population biology - wilkes university

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Population Biology

• Essential to understand human populations

• Essential to understand endangered species

• Essential to understand pests and parasites

• Essential to understand other economically important species

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Defining the individual

• Unitary

– Individuals are discrete

– Less plasticity

• Modular

– Individuals reproduce by modules

– More plasticity

– Ramets and genets

– Biomass

• Growth

Determinate

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Populations

• Number of individuals of one species

– All individuals (N)

– Sample (n)

– Within a nature boundary

• Islands

• Well defined habitats (e.g., pond)

– Within an arbitrary boundary

• State, region, county 11/8/09

Populations

N is very rarely ever known• except for the very rare!• or very obvious in well-defined boundaries •too rare… extinct?

Snowdonia hawkweed from the beautiful nation of Wales. N=1

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Population Size?

• Population Index

– Not all organisms are counted

– Some standard is chosen then changes indicate population changes (indices of abundance)

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Population Size?

• Population Index

– Not all organisms are counted

– Some standard is chosen then changes indicate population changes (indices of abundance)

– Common in wildlife management

– Example

• deer brought into check stations indicate the population when the number of deer out there are never known

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Population Indices

• Population Indices

– Require minimal work (therefore cheap)

– Provide minimal information

• Difficult to predict the consequences of management decisions

• Adaptive management

• Difficult to predict the consequences of environmental changes

– Climate change

– Introduced species (parasites, disease [West Nile], new food,

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Population Estimation

• Population estimation attempts to figure out how many individuals there are in an area

– N can refer to all individuals of a species but more often the total number of individuals in an area

– A sample of a population is n and can be used to estimate N

– Formally, n = (N hat)

– N is hardly ever known

– A good estimate is when

N̂

NN̂

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Population Estimates

• How do we get ?

• Plants

– Sample in a small area to get density then extrapolate

– Density = number of individuals/area

• Animals

– Density, but more difficult… (ex: cougar in Oregon)

– Capture-recapture

N̂

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Capture-Recapture

• Animals are captured, marked, released, and then resampled

• Labor intensive (not cheap)

• Lots of assumptions

– Animals don’t avoid trapping.…. (e.g. bears)

– Many more but… (sample representative?)

– Problems with biases….. (e.g. bird nets)

• Provide lots of information

– Survivorship, sex ratios, recruitment, health, DNA samples, etc

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Capture – Recapture

• Many animals are PIT tagged

– Passive Integrated Transponder

– used in livestock and pets as well as wildlife

– Endangered species

– Individual recognized by hand-held scanners

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• Mammals get ear tagged

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• Birds get banded, big birds get neck collars

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• Animals with complex patterns can be photographed

– Whales, jaguars, some salamanders

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• Animals with complex patterns can be photographed

– Whales, jaguars, some salamanders

• Insects and hummingbirds – white out!

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Capture – recapture

• 2 samples

)2()1(

)(1)((

1

)1)((̂2

secsecsec

recapturedrecaptured

recapturednnn

recaptured

nnN ondondinitialondinitial

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Capture – recapture

• 2 samples

)2()1(

)(1)((

1

)1)((̂2

secsecsec

recapturedrecaptured

recapturednnn

recaptured

nnN ondondinitialondinitial

Population size very useful…. But for management purposes, we also need density AND dispersion of population in environment…

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Density and Dispersion

Random Clumped Uniform

(Measure of evenness)

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DISTANCE FROM NEAREST NEIGHBOR DISTRIBUTION

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Mean

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Variance

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Density and Dispersion

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Spatial Scale: Extent

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Spatial Scale: Resolution

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Geographic Range

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Life table analysis

• Understanding the Nnow of a particular species

• So we can predict the Nfuture

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Life table analysis: patterns of birth, death and growth

• Survival (survivorship curve)

• Fecundity schedule (birth rate)

• Mortality (proportion dying at each life stages)

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Life table analysis: patterns of birth, death and growth

