NWI-BB087-2024
Population and evolutionary biology –
Matrix modelling exam
Course coordinator: Heidi Huber
Lecturer: Eelke Jongejans
The course was followed in 2024-2025 at the Radboud University, as part of the Bachelor Biology. All
information in this document is coming from the lecture slides as available on Brightspace for
students that followed this course, self study, and the practical assignments. In the R script, the
answers of Eelke Jongejans are included as # ANSWERS EJ.
Table of contents
Matrix modelling ..................................................................................................................................... 2
L1 population dynamics....................................................................................................................... 2
Why population models .................................................................................................................. 2
Life cycles ......................................................................................................................................... 3
Matrix models.................................................................................................................................. 3
Vital rates ......................................................................................................................................... 5
Sensitivity analysis ........................................................................................................................... 7
Variance decomposition .................................................................................................................. 9
Matrix population models 2 ................................................................................................................ 9
Elasticity........................................................................................................................................... 9
How to evaluate the importance of matrix transition elements for different ecological and
evolutionary questions .................................................................................................................. 11
How to compute the elasticities of vital rates ............................................................................... 12
How elasticities are used in conservation management ............................................................... 12
Caveats and limitations in the use of elasticities in management ................................................ 12
The differences between prospective and retrospective analysis, and their implications for
management ................................................................................................................................. 13
L3: Comparative demography, stochasticity, environmental drivers................................................. 13
Matrix models allow for comparative demography across species .............................................. 13
The assumptions of time-invariant matrix models and how to relax them .................................. 15
Differences between the short-term (transient) dynamics and the asymptotic dynamics, and the
values and difficulties with prediction and projection .................................................................. 15
Stochasticity ...................................................................................................................................... 16
Ways to create stochastic transition matrices and interpret their output in terms of stochastic
population growth rates and extinction probabilities ................................................................... 16
,NWI-BB087-2024
Environmental drivers ....................................................................................................................... 17
Environmental factors can be incorporated into population models............................................ 17
R code .................................................................................................................................................... 18
Exercise 1 ........................................................................................................................................... 18
Exercise 2 ........................................................................................................................................... 21
Exercise 3 ........................................................................................................................................... 23
Exercise 4 ........................................................................................................................................... 27
Exercise 5 ........................................................................................................................................... 30
Exercise 6 ........................................................................................................................................... 37
Exercise 7 ........................................................................................................................................... 41
Exercise 8 ........................................................................................................................................... 46
Exercise 9 ........................................................................................................................................... 57
References ............................................................................................................................................. 61
Matrix modelling
L1 population dynamics
Why population models
Why?
- Study life histories in organized way, simplification of complex reality
- Project population dynamics into near and distant future
o Projection: asymptotic properties (lambda, stable stage distribution, sensitivities)
o Prediction: population viability analysis, stochastic dynamics, including demographic
stochasticity and environmental variation
- Quantify how much age- and size-dependent survival, development and reproduction
contribute to population growth
- Compare life history strategies between populations and across species
- Integrate environmental effects across life cycles
- Include plastic and genetic processes
- Testing hypotheses (‘what if we kill half of the population?’, how effective will management
options be in controlling an invasive population? How likely is a population of an endangered
species to go extinct over time?)
How to do this?
,NWI-BB087-2024
- What can go wrong? Seeds in seedbanks, dormant stages (orchids seem dead), birds flying to
different places outside research area, not all time steps have the same length
- Failure to estimate reproduction properly, uncertainty/parameter estimation, failure to check
for internal consistency, missing information on stage/age structure, failure to note scope of
interference, failure to separate vital rates
- (Gascolgne
et al 2023, under consideration)
Life cycles
Diagram
- Each arrow represents the contribution of an average individual in one class to the nr of
individuals in a particular class one time step later (e.g. 1 year). Contribution can be through
survival or reproduction (sexual/asexual)
- Five age classes (fledglings, 1, 2, 3 year olds, and adults (4+ years old))
- Slide arrows = survival rates (adults)
- Interrupted arrows = reproduction rates (include adult/stage 3 survival) = average nr of
fledglings per individual per year + survival to next year + nest making + breeding success etc
- Purpose: bring together all demographic processes (at a single location, migration is often
ignored)
Matrix models
- Life cycle diagram and matrix model contain same information
- A column contains the “contributions of an average individual in that stage to the number of
individuals one time step later” (1). So: column 1 = fledglings, and it contributes 0 to
, NWI-BB087-2024
fledglings itself, 0.6 to the 1-year olds, and 0 to all other stages, so only one arrow in diagram.
Column 5 = adult, contributing to adults (0.89) and fledglings by 0.534
- “A row contains all contributions toward the nr of individuals in a particular class after one
time step” (1)
- “Once all transition arrows/nr are quantified, and if initial nr of individuals in each class is
known, the population size after one time step can be calculated by multiplying each arrow
with its corresponding initial group size” (1) → matrix
column * initial class size → sum in row to get nt+1
(1) 𝑛𝑡+1 = 𝐴 ∗ 𝑛𝑡 , in R: A %*% n(t)
- nt = nr of individuals in a stage = population vector
- A = transition matrix
- nt+1 = population vector after one time step
Types of matrices and diagrams
- Can be a combination of multiple types!
- Age-structured Leslie matrix
o Age dependent structure in classes: individuals move up one class every time step
(pooling may be present, e.g. adults = 4+ years old) (1)
o Fecundity on first row, survival rates are diagonal
o always this kind of matrix, but if multiple types, then a mix is
possible, and other matrices can also have this kind of structure
- Size-structured Lefkovitch matrix
o Classes with small and larger stages are present: individuals can skip classes, remain
in the same class (‘stasis’) or regress to a previous class (1)
o All combinations and transitions are possible
Population and evolutionary biology –
Matrix modelling exam
Course coordinator: Heidi Huber
Lecturer: Eelke Jongejans
The course was followed in 2024-2025 at the Radboud University, as part of the Bachelor Biology. All
information in this document is coming from the lecture slides as available on Brightspace for
students that followed this course, self study, and the practical assignments. In the R script, the
answers of Eelke Jongejans are included as # ANSWERS EJ.
