	Generation time: The time T required for the population to increase by a factor of Ro (net reproductive rate). e.g. for Generation time T and Net reproductive rate Ro: If T = 8.13 and Ro = 5.46 then the average individual in the study year replaced itself with about five new plants and took approximately 8 years to do so.
net_reproductive_rate	The net reproductive rate is the mean number of offspring by which a new-born individual will be replaced by the end of its life, and thus the rate by which the population increases from one generation to the next.
lambda
histogram_ci_95pc_of_lambda	Histogram generated from the analysis of the confidence interval of lambda.
confidence_interval_95pc_of_lambda	Calculate bootstrap distributions of population growth rates (lambda) by randomly sampling with replacement from a stage-fate data frame of observed transitions. Resampling transitions with equal probability.
survival_curve_plot	A plot of survival curves is produced, one point for each stage.
sensitivity_matrix	See Eigen analysis
sensitivity_plot	A sensitivity matrix plot showing the sensitivities values of the actual transitions (i.e. the sensitivity values of non-zero elements) per stage.
elasticity_matrix	See Eigen analysis
elasticity_plot	An Elasticity matrix plot showing the elasticities values per stage.
cohen_cumulative_distance	Cohen’s cumulative distance measures the difference between observed and expected vectors along the matrix path that the population would take to reach the expected population vector. It is a function of both the observed stage distribution (n0) and the structure of the matrix (A) (Williams et al 2011). Cohen’s cumulative distance will not work for reducible matrices and returns a warning for imprimitive matrices (although will not function for imprimitive matrices with nonzero imaginary components in the dominant eigenpair) (Caswell 2001).
keyfitz_delta	A distance measure between probability vectors of n (Observed Stage Distribution, abundances per stage) and w (Stable Stage Distribution). Its maximum value is 1 and its minimum is 0 when the vectors are identical.

Datalinks (55)

Source	Sink
CategoriseStages_Interaction:sortedStages	Abundance_Interaction:stages
stageMatrixFile	StageMatrix_ReadFromFile:stage_matrix_file
CategoriseStages_Interaction:sortedStages	StageMatrix_ReadFromFile:stages
stages	CategoriseStages_Interaction:unsortedStages
Abundance_Interaction:abundances	StageMatrixAnalysis:abundances
label	StageMatrixAnalysis:label
longTermYears	StageMatrixAnalysis:longTermYears
shortTermYears	StageMatrixAnalysis:shortTermYears
StageMatrix_ReadFromFile:stage_matrix	StageMatrixAnalysis:stage_matrix
CategoriseStages_Interaction:sortedStages	StageMatrixAnalysis:stages
FecundityRows_FromRecruitedStages:indices	StageMatrixAnalysis:frows
FecundityCols_FromReproductiveStages:indices	StageMatrixAnalysis:fcols
StageMatrix_ReadFromFile:stage_matrix	Confidence_interval:stage_matrix
Abundance_Interaction:abundances	Confidence_interval:abundances
iterations	Confidence_interval:iterations
FecundityCols_FromReproductiveStages:indices	Confidence_interval:fcols
FecundityRows_FromRecruitedStages:indices	Confidence_interval:frows
Confidence_interval:confidence_interval	plot_histogram:ci
Confidence_interval:y	plot_histogram:y
label	plot_histogram:plottitle
StageMatrixAnalysis:eigenanalysis	OutputEigenanalysis:input
StageMatrixAnalysis:elasticity_matrix	OutputElasticityMatrix:input
StageMatrixAnalysis:sensitivity_matrix	OutputSensitivityMatrix:input
StageMatrixAnalysis:fundamentalMatrix	OutputFundamentalMatrix:input
Confidence_interval:confidence_interval	OutputConfidenceInterval:input
StageMatrix_ReadFromFile:stage_matrix	OutputStageMatrix:input
CategoriseStages_Interaction:sortedStages	FecundityCols_FromReproductiveStages:all_values
CategoriseStages_Interaction:reproductiveStages	FecundityCols_FromReproductiveStages:some_values
CategoriseStages_Interaction:sortedStages	FecundityRows_FromRecruitedStages:all_values
CategoriseStages_Interaction:recruitedStages	FecundityRows_FromRecruitedStages:some_values
StageMatrixAnalysis:barPlot	stable_stage_distribution
StageMatrixAnalysis:projectionMatrix	projection_matrix
StageMatrixAnalysis:eigenanalysis_elasticity_matrix	eigenanalysis_elasticity_matrix
StageMatrixAnalysis:eigenanalysis_sensitivity_matrix_1	eigenanalysis_sensitivity_matrix_1
StageMatrixAnalysis:eigenanalysis_sensitivity_matrix_2	eigenanalysis_sensitivity_matrix_2
OutputStageMatrix:output	stage_matrix
OutputEigenanalysis:output	eigenanalysis
StageMatrixAnalysis:stageVectorPlotShortTerm	short_term_stage_vector_plot
StageMatrixAnalysis:stageVectorPlotLongTermLogarithmic	long_term_logarithmic_stage_vector_plot
StageMatrixAnalysis:populationProjection	population_projection
StageMatrixAnalysis:stageVectorPlotLongTermProportional	long_term_proportional_stage_vector_plot
StageMatrixAnalysis:dampingRatio	damping_ratio
OutputFundamentalMatrix:output	fundamental_matrix
StageMatrixAnalysis:generationTime	generation_time
StageMatrixAnalysis:netReproductiveRate	net_reproductive_rate
Confidence_interval:lambda	lambda
plot_histogram:output	histogram_ci_95pc_of_lambda
OutputConfidenceInterval:output	confidence_interval_95pc_of_lambda
StageMatrixAnalysis:survivalCurvePlot	survival_curve_plot
OutputSensitivityMatrix:output	sensitivity_matrix
StageMatrixAnalysis:sensitivityPlot	sensitivity_plot
OutputElasticityMatrix:output	elasticity_matrix
StageMatrixAnalysis:ElasticityPlot	elasticity_plot
StageMatrixAnalysis:cohenCumulative	cohen_cumulative_distance
StageMatrixAnalysis:keyfitzDelta	keyfitz_delta

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Maria Paula Balcazar-Vargas

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Matrix Population Models analysis documentation
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Stage Matrix of Gentiana pneumonanthe 1987-88
Cohen’s cumulative distance
Keyfitz delta
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Parametric Bootstrap or Resample a projection matrix Workflow
Transient Dynamics.
Net reproductive rate (Ro)
Generation time (T)
Eigen analysis
Age specific analysis

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Version History

In chronological order:

Matrix Population Model analysis v10

Created by Maria Paula Balcazar-Vargas on Tuesday 13 August 2013 09:18:06 (UTC)
Matrix Population Model analysis v12

Created by Maria Paula Balcazar-Vargas on Wednesday 09 July 2014 12:32:27 (UTC)

Revision comment:

Use options(digits=16) in R scripts, and sink() to keep sink stack clean.
Fix bug, selection of the Fecundity matrix values.