Matrix Population Model construction and analysis

Created: 2012-11-02 11:48:53 Last updated: 2014-09-17 10:42:43

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

- Eigen analysis; - Age specific survival; - Generation time (T); - Net reproductive rate (Ro); - Transient Dynamics; - Bootstrap of observed census transitions (Confidence intervals of Î»).

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

This workflow was created using and based on Package â€˜popbioâ€™ in R. (Stubben & Milligan 2007; Stubben, Milligan & Nantel 2011).

Literature ==========

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: 959â€“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 â€˜popbioâ€™. 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 create a stage-matrix with no density dependence and to perform several analyses on it: - Eigen analysis; - Age specific survival; - Generation time (T); - Net reproductive rate (Ro); - Transient Dynamics; - Bootstrap of observed census transitions (Confidence intervals of λ). 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 was created using and based on Package ‘popbio’ in R. (Stubben & Milligan 2007; Stubben, Milligan & Nantel 2011). Literature ========== 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 (5)

Name	Description
censusDataFile	The input data file must have the format of a table containing the demographic data on a series of individuals over several years. Each individual has a table row for each year and is identified by a number (plant column in example). For the year specified in the column 'year', each individual has a certain life stage ('stage'). The stage codes can be chosen by the user, but they cannot be longer than 8 characters. For all individuals, the number of offspring is specified in a chosen way (in the example, 'reprstr' gives the number of flowers/fruits for each plant. In the example, it can be seen that only generative adults (stage=G) have flowers. Individuals without offspring must be indicated by filling in '0' in this column (i.e. the cannot have empty cells). The column 'recruitment' specifies those individuals that are new in the second year census to the population by means of a code. In the example, 'RS' is a new seedling, and 'RJ' a new juvenile plant. As can be seen in the example, no code is needed in this column for individuals that were already present.
label	Descriptive title for labelling generated outputs.
bootstrapIterations	Number of iterations for calculation of bootstrap distributions.
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.

Processors (15)

Name	Type	Description
DisplayEigenanalysis	rshell	Script sink(output) print(input) R Server localhost:6311
CensusData_ReadFromFile	rshell	Script census_data <- read.table(census_data_file, header=TRUE, fill=TRUE) R Server localhost:6311
StageMatrixFromCensu	workflow
Eigen_analysis	workflow	Eigen analysis of a stage matrix model. This component of the workflow performs the Eigen analysis. This analysis results are a set of demographic statistics: Lambda or dominant eigenvalue (λ), the stable stage distribution, sensitivity and elasticity matrix, reproductive value and damping ratio (see outputs: Eigen_analysis).
DisplayStageMatrix	rshell	Script sink(output) print(input) R Server localhost:6311
Bootstrap_of__observ	workflow	Bootstrap of observed census transitions: This workflow calculates bootstrap distributions of population growth rates (λ), stage vectors, and projection matrix elements by randomly sampling with replacement from a stage-fate data frame of observed transitions. The goal of a demographic analysis is very often to estimate lambda, because lambda is estimated from imperfect data, such estimation are uncertain. Therefore, when the results have policy implications it is important to quantify that uncertainty. Confidence interval is one of the traditional tools to doing so (see outputs: Bootstrap analysis).
DisplayConfidenceInterval	rshell	Script sink(output) print(input) R Server localhost:6311
TransientDynamics	workflow	Transient Dynamics: This workflow produces plots of the short-term dynamics and convergence to stable stage distribution using stage vector projections (see outputs: Transient Dynamics).
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/popmod-20121101/set_abundance.html
FecundityRows_FromRecruitedStages	rshell	Script indices = match(some_values, all_values) R Server localhost:6311
FecundityCols_FromReproductiveStages	rshell	Script indices = match(some_values, all_values) R Server localhost:6311
Fundamental_matrix	workflow	Age specific survival or fundamental analysis. This workflow gives the basic information on age-specific survival, this includes the fundamental matrix (N): is the mean of the time spent in each stage class, the Variance (var): is the variance in the amount of time spent in each stage class, the 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 and Vareta: is the variance of time to death. (see outputs: Age specific survival).
DisplayFundamentalMatrix	rshell	Script sink(output) print(input) R Server localhost:6311
Generation_time__T_	workflow	Generation time (T): This component of the workflow calculates the generation time. The time T required for the population to increase by a factor of Ro (net reproductive rate). In other words, T calculates how much time takes to a plant/animal to replace itself by a factor of Ro (see outputs: Generation time).
Net_reproductive_rat	workflow	Net reproductive rate (Ro): This section of the workflow calculates the net reproductive rate (Ro). Ro 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 (see outputs: Net reproductive rate).

