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From individuals to populations
The basic entities of ecological research
Single celled
Bacteria
Single information
coding strand of DNA
Unitary
organisms
have
genetically
prescribed
longivity
Modular organized
Brown fungi
A modular organism has an indeterminate
structure wherein modules of various complexity
(e.g., leaves, twigs) may be assembled without
strict limits on their number or placement.
Bees, ants, and
other insect
societies form
superorganisms
that behave as an
ecological unit.
Clonal
organisms
might have
extreme
longivity
Clonal Populus
tremuloides forests
A clonal colony or genet is a group
of genetically identical individuals, such as plants, fungi,
or bacteria, that have grown in a given location, all
originating vegetative, not sexually, from a single
ancestor. In plants, an individual in such a population is
referred to as a ramet.
Life cycles
All organisms have life cycles from single celled zygotes through ontogenetic stages to adult
forms. All organsims finally die.
k2
k1
k3
k4
k5=1
Often stages of dormancy
Type I
Type II
Type III
Individual age
Type I, high survivorship of
young individuals: Large
mammals, birds
Type II, survivorship
independent of age, seed
banks
Type III, low survivorship of
young individuals, fish, many
insects
Surviving individuals
Surviving individuals
Mortality
Age dependent
survival in annual
plants
Individual age
K-factor analysis
k2
k1
k4
k3
k5
Each life stage t has a certain mortality rate dt.
𝑁 𝑡 + 1 = 𝑁 𝑡 − 𝑑 𝑡 𝑁 𝑡 = 𝑁(𝑡)(1 − 𝑑(𝑡)
𝑘 𝑡 = ln 𝑁𝑡+1 − ln(𝑁𝑡 )
The k-factor is the difference of the logarithms of the number of
surviving indiiduals at the beginning and the end of each stage.
Stage
Year
Number of eggs
Stage 1
Stage 2
Stage 3
Stage 4
Stage 5
Number
surviving
2000
800
120
30
5
1
Number
Mortality
of
rate
deaths
1200
680
90
25
4
0.60
0.85
0.75
0.83
0.80
A simple life table
Kfactor
k1
k2
k3
k4
k5
2000
3500
0.92
1.90
1.39
1.79
1.61
k-factor
2001
3300
0.13
1.49
2.50
1.37
2.12
2002
1500
0.32
2.83
0.61
0.13
3.71
2003
1300
0.05
2.50
2.74
1.45
0.86
2004
1500
0.12
2.10
0.88
1.08
3.42
2005
1000
0.02
2.00
1.83
0.77
2.98
2006
1200
0.02
1.10
1.19
1.47
3.83
2007
1050
0.08
1.97
1.20
0.26
4.09
2008
980
0.11
2.25
1.04
0.06
4.14
2009
1100
0.16
1.78
1.21
0.05
4.40
k-factors calculated for a number of years
Time series
Density series
2000
2001
Density
Clear temporal trends in mortality rates
No density dependence in mortality rates
Mammals
Allometric constraints on life history parameters
Body size is an important determinant on life history.
Insects
Microorganisms
Birds
Various
vertebrates
𝑌 = 𝑌0 𝑊 𝑧
Degree of starvation
Optimal food
intake time
Number of offspring
Quality of food
Optimal
offspring
number
Fitness
Survival probability
Life history trade-offs
Time
Trade-offs: Organisms allocate limited energy or resources to one structure or
function at the expense of another.
All species face trade-off.
Trade-offs shape and constrain life history evolution.
Complex life histories appear to be one
way to maximize reproductive success in
such highly competitive environments.
The value of food is the
product of food quality and
the difference of total
amount N and amount
consumed C).
𝑉 = (𝑁 − 𝐶)𝑄
X
Trade-offs between resource quality and resource
availability at a given point of time mark the beginn of
individualistic behaviour.
Individualistic behaviour is already observable in bacteria.
Food value
Food quality
Amount of food
consumed
The importance of individualistic behaviour
Th perceived food
value migh
remain more
stable than food
quality
Food quality
For different individuals it
pays to use resources of
different quality.
The precise estimation of resource value is one of the motors of brain evolution.
Trade-off decisions during life history
How large to grow?
