Methods in population biology
The aim of the course is to provide introduction into techniques used to model population
dynamics. The course will consist of introduction to each topic, followed by
practical analysis of example datasets and discussion of the results. Own data
of the participants are welcome. The examples used will be based on plants, but
the techniques apply to all other organisms as well.
4.-8. February, 2008, Department of Botany,
The course will last for 5
full days and will cover the following topics:
· Construction of population transition matrices, techniques used to collect
data needed to construct population transition matrices
· Matrix models of population dynamics: stable stage distribution, population
growth rate, variation in population growth rate
· Introduction to programs POPTOOLS and MATLAB
· Elasticity, importance of single transitions in the life cycle to population
growth rate
· Reliability of matrix models
Sample papers to get an
idea what we are going to talk about (there are of course many others):
· Ehrlén J. (1995) Demography of the perennial herb Lathyrus vernus .2. Herbivory and population-dynamics. - Journal of Ecology 83
(2): 297-308.
· Crowder et al. (1994): Predicting the impact of turtle excluder devices on
loggerhead sea turtle populations. - Ecological Applications 4: 437-445.
· Oostermeijer J.G.B., Brugman M.L., DeBoer E.R. et al. (1996) Temporal and
spatial variation in the demography of Gentiana pneumonanthe, a rare perennial
herb. - Journal of Ecology 84 (2): 153-166.
Program of the course:
1. day
Lecture: Population projection matrices: principles and possibilities of the
method. Construction of matrices from field data.
Practicals: Construction of matrices from field data. Introduction
to POPTOOLS and MATLAB.
2. day
Lecture: Matrix models of population dynamics: stable stage distribution,
population growth rate and its variation, average life span. Sensitivity and
elasticity, contribution of single life cycle transitions to population growth
rate, identification of critical life transitions in the life cycle.
Practicals: Further practice with program MATLAB. Matrix
modeling of population dynamics. Writing simple
scripts in MATLAB.
3. day
Lecture: Life table response experiments in more detail.
Practicals: Modeling of population dynamics, analysis of your own data or other
example datasets (mainly construction of transition matrices).
4. day
Lecture: Statistical testing, reliability of predictions of matrix
models. Practicals: Analysis of other datasets, writing simple scripts in
MATLAB.
5. day
Practicals: Analysis of other datasets, writing simple scripts in MATLAB. Presentation of the results by the participants. The last
day depends on how many participants will have their own
data, so we will see, if the last day will be used.
Syllabus
Population projection
matrices: principals and possibilities of the method.
Life cycle: parameters used to describe individuals (age, size, fertility). Life history stages.
Population size and distribution of individuals among stages
(population vector).
Interpretation of life cycle into a transition matrix.
Columns: initial category, rows: target category. Meaning of single matrix
elements, what elements represent what parts of the life cycle. Length of the transitional interval.
Condition: classification of individual by a given criteria (size/age). Leslie
matrices - individuals classified by age.
Reproductive value of single matrix elements. Contribution of single matrix elements to population growth rate.
Population projection by multiplying a population vector by a
matrix. Dynamical modeling of population size (size in time t determines
size in time t+1).
Properties of transition matrices: stable stage distribution, population growth
rate.
Eigenvalues, right and left eigenvectors. (Small reminder of matrix algebra). Dominant eigenvalue:
determines asymptotic behavior of the matrix (if the whole process last for
long enough).
Dominant eigenvalue for a nonnegative matrix is a real number. Strange
matrices: matrices without closed cycles (postreproductive individuals) -
reducible matrices; matrices with 0 elements on the diagonal (do not allow
transition between stages) - imprimitive matrices.
What can we gain from a matrix model: (i) stable stage/age distribution, (ii)
theoretical population growth rate of a population in a stable stage
distribution, (iii) sensitivity/elasticity analysis, (iv) reproductive values.
Restrictions: (i) Classical transitional matrices are independent of density
and spatial structure. (ii) Population can grow to infinity; growth is strictly
deterministic. It is possible to deal with both of the issues, but it is not
usually done.
Stochastic matrices: random, periodic. Stochasticity of
matrix elements, stochasticity of matrices. Most properties of single
matrix can be generalized for stochastic matrices.
Construction
of matrices from field data.
Construction of population projection matrices.
Construction of single life history stages. Rules for constructing matrices in data from multiple years or when
comparing multiple populations. When the stages must be the same in all
populations and years and when it is not necessary.
Two types of data used for constructing matrices: data on transitions between
stages and survival, vs. data on reproduction.
Data on transitions between stages and survival: how to mark individuals
Data on seed production, germination and seedling survival. Problem: we need to
know how many adults produced our seedlings. (i) sowing
experiment: control plots, problems with density dependence, (ii) natural
regeneration: what was the original number of seeds, problems with locating
seedlings in the field.
How to deal with clonal plants: what is an individual, natality of vegetative
ramets vs. natality of seedlings, ramet/genet dynamics.
Construction of stochastic matrices (variation in time and
space). Different sources of variability for different matrix elements.
Combined matrices (herbivory)
Using matrix models
for modeling population dynamics: projection of populations, stable stage/age
distribution, population growth rate, variation in population growth rate, mean
life span
Population projection based on population vector
Calculation of stable stage/age distribution and its comparison with actual
distribution. Calculation of population growth rate assuming stable stage
distribution given, its interpretation (density independence, no spatial
structure)
Population viability analysis
Calculation of mean life span
Calculation of reproductive value and its use
Sensitivity and elasticity, contribution of single life cycle transitions to population
growth rate, identification of critical life history transitions
Prospective and
retrospective analysis
Sensitivity to a given element: change of the population growth rate given
small change in size of the element.