Beanshells (0)

Outputs (17)

Name	Description
barPlot	A bar plot which shows the stable stage distribution (w) of the analysed matrix. In other words, the proportion of individuals per stage.
projectionMatrix	The stage matrix, displayed with coloured squares to highlight the magnitude of the values.
elasticityMatrix	The elasticity matrix, displayed with coloured squares to highlight the magnitude of the values.
sensitivityMatrix1	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.
sensitivityMatrix2	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).
confidence_interval	95% Confidence interval of Lambda In statistics, a confidence interval (CI) is a type of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest if the experiment is repeated. How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient.
ci_histogram	Confidence interval histogram Histogram plotting the frequencies of the lambda values and the 95% confidence intervals resulting from the bootstrap analysis.
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.
stageVectorPlotShortTerm	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.
stageVectorPlotLongTermLogarithmic	The number of individuals per stage in the long term
populationProjection	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).
stageVectorPlotLongTermProportional	Stage vector plot long term proportional: Is the proportion of individuals per stage in the long term (e.g.: 50 years)
dampingRatio	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.
fundamentalMatrix	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.
generationTime	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.
netReproductiveRate	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.

Datalinks (50)

Source	Sink
Eigen_analysis:eigenanalysis	DisplayEigenanalysis:input
censusDataFile	CensusData_ReadFromFile:census_data_file
CensusData_ReadFromFile:census_data	StageMatrixFromCensu:census_data
StageMatrixFromCensu:stage_matrix	Eigen_analysis:stage_matrix
label	Eigen_analysis:plotTitle
StageMatrixFromCensu:stage_matrix	DisplayStageMatrix:input
StageMatrixFromCensu:recruitedStages	Bootstrap_of__observ:recruitedStages
StageMatrixFromCensu:initialYear	Bootstrap_of__observ:startYear
StageMatrixFromCensu:matrixStages	Bootstrap_of__observ:stages
label	Bootstrap_of__observ:plotTitle
bootstrapIterations	Bootstrap_of__observ:bootstrapIterations
CensusData_ReadFromFile:census_data	Bootstrap_of__observ:census_data
Bootstrap_of__observ:confidence_interval	DisplayConfidenceInterval:input
StageMatrixFromCensu:stage_matrix	TransientDynamics:stage_matrix
Abundance_Interaction:abundances	TransientDynamics:abundances
label	TransientDynamics:plot_title
shortTermYears	TransientDynamics:short_term_years
longTermYears	TransientDynamics:long_term_years
StageMatrixFromCensu:matrixStages	Abundance_Interaction:stages
StageMatrixFromCensu:recruitedStages	FecundityRows_FromRecruitedStages:some_values
StageMatrixFromCensu:matrixStages	FecundityRows_FromRecruitedStages:all_values
StageMatrixFromCensu:reproductiveStages	FecundityCols_FromReproductiveStages:some_values
StageMatrixFromCensu:matrixStages	FecundityCols_FromReproductiveStages:all_values
StageMatrixFromCensu:stage_matrix	Fundamental_matrix:stage_matrix
FecundityCols_FromReproductiveStages:indices	Fundamental_matrix:columns_fecundity
FecundityRows_FromRecruitedStages:indices	Fundamental_matrix:rows_fecundity
Fundamental_matrix:fundamental_matrix	DisplayFundamentalMatrix:input
StageMatrixFromCensu:stage_matrix	Generation_time__T_:stage_matrix
FecundityCols_FromReproductiveStages:indices	Generation_time__T_:columns_fecundity
FecundityRows_FromRecruitedStages:indices	Generation_time__T_:rows_fecundity
StageMatrixFromCensu:stage_matrix	Net_reproductive_rat:stage_matrix
FecundityCols_FromReproductiveStages:indices	Net_reproductive_rat:columns_fecundity
FecundityRows_FromRecruitedStages:indices	Net_reproductive_rat:rows_fecundity
Eigen_analysis:barPlot	barPlot
Eigen_analysis:projectionMatrix	projectionMatrix
Eigen_analysis:elasticityMatrix	elasticityMatrix
Eigen_analysis:sensitivityMatrix1	sensitivityMatrix1
Eigen_analysis:sensitivityMatrix2	sensitivityMatrix2
DisplayStageMatrix:output	stage_matrix
DisplayConfidenceInterval:output	confidence_interval
Bootstrap_of__observ:histogram	ci_histogram
DisplayEigenanalysis:output	eigenanalysis
TransientDynamics:stage_vector_plot_short_term	stageVectorPlotShortTerm
TransientDynamics:stage_vector_plot_long_term_logarithmic	stageVectorPlotLongTermLogarithmic
TransientDynamics:pop_projection	populationProjection
TransientDynamics:stage_vector_plot_long_term_proportional	stageVectorPlotLongTermProportional
TransientDynamics:damping_ratio	dampingRatio
DisplayFundamentalMatrix:output	fundamentalMatrix
Generation_time__T_:generation_time	generationTime
Net_reproductive_rat:netReproductiveRate	netReproductiveRate