Table of contents
Matrix modelling ..................................................................................................................................... 2
L1 population dynamics....................................................................................................................... 2
Why population models .................................................................................................................. 2
Life cycles ......................................................................................................................................... 3
Matrix models.................................................................................................................................. 3
Vital rates ......................................................................................................................................... 5
Sensitivity analysis ........................................................................................................................... 7
Variance decomposition .................................................................................................................. 9
Matrix population models 2 ................................................................................................................ 9
Elasticity........................................................................................................................................... 9
How to evaluate the importance of matrix transition elements for different ecological and
evolutionary questions .................................................................................................................. 11
How to compute the elasticities of vital rates ............................................................................... 12
How elasticities are used in conservation management ............................................................... 12
Caveats and limitations in the use of elasticities in management ................................................ 12
The differences between prospective and retrospective analysis, and their implications for
management ................................................................................................................................. 13
L3: Comparative demography, stochasticity, environmental drivers................................................. 13
Matrix models allow for comparative demography across species .............................................. 13
The assumptions of time-invariant matrix models and how to relax them .................................. 15
Differences between the short-term (transient) dynamics and the asymptotic dynamics, and the
values and difficulties with prediction and projection .................................................................. 15
Stochasticity ...................................................................................................................................... 16
Ways to create stochastic transition matrices and interpret their output in terms of stochastic
population growth rates and extinction probabilities ................................................................... 16
,NWI-BB087-2024
Environmental drivers ....................................................................................................................... 17
Environmental factors can be incorporated into population models............................................ 17
R code .................................................................................................................................................... 18
Exercise 1 ........................................................................................................................................... 18
Exercise 2 ........................................................................................................................................... 21
Exercise 3 ........................................................................................................................................... 23
Exercise 4 ........................................................................................................................................... 27
Exercise 5 ........................................................................................................................................... 30
Exercise 6 ........................................................................................................................................... 37
Exercise 7 ........................................................................................................................................... 41
Exercise 8 ........................................................................................................................................... 46
Exercise 9 ........................................................................................................................................... 57
References ............................................................................................................................................. 61
Matrix modelling
L1 population dynamics
Why population models
Why?
- Study life histories in organized way, simplification of complex reality
- Project population dynamics into near and distant future
o Projection: asymptotic properties (lambda, stable stage distribution, sensitivities)
o Prediction: population viability analysis, stochastic dynamics, including demographic
stochasticity and environmental variation
- Quantify how much age- and size-dependent survival, development and reproduction
contribute to population growth
- Compare life history strategies between populations and across species
- Integrate environmental effects across life cycles
- Include plastic and genetic processes
- Testing hypotheses (‘what if we kill half of the population?’, how effective will management
options be in controlling an invasive population? How likely is a population of an endangered
species to go extinct over time?)
How to do this?
,NWI-BB087-2024
- What can go wrong? Seeds in seedbanks, dormant stages (orchids seem dead), birds flying to
different places outside research area, not all time steps have the same length
- Failure to estimate reproduction properly, uncertainty/parameter estimation, failure to check
for internal consistency, missing information on stage/age structure, failure to note scope of
interference, failure to separate vital rates
- (Gascolgne
et al 2023, under consideration)
Life cycles
Diagram
- Each arrow represents the contribution of an average individual in one class to the nr of
individuals in a particular class one time step later (e.g. 1 year). Contribution can be through
survival or reproduction (sexual/asexual)
- Five age classes (fledglings, 1, 2, 3 year olds, and adults (4+ years old))
- Slide arrows = survival rates (adults)
- Interrupted arrows = reproduction rates (include adult/stage 3 survival) = average nr of
fledglings per individual per year + survival to next year + nest making + breeding success etc
- Purpose: bring together all demographic processes (at a single location, migration is often
ignored)
Matrix models
- Life cycle diagram and matrix model contain same information
- A column contains the “contributions of an average individual in that stage to the number of
individuals one time step later” (1). So: column 1 = fledglings, and it contributes 0 to
, NWI-BB087-2024
fledglings itself, 0.6 to the 1-year olds, and 0 to all other stages, so only one arrow in diagram.
Column 5 = adult, contributing to adults (0.89) and fledglings by 0.534
- “A row contains all contributions toward the nr of individuals in a particular class after one
time step” (1)
- “Once all transition arrows/nr are quantified, and if initial nr of individuals in each class is
known, the population size after one time step can be calculated by multiplying each arrow
with its corresponding initial group size” (1) → matrix
column * initial class size → sum in row to get nt+1
(1) 𝑛𝑡+1 = 𝐴 ∗ 𝑛𝑡 , in R: A %*% n(t)
- nt = nr of individuals in a stage = population vector
- A = transition matrix
- nt+1 = population vector after one time step
Types of matrices and diagrams
- Can be a combination of multiple types!
- Age-structured Leslie matrix
o Age dependent structure in classes: individuals move up one class every time step
(pooling may be present, e.g. adults = 4+ years old) (1)
o Fecundity on first row, survival rates are diagonal
o always this kind of matrix, but if multiple types, then a mix is
possible, and other matrices can also have this kind of structure
- Size-structured Lefkovitch matrix
o Classes with small and larger stages are present: individuals can skip classes, remain
in the same class (‘stasis’) or regress to a previous class (1)
o All combinations and transitions are possible