Sensitivity: effect of single matrix elements on population growth rate. Change
in population growth rate given a change in the matrix element. Definition, meaning, calculation, use.
Elasticity: correction for the fact that different matrix elements have
different meaning (probability of transition vs. natality).
Elasticity: standardized sensitivity. Meaning: what is the contribution of
single matrix elements to population growth rate - change in population growth
rate given a proportional change in the matrix element. Sum of all elasticities
is 1.
Retrospective analysis: Variability of single elements (based on field data -
on multiple matrices), relationship between variability and sensitivity,
decomposition of variation in population growth rate - random and fixed
effects. Life table response "experiments".
Identification of critical life history stages.
How to study effect of frugivory on population growth and
population size.
Classification of plants by elasticity: Growth (i.e. increase in size),
Survival (stasis in the same size category), and Fecundity (i.e. reproduction).
Relationship to life history strategies, types of habitats.
Retrospective and prospective analysis: relationship between sensitivity and
variability.
Loop analysis.
Statistical
testing, reliability of matrix models.
Matrix modeling is about dynamical modeling, thus statistics is just secondary.
Statistical questions, when estimating the parameters;
consequences of these estimates for matrix predictions.
Methods for estimating parameters/test criteria: analytical (assuming sampling
from a known distribution), numerically.
Bootstrap techniques. Construction of large number of replicates by repeated
resampling of the original sample. Information about initial
distribution of the studied variable. Bootstrap is a type of
Estimating confidence intervals using bootstrap, for reliable estimate we
usually need about 2000 replicates for 95% confidence interval.
Using bootstrap for matrix modeling: different elements come from different
data files (germination vs. survival/growth/flowering). Independent
bootstrap analysis of different files. What to do with elements that
cannot be bootstrapped.
Using bootstrap for interpreting results of matrix models: each prediction
(population growth rate, stable stage distribution, reproductive value,
elasticity) can be accompanied by information on its reliability: statistical
envelopes.
Comparison of real size distribution with stable stage distribution
Does the confidence interval overlaps with some theoretically interesting value
(e.g. lambda = 1).
Using bootstrap techniques to compare two samples: analogy of
t-test.
Permutation tests (another type of
Advantages of randomization tests: no assumptions about underlying
distribution, easy and flexible implementation.
Data (All the data are only for teaching
purposes, any other use conditional on the agreement of the author)
Data for manual
construction of transitional matrices for Linum tenuifolium are here.
Data for manual construction of transitional matrices for Cirsium
acaule are here.
Two simple matrices of Succisa pratensis are here.
One matrix for Linum tenuifolium is here.
Set of matrices of Succisa pratensis for stochastic modeling
is here. All the matrices
in one file are here.
Set of matrices of Succisa pratensis for retrospective analysis
is here.
Transitional matrices for larger number of species from
the literature is here.
Data for calculation of mean life span are here (for
script Cochran).
Sample data for bootstrap analysis for Linum tenuifolium are here.
Data on population dynamics of Dracocephalum austriacum that can
be used for aswering a range of questions on dynamics of the species. Data and
suggestions for the analysis are here.
Data for studying effects of seed herbivory on population dynamics
of a thistle Cirsium acaule are here.
Data on population dynamics of Pinus strobus - an invasive
tree in the
Matlab scripts
Simple script for population
projection in time is here.
Simple script for calculation of population growth rate, stable stage/age
distribution, reproductive values, sensitivity and elasticity is here.
Script for calculation of population growth rate, stable stage/age
distribution, reproductive values, sensitivity and elasticity for multiple
matrices using stochastic simulations is here.
Script for studying shape of relationship between matrix element and
population growth rate is here.
Script for studying mean life span based on a
transitional matrix acording to Cochran and Ellner. Description to the file
and to the data above is here.
Script for retrospective analysis (variance decomposition) based
on larger number of matrices (life table response experiment) for a single
species is here.
Script for analysis of matrices including calculation of confidence
intervals (using bootstrap) is here. Explanations are here.
Script for analysis of extinction probability of a population is here.
Scrip for analysis of effect of harvesting on population dynamics
- use the extinction script above.
Script for studying effects of seed herbivory on population
dynamics of a species is here.
Basic programs
PopTools - Excel Add-Ins enabeling simple
analysis of matrix models. It can be downloaded here.
Matlab. Universal program
for matrix modelling. See
the webpage of Matlab for details..
Matlab
clones. Freeware clones of Matlab. Link to instalation of program Octave.
This program is one of the Matlab clones with highest compatibility
with Matlab. It allows using scripts written in Matlab.
Basic literature
Caswell H.: Matrix Population Models: Construction, Analysis, and
Interpretation. The bible of matrix population models.
Gibson, D. J. 2002. Methods in comparative plant population
ecology.
Scheiner S.M. & Gurevitch J. (1993): Design and analysis of ecological
experiments.
Place of the course:
Department of Botany,
Benátská 2
The map is here. The lower arrow marks the
Department building; the upper two arrows mark the 2 closest Metro stations –
Karlovo namesti on the B line and I.P. Pavlova on the C line.
The accommodation is available at Benátská 4 – it is just above (uphill)
the building of the Department.