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From individuals to populationsThe basic entities of ecological research
Modular organized Brown fungi
Clonal Populus tremuloides forests
Unitary organisms have genetically prescribed longivity
Single celled Bacteria
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.
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.
Bees, ants, and other insect
societies form superorganisms
that behave as an ecological unit.Single information
coding strand of DNA
Clonal organisms
might have extreme longivity
Life cycles
All organisms have life cycles from single celled zygotes through ontogenetic stages to adult forms. All organsims finally die.
Mortality
k1 k2 k3 k4 k5=1
Surv
ivin
g in
divi
dual
s
Individual age
Type I
Type II
Type III
Type I, high survivorship of young individuals: Large mammals, birdsType II, survivorship independent of age, seed banksType III, low survivorship of young individuals, fish, many insects
Surv
ivin
g in
divi
dual
sIndividual age
Age dependent survival in annual plants
Often stages of dormancy
K-factor analysis
k1 k2 k3 k4 k5
Each life stage t has a certain mortality rate dt.
Stage Number surviving
Number of
deathsMortality
rateK-
factor
Year 2000Number of eggs 2000 3500Stage 1 800 1200 0.60 k1 0.92Stage 2 120 680 0.85 k2 1.90Stage 3 30 90 0.75 k3 1.39Stage 4 5 25 0.83 k4 1.79Stage 5 1 4 0.80 k5 1.61
k-factor
2001 2002 2003 2004 2005 2006 2007 2008 20093300 1500 1300 1500 1000 1200 1050 980 11000.13 0.32 0.05 0.12 0.02 0.02 0.08 0.11 0.161.49 2.83 2.50 2.10 2.00 1.10 1.97 2.25 1.782.50 0.61 2.74 0.88 1.83 1.19 1.20 1.04 1.211.37 0.13 1.45 1.08 0.77 1.47 0.26 0.06 0.052.12 3.71 0.86 3.42 2.98 3.83 4.09 4.14 4.40
𝑁 (𝑡+1 )=𝑁 (𝑡 )−𝑑 (𝑡 )𝑁 (𝑡 )=𝑁 (𝑡)¿
𝑘 (𝑡 )=ln (𝑁 𝑡+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.
k-factors calculated for a number of years A simple life table
Time series Density series2000
2001
No density dependence in mortality ratesClear temporal trends in mortality ratesDensity
Birds
Various vertebrates
Allometric constraints on life history parameters
Body size is an important determinant on life history.
Mammals
Insects
Microorganisms
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.
Number of offspring
Surv
ival
pro
babi
lity
Fitness
Optimal offspring number
Life history trade-offs
Complex life histories appear to be one way to maximize reproductive success in such highly competitive environments.
Time
Deg
ree
of st
arva
tion Q
uality of foodOptimal food intake time
The importance of individualistic behaviour
X
Amou
nt o
f foo
d co
nsum
ed
Food quality
Food quality
Food
val
ue
𝑉=(𝑁−𝐶)𝑄
The value of food is the product of food quality and
the difference of total amount N and amount
consumed C).
Th perceived food value migh
remain more stable than food
quality
For different individuals it pays to use resources of
different quality.
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.
The precise estimation of resource value is one of the motors of brain evolution.
Trade-off decisions during life history
How long to live?
How often to breed?(semelparous, iteroparous)
Caring for offspring?
When to begin reproducing?
How many offspring?
How fast to develop?
When to change morphology?
How fast to grow?
How large to grow?
What size of offspring?
At each time step in life animals take decisions.These decisions determine future reproductive success and ae objects of selective forces
Different selective forces might act on different stages of life. Contrary forces might cause the development of subpopulations.
Each step is a decision on
resource allocation.
How long to live after reproduction?
Contrasting selective forces on life history
Brookesia desperataRana temporaria
High reproduction rateHigh population growth Low parental investmentNo care of offspringOften unstable habitats
Low reproduction rateLow population growth
High parental investmentIntensive care of offspring
Often stable habitats
r-selection and K-selection describe two ends of a continuum of reproductive patterns.
Continuum
K selectedmature more slowly and have a
later age of first reproductionhave 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 investmentHave often relatively stable
populations
r selected speciesmature rapidly and have an early age of first reproductionhave a relatively short lifespanhave few reproductive events, or are semelparoushave a high mortality rate and a low offspring survival ratehave minimal parental care/investmentare often highly variable in population size
r refers to the high reproductive rate.K refers to the carrying capacity of the habitat
In many species different developmental stages,the sexes and particulalry subpopulations
range differently on the r/K continuum!
