Basics in population ecology
It is not the strongest of the species that survives, nor the
most intelligent, but the one most responsive to change.
Charles Darwin.
Our program
1. Simple growth processes
2. Outbreaks
3. Age structured populations
4. Harvesting and viability analysis
5. Competition , predation and parasitism
6. Populations in space: Metapopulation and
spatial dynamics
7. Populations in space: Metapopulation and
spatial dynamics
Literature
What is a population?
A population is a group of potentially interbreeding individuals of the same species living
in the same area at the same time and sharing a common gene pool.
Population ecology is a sub-field of ecology that deals with the dynamics
of species populations and how these populations interact with the environment.
It is the study of how the population sizes of species living together in groups change over
time and space.
Carabus coriaceus in a forest
Carabidae in a forest
Basic characteristics of populations:
Absolute density (individuals per unit area)
Relative density (Proportion of individuals with respect to some standard)
Abundance (size; total number of individuals)
Age structure (triggered by natality and age dependent mortality)
Dispersal (spatial dynamics)
Main axiom of population ecology:
Organisms in a population are ecologically equivalent.
Ecological equivalency means:
Organisms undergo the same life-cycle
Organisms in a particular stage of the life-cycle are involved in the same set of
ecological processes
The rates of these processes (or the probabilities of ecological events) are
basically the same if organisms are put into the same environment (however
some individual variation may be allowed)
Sometimes species of different species interbred.
These do not form a population per definition
In Sulawesi seven species of
macaques (Macaca spp.)
interbreed where their home
ranges overlap.
Interbreedin is the cause of
endangerment of Macaca nigra.
Adapted from Riley (2010) The endemic
seven: four decades of research onth
Sulawesi Macaques. Evol. Anthr. 19: 22.
Spatially separated individuals do not
form true populations
A species occurring on four islands that
are isolated is divided into four
independently evolving populations.
Due to limited gene flow populations on
two islands might be considerd as foring a
single genet
ically structured
populations
Raven
(Corvus corax)
Ravens in different continents
do not form a single population.
There is no (or only limited)
gene flow.
Temporary separated individuals do not form populations
Omphale lugens
Mikiola fagi
N
Macrotera arcuata
Number of bees hatching from eggs
N
Spring
Summer
Summer
Spring
Summer
Spring and summer generations have only
limited overlap and thus form partly
separated populations.
Overlaying is connected with host change.
M. fagi is univoltine.
0
Eggs
2
1
Hatching year
3
Overlaying is a strategy to reduce risk due to
unfavourable conditions.
If overlaying is genetically fixed the genotypes of
the three hatching cohorts never meet.
Life cycles
North atlantic salmon is
semelparous
Important questions:
• What is the population rate of growth or
decline?
• To what factor is the population growth
• rate most responsive?
• Will the population eventually go extinct?
• What happened to the population in the
• past?
Man is iteroparous
Iteroparous populations are of age
structured with each age cohorte
having a different reproductive output.
Differences in life history
Egg
Larva 1
Larva n
Semelparous species
reproduce only once
and can be described
by simple growth
models
Iteroparous
species reproduce
at least two times
and might form
age structured
populations
Adult
Fertility =
number of
eggs per
female
Egg
Juvenile
Adults 1
Fertility =
number of eggs
per female
Adult n
Some species have age
cohorts after the
reproductive phase
Senex
Why
grandparents?
Some basic definitions
Females only
Total fertility rate (TFR) is the total number of children a female would bear during her
lifetime.
Gross Reproduction Rate (GRR) is the potential average number of female offspring per
female.
Net Reproduction Rate (NRR) is the observed average number of female offspring per
female. NRR is always lower than GRR. When NRR is less than one, each generation is
smaller than the previous one. When NRR is greater than 1 each generation is larger than
the one before.
In semelparous species age specific fertility (ASF) is the average number of offspring per
female of a certain age class.
Males and females
Population growth is the change in population size over time. Growth can be negative.
Population growth rate is the multiplication factor that describes the magnitude of
population growth. Growth rate is always positive.