When to begin
reproducing?
How fast to grow?
Each step is a
decision on
resource allocation.
How many
offspring?
When to change
morphology?
How fast to
develop?
Caring for
What size of offspring?
offspring?
At each time step in life
animals take decisions.
These decisions
determine future
reproductive success and
ae objects of selective
forces
How often to breed?
(semelparous, iteroparous)
How long to live?
How long to live after
reproduction?
Different selective forces might act on different stages of life.
Contrary forces might cause the development of subpopulations.
Contrasting selective forces on life history
r-selection and K-selection describe two ends
of a continuum of reproductive patterns.
Rana temporaria
r refers to the high reproductive rate.
K refers to the carrying capacity of the habitat
Brookesia desperata
High reproduction rate
High population growth
Low parental investment
No care of offspring
Often unstable habitats
Low reproduction rate
Low population growth
r
Continuum
K
High parental investment
In many species different
Intensive care of offspring
developmental stages,the sexes
Often stable habitats
and particulalry subpopulations
range differently on the r/K
r selected species
K selected
continuum!
mature rapidly and have an early age
mature more slowly and have a
of first reproduction
have a relatively short lifespan
have few reproductive events, or are
semelparous
have a high mortality rate and a low
offspring survival rate
have minimal parental care/investment
are often highly variable in population
size
Literature: Reznick et al. 2002, Ecology 83.
later age of first reproduction
have a longer lifespan
have few offspring at a time and
are iteroparous
have a low mortality rate and a
high offspring survival rate
have high parental investment
Have often relatively stable
populations
Number of deaths
Equilibrium
Birth
excess
∆𝑁 𝑡 = 𝐵 𝑡 − 𝐷(𝑡)
Number of births
Population size
The growth of populations
Time
𝑁 𝑡 + 1 = 𝐵 𝑡 − 𝐷(𝑡)+N(t)
Birth rate:
Death rate:
𝐵(𝑡)
𝑏(𝑡) = 𝑁(𝑡)
𝐷(𝑡)
𝑑(𝑡) = 𝑁(𝑡)
𝑁 𝑡 + 1 = 𝑏(𝑡)𝑁 𝑡 − 𝑑 𝑡 𝑁(𝑡) + N(t)
𝑁 𝑡 + 1 = (b(t) − d(t) + 1)N(t)
𝑅(t) = b(t) − d(t) + 1
The net reproductive rate R is the number of reproducing female offspring produced per
female per generation.
If R > 1: population size increases
If R = 1: population remains stable
If R < 1: population size decreases
The density of a population is the average
number of individuals per unit of area.
Abundance is the total number of individuals
in a given habitat.
Population size
Population fluctuations
Equilibrium
density
Amplitude
Time
The exponential growth of
populations
∆𝑁 𝑡 = 𝑏 𝑡 − 𝑑 𝑡 𝑁(𝑡)
Population
size
𝑁 𝑡 + 1 − 𝑁 𝑡 = ∆𝑁 𝑡 = 𝑏 𝑡 − 𝑑 𝑡 𝑁(𝑡) = 𝑟𝑁(𝑡)
𝑑𝑁
=𝑟
𝑑𝑡
Time
North atlantic gannets in north-western
England (Nelson 1978)
The intrinsic rate of population
growth r (per-capita growth rate) is
fraction of population change per
unit of time.
If r > 0: population size increases
If r = 0: population remains stable
If r < 0: population size decreases
ln(2)
𝑡=
𝑟
The growth
rate is
r = 0.057
Population
doubling time
Under exponential growth there is no
equilibrium density.
Exponential growth is not a realistic model
since populations cannot infinite sizes.
The logistic growth of populations
Populations do not increase to infinity. There is an upper boundary, the carrying capacity K.
