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The Matrix Population Models construction and analyses Workflow provides an environment for create a stage-matrix with no density dependence and 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).
  • Survival curve
  • Keyfitz’s Δ
  • Cohen’s cumulative distance


 Biovel Portal Tutorial

To run this workflow in the Biovel Portal please refer to Tutorial Manual 



Name of the workflow and its myExperiment identifier

Name: Matrix Population Models construction and analyses v20

The workflow pack can be downloaded from myExperiment pack 483  or only the workflow:   myExperiment workflow 3684

Date, version and licensing

Last updated: 4th July 2014
Version: 20
Licensing: CC-BY-SA

How to cite this workflow

To report work that has made use of this workflow, please add the following credit acknowledgement to your research publication:

The input data and results reported in this publication (tutorial) come from data (Dr. Gerard Oostermeijer unpublished results and publication: Oostermeijer, J.G.B. M.L. Brugman, E.R. de Boer; H.C.M. Den Nijs. 1996.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, 84: 153-166.) using BioVeL workflows and services ( Matrix Population Models construction and analyses (Integrated) workflow was run on <date of the workflow run>. BioVeL is funded by the EU’s Seventh Framework Program, grant no. 283359.


Scientific specifications


Matrix Population Models, stage matrix with not density dependence, Lambda (λ), Sensitivity analysis, Elasticity analysis, Age specific survival, Generation time (T), Net reproductive rate (Ro), Transient Dynamic, Damping ratio, Bootstrap of observed census transitions (Confidence intervals of λ), Survival curve, Keyfitz delta, Cohen’s cumulative distance.


Scientific workflow description

The aim of the Matrix Population Models Workflow is to provide a connected environment for construct a stage matrix and analyse it. The workflow accepts input data in a .txt format. The output is provided as a set of R results and graphic plots. Currently, the workflow is composed of seven distinct parts and/or analyses (Fig 1a and B):


1. Construction of a stage matrix model from demographic monitoring of individuals in an animal or plant population with not density dependence. This section of the workflow create a stage matrix based on two years census data that contains transitions probabilities from each stage to the next, of tagged or mapped individuals of an animal or plant population. The resulting stage matrix is then used to perform the other components of the workflow: the Eigen analyses, age specific survival, generation time (T), net reproductive rate (Ro) and transient dynamics.


2. 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:

a) Lambda or dominant eigenvalue (λ).

b) The stable stage distribution.

c) The sensitivity matrix

d) The Elasticity matrix.

e) Reproductive value.

f) The damping ratio.


3. Age specific survival. This component of the workflow calculates the age-specific survival: that 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.

a) Fundamental matrix (N): is the mean of the time spent in each stage class.

b) Variance (var): is the variance in the amount of time spent in each stage class.

c) 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).

d) Meaneta: is the mean of time to death, of life expectancy of each stage.

e) Vareta: is the variance of time to death.


4. 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.


5. 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.


6. Transient Dynamics: This workflow produces plots of the short-term dynamics and convergence to stable stage distribution using stage vector projections.


7. 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.


8. Survival curve: Plots the survival curve.


9. Keyfitz’s Δ: This workflow produces the Keyfitz’s delta and the Cohen’s cumulative distance, which are measurements of the distance to the stable stage distribution (SSD). Keyfitz’s delta is a measure of the distance between any two probability vectors (Keyfitz 1968). In this case is a measure of the distance between n (Observed Stage Distribution) and w (Stable Stage Distribution). The maximum value is 1 and the minimum value is 0 (when the vectors are identical).


10. Cohen’s cumulative distance: The 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 or imprimitive matrices with nonzero imaginary components in the dominant eigenpair, and returns a warning for other imprimitive matrices (Caswell 2001).



Technical specifications

The Workflow requires a Taverna Engine. The simplest way to install a Taverna Engine is to install Taverna Workbench. The workflow also requires an Rserve installation with popbio and popdemo packages installed. It is possible to setup the workflow to use a remote Rserve. However, instructions for installing a local Rserve are provided below.


Install R software in your computer. See:

How it works

First, open R, once R is opened, type library(Rserve) and press enter; then type Rserve() and press enter again. You will see then something similar to the following message:

After this operation you can open Taverna and run the workflow.



This workflow was created using and based on Packages ‘popbio’ in R. (Stubben & Milligan 2007; Stubben, Milligan & Nantel 2011) and popdemo (Stott, Hodgson and Townley, 2013)

  • 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.
  • Cohen, J.E. 1979. The cumulative distance from an observed to a stable age structure. SIAM Journal on Applied Mathematics, 36:169–175.
  • 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
  • Keyfitz, N. 1968. Introduction to the Mathematics of Populations. Addison-Wesley, Reading, MA, USA.
  • 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
  • Stott, I., D.J. Hodgson and S. Townley. 2013. popdemo: Provides Tools For Demographic Modelling Using Projection Matrices. Version 0.1-3.
  • 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.
  • Williams, J.L., M.M. Ellis, M.C. Bricker, J.F. Brodie and E.W. Parsons. 2011. Distance to stable stage distribution in plant populations and implications for near-term population projections. Journal of Ecology, 99, 1171–1178.
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