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

Degr

ee o

f sta

rvati

on Quality of food

Optimal 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 siz

e

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 siz

e

Time

𝑁 (𝑡+1 )−𝑁 (𝑡 )=∆𝑁 (𝑡 )=(𝑏 (𝑡 )−𝑑 (𝑡 ) )𝑁 (𝑡 )=𝑟𝑁 (𝑡 )

𝑡= ln (2)𝑟Population

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 =𝑟𝑁 𝐾 −𝑁

𝐾 =rN − 𝑟𝐾 𝑁2

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)

00.20.40.60.8

11.2

0 5 10 15 20 25

Popu

latio

n si

ze

Time

)10(5.011)(

tetN

)10(5.011)(

tetN

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 0t0=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 =𝑟𝑁 𝐾 −𝑁

𝐾If 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 =𝑟𝑁 𝐾 −𝑁

𝐾 =𝑟𝑁− 𝑟𝐾 𝑁 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 =𝑟𝑁 𝑁− 𝐴

𝐴𝐾 −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

)(KNK

rNdtdN

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 ) 𝑑𝑄

𝑑𝑡 =−𝑒𝑃

𝑑𝑃𝑑𝑡 =𝑚𝑃 (𝐾−𝑃𝐾 )−𝑒𝑃

The 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