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒
𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒
=
𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑜𝑛
𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 𝑟𝑎𝑡𝑒 𝑜𝑓
𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
×
× 𝑢𝑛𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑔𝑟𝑜𝑤𝑡ℎ
𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑔𝑟𝑜𝑤𝑡ℎ
𝑠𝑖𝑧𝑒
𝑑𝑁
𝐾−𝑁
𝑟 2
= 𝑟𝑁
= rN − 𝑁
𝑑𝑡
𝐾
𝐾
The logistic model of population growth
𝑁 𝑡 =
𝐾
1 + 𝑒 −𝑟(𝑡−𝑡0)
=
𝐾
𝐾
1 − 𝑁 − 1 𝑒 −𝑟𝑡
0
The logistic growth function is the
standard model in population ecology
Raymond Pearl
(1879-1940)
Pierre Francois Verhulst
(1804-1849)
The logistic growth of populations
𝑁 𝑡 =
𝐾
1 + 𝑒 −𝑟(𝑡−𝑡0)
=
𝐾
𝐾
1 − 𝑁 − 1 𝑒 −𝑟𝑡
0
Population size
1.2
1
N (t ) 
0.8
Maximum
population
growth
0.6
0.4
N (t ) 
0.2
1
1 e
 0.5 ( t 10)
1
1  e 0.5(t 10 )
0
0
5
10
15
20
25
Time
The equilibrium population size
𝑑𝑁
𝐾−𝑁
= 0 = 𝑟𝑁
𝑑𝑡
𝐾
Time t0 of maximum growth
𝑑2𝑁
2𝑟
=0=𝑟− 𝑁
𝑑𝑡 2
𝐾
The logistic growth of populations
Growth of yeast cells
(data from Carlson 1913)
How to estimate the
population parameters?
K = 665
K/2
𝑁=
665
1 + 𝑒 −0.54(𝑡−7.70)
t0
𝑁=
𝐾
1 + 𝑒 −𝑟(𝑡−𝑡0)
𝐾
− 1 = 𝑒 −𝑟(𝑡−𝑡0)
𝑁
𝐾
𝑙𝑛
− 1 = −𝑟𝑡 + 𝑟𝑡0
𝑁
t0=7.70
Logistic growth occurs particularly in organisms with non-overlapping (discrete)
populations, particularly in semelparous species:
e.g. bacteria, protists, single celled fungi, insects.
Logistic population growth implies a density dependent regulation of population size
𝑑𝑁
𝐾−𝑁
= 𝑟𝑁
𝑑𝑡
𝐾
If N > K, dN/dt < 0:
the population decreases
Natural variability in
population size
Density dependence means that the increase or decrease in population size is regulated
by population size.
The mechanism of regulation is intraspecific competition.
The number of offspring decrease with increasing population size due to resource
shortage.
The Allee effect
𝑑𝑁
𝐾−𝑁
𝑟
= 𝑟𝑁
= 𝑟𝑁 − 𝑁 2
𝑑𝑡
𝐾
𝐾
Logistic growth is equivalent to a quadratic function of population growth
Population growth
No Allee effect
K/2
Weak Allee effect
K
N
K
Strong Allee effect
N
K
At low population size propolation growth is in many cases lower than
predicted by the logistic growth equation.
Allee extension of the logistic function
𝑑𝑁
𝑁 − 𝐴𝐾 − 𝑁
= 𝑟𝑁
𝑑𝑡
𝐴
𝐾
A is an empitical factor that determines
the strength of the Allee effect
Most often Allee effects are caused
by mate limitation at low
population densities
N
Variability in population size
Proportional rescaling
Poisson random
Density regulated
J=1.14
J=0.91
We use the variance mean ratio
as a measure of the type of
Proportional
density fluctuation
rescaling
2
𝜎
1
𝐽 = 2 − +1
𝜎 2 ~𝜇2
𝜇
𝜇
𝜎 2 ~𝜇 𝑧
The Lloyd index
Taylor’s power law
of aggregation
needs m > > 1.
J=0.82
Aphids
Birds
Butterflies
Fragmented landscapes
Landscape ecology
Agroecology
The metapopulation of Melitaea cinxia
Illka Hanski
In fragmented landscapes populations are dived into
small local populations separated by an inhostile matrix.
Between the habitat patches migration occurs.
Such a fragmented population structure connected by
Glanville fritillary
dispersal is called a metapopulation.
Melitaea cinxia
Different types of metapopulations
The Lotka – Volterra model of population growth
Dispersal in a fragmented landscape
dN
KN
 rN (
)
dt
K
Levins (1969) assumed that the change in the
occupancy of single spatially separated habitats
(islands) follows the same model.