Fertility versus population growth rate
Bacterial growth
Animal growth
𝑁𝑡+1 = 2𝑁𝑡
𝑁𝑡+1 = 𝑅𝑁𝑡
𝑁𝑡+1 = 𝑅𝑁𝑡
Males
Females
𝐹𝑡+1 = 𝑅𝐹𝑡
R describes the population growth rate
R describes the net reproduction rate
In demographic analysis only females are
counted.
The number of females in reproductive age
is called the effective population size.
R is the average number of daughters of
each female in the population
Net refers to the number of daughters,
which reach reproductive age.
Birth and death dynamics
Discrete population growth
A population growth process considers four
basic variables (BIDE model)
B: number of births D: number of deaths
I: number of immigrations
E: number of emigration
𝑁𝑡+1 = 𝑁𝑡 + 𝐵𝑡 − 𝐷𝑡 = 𝑁𝑡 + 𝑏𝑡 𝑁𝑡 − 𝑑𝑡 𝑁𝑡
𝐵𝑡
𝑏𝑡 =
𝑁𝑡
𝐷𝑡
𝑑𝑡 =
𝑁𝑡
Natality
Immigration
N
Emigration
Mortality
I, E = 0
𝑁𝑡+1 = 𝑁𝑡 (1 + 𝑏𝑡 − 𝑑𝑡 ) = 𝑅𝑁𝑡
𝑁𝑡+1
= 𝑅𝑡
𝑁𝑡
𝑁𝑡+1 = 𝑅𝑡 𝑁𝑡 = 𝑏𝑡 − 𝑑𝑡 𝑁𝑡 + 𝑁𝑡
𝑅𝑡 = 1 + 𝑏𝑡 -𝑑𝑡
R: fundamental net population growth rate
𝑟𝑡 = (𝑏𝑡 -𝑑𝑡 )
r: intrinsic rate of population change
The population increases if Rt > 1.
The population decreases if Rt < 1.
The population increases if rt > 0.
The population decreases if rt < 0.
Simple population growth processes
𝑁𝑡+1 = 𝑅𝑁𝑡 = 𝑏𝑡 − 𝑑𝑡 𝑁𝑡 + 𝑁𝑡
Discrete growth model
The growth model has only one free parameter:
R: fundamental net growth rate
• The model is simple.
• The model parameter has a clear and logical ecological interpretation.
• The parameter r can be estimated from field data.
𝑁𝑡 = 𝑓(𝑁𝑡−1 )
Change equation
∆𝑁 = 𝑁𝑡 − 𝑁𝑡−1 = 𝑓(𝑁𝑡−1 ) Difference equation
𝑁𝑡
= 𝑓(𝑁𝑡−1 )
𝑁𝑡−1
Ratio equation
Recurrence functions
Recurrence functions
𝑓 𝑥 = 𝑓(𝑥 − 𝑛)
𝑓 𝑥 =𝑓 𝑥−1 +𝑓 𝑥−2
Leonardo Pisano (Fibonacci; 1170-1250)
developed this model to describe the
growth of rabbit populations.
This is the first model in population ecology.
Fibonacci series
1=1+0
2=1+1
3=2+1
5=3+2
8=5+3
13=8+5
Assume a couple of immortal rabbits that
five birth to a second couple every month.
13
8
2
1
3
5
Start
1
1. month
1
2. month
2
3. month
3
4. month
5
𝑁𝑡 = 𝑅𝑁𝑡−1 = 𝑅2 𝑁𝑡−2 … = 𝑅 𝑡 𝑁𝑜
𝑁𝑡 = 𝑅 𝑡 𝑁𝑜 = 𝑁𝑜 𝑒 𝑙𝑛𝑅×𝑡
The discrete form of N
the exponential
growth model
Exponental
growth is a very
fast increase in
population size.
R: fundamental net population growth rate
𝑅0 = 𝑅𝑡
Basic reproductive rate
𝑟 = 𝑙𝑛𝑅 = 𝑏 − 𝑑 =
N0
𝑙𝑛𝑅0 Intrinsic rate of increase
per unit of time
𝑡
Scots pine (Pinus sylvestris) population in
Great Britain after introduction (7500 BC)
t
Whooping crane (Grus americana) population
in North America after protection in 1940
www.whoopingcrane.com
The Human population growth
Human growth was hyperexponential
until about 1970.