Literature: Reznick et al. 2002, Ecology 83.
r K
The growth of populations
Time
Num
ber o
f dea
ths Equilibrium N
umber of birthsPo
pula
tion
size
∆𝑁 (𝑡 )=𝐵 (𝑡 )−𝐷(𝑡)+N(t)
Birth rate: Death rate:
𝑁 (𝑡+1 )=𝑏 (𝑡)𝑁 (𝑡 )−𝑑 (𝑡 )𝑁 (𝑡)+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.
Birth excess
If R > 1: population size increasesIf R = 1: population remains stableIf R < 1: population size decreases
Popu
latio
n si
ze
Time
Population fluctuations
Amplitude
Equilibrium density
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.
North atlantic gannets in north-western England (Nelson 1978)
The exponential growth of populations
Popu
latio
n si
ze
Time
𝑁 (𝑡+1 )−𝑁 (𝑡 )=∆𝑁 (𝑡 )=(𝑏 (𝑡 )−𝑑 (𝑡 ) )𝑁 (𝑡)=𝑟𝑁 (𝑡)
𝑡=ln (2)𝑟
Population doubling time
∆𝑁 (𝑡 )= (𝑏 (𝑡 )−𝑑 (𝑡 ) )𝑁 (𝑡)
If r > 0: population size increasesIf r = 0: population remains stableIf r < 0: population size decreases
The intrinsic rate of population growth r (per-capita growth rate) is
fraction of population change per unit of time.
Under exponential growth there is no equilibrium density.
Exponential growth is not a realistic model since populations cannot infinite sizes.
The growth rate is r = 0.057
(𝑅𝑎𝑡𝑒𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒 )=( 𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐𝑟𝑎𝑡𝑒𝑜𝑓𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 h𝑔𝑟𝑜𝑤𝑡 )×(𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛𝑠𝑖𝑧𝑒 )×( 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑜𝑛
𝑢𝑛𝑏𝑜𝑢𝑛𝑑𝑒𝑑 h𝑔𝑟𝑜𝑤𝑡 )
The logistic growth of populations
Populations do not increase to infinity. There is an upper boundary, the carrying capacity K.
𝑑𝑁𝑑𝑡
=𝑟𝑁𝐾 −𝑁𝐾
=rN −𝑟𝐾𝑁2
The logistic model of population growth
The logistic growth function is the standard model in population ecology
𝑁 (𝑡 )= 𝐾1+𝑒−𝑟 (𝑡− 𝑡0 )
=𝐾
1−( 𝐾𝑁0
−1)𝑒−𝑟𝑡
Raymond Pearl (1879-1940)
Pierre Francois Verhulst (1804-1849)
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25
Popu
lati
on si
ze
Time
)10(5.011
)( te
tN
)10(5.011
)( te
tN
The equilibrium population size
Maximum population
growth
Time t0 of maximum growth
The logistic growth of populations
𝑁 (𝑡 )= 𝐾1+𝑒−𝑟 (𝑡− 𝑡0 )
=𝐾
1−( 𝐾𝑁0
−1)𝑒−𝑟𝑡
The logistic growth of populations
Growth of yeast cells (data from Carlson 1913)
K = 665
𝑁=665
1+𝑒−0.54 (𝑡−7.70 )K/2
t0
𝑁=𝐾
1+𝑒−𝑟 (𝑡− 𝑡0 )𝐾𝑁−1=𝑒−𝑟 (𝑡 −𝑡 0) 𝑙𝑛(𝐾𝑁 −1)=−𝑟𝑡+𝑟 𝑡 0
t0=7.70
How to estimate the population parameters?
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
𝑑𝑁𝑑𝑡
=𝑟𝑁𝐾 −𝑁𝐾
If N > K, dN/dt < 0: the population decreases
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.
Natural variability in population size
The Allee effect𝑑𝑁𝑑𝑡
=𝑟𝑁𝐾 −𝑁𝐾
=𝑟𝑁−𝑟𝐾𝑁 2
Logistic growth is equivalent to a quadratic function of population growth
Popu
latio
n gr
owth
No Allee effect Weak Allee effect Strong Allee effect
NNN KKKK/2
At low population size propolation growth is in many cases lower than predicted by the logistic growth equation.