Assume P being the number of islands (total K)
occupied. Q= K-P is then the proportion of not
occupied islands. m is the immigration and e the local
extinction probability.
Colonisations
𝑑𝑃
𝐾−𝑃
= 𝑚𝑃
𝑑𝑡
𝐾
Emigration/Extinction
𝑑𝑄
= −𝑒𝑃
𝑑𝑡
The Levins model of meta-populations
𝑑𝑃
𝐾−𝑃
= 𝑚𝑃
− 𝑒𝑃
𝑑𝑡
𝐾
Fragments differ in population size
The higher the population size is, the
lower is the local extinction probability
and the higher is the emigration rate
𝑑𝑃
𝐾−𝑃
1
−𝑐𝐼
= 𝑎𝑒 𝑃
−𝑏 𝑃
𝑑𝑡
𝐾
𝐴
150
100
Colonisation probability is exponentially
dependent on the distance of the islands and
extinction probability scales proportionally to
island size.
1
𝑒∝
𝑚 ∝ 𝑒 −𝑐𝐼
𝐴
Distance
90
80
If we deal with the fraction of fragments colonized
𝑑𝑝
= 𝑚𝑝 1 − 𝑝 − 𝑒𝑝
𝑑𝑡
The canonical model of metapopulation ecology
200
Distance
Metapopulation modelling allows for
an estimation of species survival in
fragmented landscapes and provides
estimates on species occurrences.
Extinction times
𝑑𝑝
= 𝑚𝑝 1 − 𝑝 − 𝑒𝑝
𝑑𝑡
If we know local extinction times TL we can estimate
the regional time TR to extinction
When is a metapopulation stable?
𝑒
𝑝 =1−
𝑚
The meta-population is
only stable if m > e.
Median time to extinction
𝑑𝑝
= 0 = 𝑚𝑝 1 − 𝑝 − 𝑒𝑝
𝑑𝑡
𝑇𝑅 =
𝑃2
𝑇𝐿 𝑒 2𝐾−2𝑃
1200
1000
800
600
400
200
0
0
1
2
3
4
5
6
7
p K 0.5
The condition for long-term survival
𝑃
3
=𝑝>
𝐾
𝐾
What does metapopulation ecology predict?
Occurrences of Hesperia comma in fragmented
landscapes in southern England (from Hanski 1994)
3
Occurrences
Absences
2
1
ln Area
0
-1
-2
-3
-4
Predicted extinction threshold
-5
-6
0
1
2
3
4
5
Connectivity
In fragmented landscapes occupancy declines
nonlinear with decreasing patch area and with
decreasing conncetivity (increasing isolation)
Extinction times of ground beetles
on 15 Mazurian lake islands
Local extinction times (generations) are roughly
proportional to local abundances
Species
Pterostichus oblongopunctatus
Pseudoophonus rufipes
Pterostichus nigrita
Patrobus atrorufus
Platynus assimilis
Carabus nemoralis
Harpalus 4-punctatus
Amara brunea
Badister bullatus
Oodes gracilis
Loricera pilicornis
Amara communis
Notiophilus biguttatus
Badister sodalis
Carabus hortensis
Harpalus solitaris
Lasiotrechus discus
Amara aulica
𝑃
3
=𝑝>
𝐾
𝐾
Occurrences
14
13
13
12
11
11
10
9
8
7
6
6
5
4
3
2
2
1
𝑁 > 3 15 = 11.6
Local time to extinction
1
5
10
20
50
Regional time to extinction
1097
5483 10966 21933 54832
26
129
258
516
1290
26
129
258
516
1290
7
37
74
148
369
4
20
40
79
198
4
20
40
79
198
3
14
27
54
136
2
11
21
42
106
2
9
18
35
89
2
8
15
31
77
1
7
14
28
70
1
7
14
28
70
1
6
13
26
64
1
6
12
24
60
1
6
11
23
57
1
5
11
22
54
1
5
11
22
54
1
5
10
21
52
Population should be save if they
occupy at least 12 islands.
SPOMSIM
Population ecology needs long-term data sets