Net growth rate was not constant but
increase until about 1970
Since 1970 net growth rate declined
Continuous population growth
𝑁𝑡 = 𝑅𝑡 𝑁𝑜 = 𝑁𝑜 𝑒 𝑟𝑡
𝑑𝑁
= 𝑟𝑁0 𝑒 𝑟𝑡 = 𝑟𝑁
𝑑𝑡
𝑁𝑡+1 = 𝑅𝑁𝑡 = 𝑏𝑡 − 𝑑𝑡 𝑁𝑡 + 𝑁𝑡
𝑁𝑡+1 − 𝑁𝑡 = ∆𝑁𝑡 = 𝑏𝑡 − 𝑑𝑡 𝑁𝑡
Exponential growth model
If r > 0: population increases
If r < 0: population decreases
Intrinsic rate of increase
𝑟 = 𝑏𝑡 − 𝑑𝑡 = 𝑙𝑛𝑅
In the lack of resource limitation a population will exponentially grow.
In this case population grows is density independent.
N
ln N
ln 𝑁𝑡 = 𝑙𝑛𝑁0 + 𝑟𝑡
ln N0
a
a
tan a = (r-1)
tan a = (r-1)t
N0
t
t0
t
Logistic growth
Discrete logistic growth
N
𝐾 − 𝑁𝑡−1
𝑁𝑡 = 𝑁𝑡−1 + 𝑟𝑁𝑡−1
𝐾
K
The Pearl – Verhulst model of
logistic population growth
K/2
t0
t1/2
t
Continuous logistic growth
𝑑𝑁
𝐾−𝑁
= 𝑟𝑁
𝑑𝑡
𝐾
𝑁 𝑡 =
𝐾
1 + 𝑒 −𝑟(𝑡−𝑡0 )
Solution to this differential equation
=
𝐾
𝐾
1 − 𝑁 − 1 𝑒 −𝑟𝑡
0
𝑑𝑁
𝐾−𝑁
= 𝑟𝑁
𝑑𝑡
𝐾
The logistic growth model has only two free parameters:
r: net reproductive rate
K: the carrying capacity.
• The model is simple.
• The model parameters have a clear and logical ecological interpretations.
• The parameters can be estimated from field data.
• The model does not refer to a specific group of species, but applies to all
populations from Bacteria to vertebrates amd plants.
• The model is based on realistic assumptions about population growth.
• The model is sufficiently precise.
Constraints:
• The model refers to homogeneous environments.
• Reproductive rates are supposed to be constant.
• Carrying capacity is supposed to be constant.
• Generations do not overtlap.
Limitation:
The model is symmetrical
around the point of inflection.
The discrete version of logistic growth
𝑁𝑡 = 𝑁𝑡−1 + 𝑟𝑁𝑡−1
𝐾 − 𝑁𝑡−1
𝐾
The logistic growth function is a
discrete recursive model
r = -0.05
K = 500
r = 0.1
K = 500
𝐾 − 𝑁𝑡−1
𝑁𝑡 = 𝑁𝑡−1 + 𝑟𝑁𝑡−1
𝐾
r=1
K = 500
r = 2.099
K = 500
Density dependent population regulation
Stable cycling
r = 1.95
K = 500
r = 2.70
K = 500
Pseudochaos
r = 2.85
K = 500
r = 2.87
K = 500
r = 3.01
K = 500
High reproductive rates imply:
• high population fluctuations
• pseudochatotic population size
Local
extinction
• no density dependent population
regulation
Pseudochaos does not mean that population size is unpredictable.
Very simple determinstic processes might cause pseudochaos.
r-strategists often have pseudochaotic population fluctuations.
A random walk is a pure
stochastic process that causes
unpredictable population sizes.
𝑁𝑡+1 = 𝑁𝑡 + 𝑟𝑎𝑛(−𝑥, 𝑥)