𝑑𝑁𝑑𝑡
=𝑟𝑁𝑁− 𝐴𝐴
𝐾 −𝑁𝐾
Allee extension of the logistic function
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
Variability in population size
Proportional rescaling Poisson random Density regulated
We use the variance mean ratio as a measure of the type of density fluctuation
𝐽=𝜎 2
𝜇2−1𝜇
+1
The Lloyd index of aggregation needs m > > 1.
J=1.14
J=0.91 J=0.82
𝜎 2 𝜇2
Proportional rescaling
Taylor’s power law
Aphids
ButterfliesBirds
𝜎 2 𝜇𝑧
Fragmented landscapes
Landscape ecologyAgroecology
The metapopulation of Melitaea cinxia
Glanville fritillary
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
dispersal is called a metapopulation.
Different types of metapopulations
)(K
NKrN
dt
dN
The Lotka – Volterra model of population growth
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 Emigration/Extinction
𝑑𝑃𝑑𝑡
=𝑚𝑃 (𝐾−𝑃𝐾 ) 𝑑𝑄𝑑𝑡
=−𝑒𝑃
𝑑𝑃𝑑𝑡
=𝑚𝑃 (𝐾−𝑃𝐾 )−𝑒𝑃The Levins model of meta-populations
Dispersal in a fragmented landscape
80
200
90
100 150
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
Colonisation probability is exponentially dependent on the distance of the islands and extinction probability scales proportionally to
island size.
𝑒∝1𝐴𝑚∝𝑒−𝑐𝐼
𝑑𝑃𝑑𝑡
=𝑎𝑒−𝑐𝐼 𝑃 (𝐾−𝑃𝐾 )−𝑏 1𝐴 𝑃
The canonical model of metapopulation ecology
Distance
Distance
Metapopulation modelling allows for an estimation of species survival in
fragmented landscapes and provides estimates on species occurrences.
If we deal with the fraction of fragments colonized
𝑑𝑝𝑑𝑡
=𝑚𝑝 (1−𝑝 )−𝑒𝑝
Extinction times
When is a metapopulation stable?
𝑑𝑝𝑑𝑡
=0=𝑚𝑝 (1−𝑝 )−𝑒𝑝
𝑝=1−𝑒𝑚
The meta-population is only stable if m > e.
𝑇 𝑅=𝑇 𝐿𝑒𝑃2
2 𝐾−2 𝑃
If we know local extinction times TL we can estimate the regional time TR to extinction
0
200
400
600
800
1000
1200
0 1 2 3 4 5 6 7
p K 0.5
Med
ian
time
to e
xtinc
tion
𝑃𝐾
=𝑝>3
√𝐾
The condition for long-term survival
𝑑𝑝𝑑𝑡
=𝑚𝑝 (1−𝑝 )−𝑒𝑝
What does metapopulation ecology predict?
-6
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5
Connectivity
ln A
rea
Occurrences of Hesperia comma in fragmented landscapes in southern England (from Hanski 1994)
OccurrencesAbsences
In fragmented landscapes occupancy declines nonlinear with decreasing patch area and with decreasing conncetivity (increasing isolation)
Predicted extinction threshold
Local time to extinction1 5 10 20 50
Species Occurrences Regional time to extinctionPterostichus oblongopunctatus 14 1097 5483 10966 21933 54832Pseudoophonus rufipes 13 26 129 258 516 1290Pterostichus nigrita 13 26 129 258 516 1290Patrobus atrorufus 12 7 37 74 148 369Platynus assimilis 11 4 20 40 79 198Carabus nemoralis 11 4 20 40 79 198Harpalus 4-punctatus 10 3 14 27 54 136Amara brunea 9 2 11 21 42 106Badister bullatus 8 2 9 18 35 89Oodes gracilis 7 2 8 15 31 77Loricera pilicornis 6 1 7 14 28 70Amara communis 6 1 7 14 28 70Notiophilus biguttatus 5 1 6 13 26 64Badister sodalis 4 1 6 12 24 60Carabus hortensis 3 1 6 11 23 57Harpalus solitaris 2 1 5 11 22 54Lasiotrechus discus 2 1 5 11 22 54Amara aulica 1 1 5 10 21 52
Extinction times of ground beetles on 15 Mazurian lake islands
Local extinction times (generations) are roughly proportional to local abundances
𝑁>3√15=11.6 Population should be save if they occupy at least 12 islands.
𝑃𝐾
=𝑝>3
√𝐾
SPOMSIM
Population ecology needs long-term data sets