• For species with annual life cycle…

• Semelparity vs. iteroparity

• Easy for species with distinct life stages

• Generation only overlap between breeding adults and offspring (distinct generations)

• Cohort: group of individuals born within the same short interval of time

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Simple Cohort Life TableInterval (days) Number

surviving (ax)Proportion of original

surviving (lx)Proportion of orginal cohert dying during interval (dx)

Mortality rate per day (qx)

Log10 lx Cohort fecundity Fx

Average fecundity per survivor (mx)

Proportion of original fecundity

(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Various stages of the life cycle that have been distinguishedCan be divided by age-classes…

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Raw data: number of individuals observed at each stagea0 for the first stage, a1 for the second, etc

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

ax good only for that population, lx can be compared, since it’s a proportion

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Proportion of original cohort dying at each stage

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Fraction dying during each stage: probability of an ind. dying.“intensity of mortality”

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Used for survivorship curvesCaptures the biologically meaningful observations instead of just arithmetic

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(Parenthesis: the use of log)

• Ex.: If pop. goes down 1000 to 500 in a single time interval

• Then, goes down from 100 to 50

• One is down 500 ind. (- 500)

• The other is only 50 (-50)

• BOTH are identical biologically: they lost half their population (mortality rate is same)

• Both provide a slope of -0.301 in log scale!

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Raw data: number of eggs deposited during each stage

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Bit more useful: number of eggs produced by surviving individual in each stage

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Basic reproductive rateMean number of offspring produced by original ind. by the end of the cohort 11/8/09

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Cohort Life Table: Red Deer of the Isle of Rhum

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Cohort Life Table: Red Deer of the Isle of Rhum

All individuals known, all calves counted form 1957 until 1966

All cause of death known

Monitored all events

Problems: we have overlapping generations, very few are sessile…

so not easy… Too hard for long lived animals?

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Static Life Table (or time-specific)

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Static Life Table (or time-specific)

Try to find a solution…

Based on reconstructed age-structure of the population (looking at those who died)

Also criticized…. Provide negative mortality rates…Same data as a single cohort followed.

Not quite perfect for species with overlapping generations...

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R0 (basic reproductive rate)

calculated for overlapping generation species is almost the same as species with discrete or distinct generations

What happens with varying densities from year to year?

What causes changes in density?

Can we include that in life table?

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Iteroparity and the age-dependent reproduction

Static fecundity schedules for multiple breeding seasonsAge- or stage-related patterns...

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Reproductive rates (R0, λ, r)

• Basic reproductive rate

For species with distinct generationsSums of all proportion of ind. surviving with average fecundity per survivor

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Becomes average number of offspring produced by an individual in its lifetimeUnit of time IS a generation in distinct generations

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• Fundamental net reproductive rate

– If prefer the symbol: λ

– If λ > 1 the population increases if < 1 the population decreases

– Does not separate between survival and reproductionPop size at one interval of time

Initial pop size11/8/09


• Note that R0 = λ T

• And ln(R0)= T ln(λ)

• And ln(λ) = ln(R0)/T

• ln(λ) = r

• So r=ln(R0)/T

• r is the intrinsic rate of natural increase

(increase in pop size per unit time)

• If r > 0 the population grows, if < 0 then population declines and if r = 0 then?

Generation time

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(lxmx)

0-63 996 1.000 0.329 0.006 0.000

63-124 668 0.671 0.374 0.013 -0.173

124-184 295 0.296 0.105 0.007 -0.528

184-215 190 0.191 0.014 0.003 -0.720

215-264 176 0.177 0.004 0.002 -0.753

264-278 172 0.173 0.005 0.002 -0.763

278-292 167 0.168 0.008 0.004 -0.776

292-306 159 0.160 0.005 0.002 -0.797 53 0.333 0.053

306-320 154 0.155 0.007 0.003 -0.811 485 3.149 0.487

320-334 147 0.148 0.042 0.025 -0.831 803 5.461 0.806

334-348 105 0.105 0.083 0.106 -0.977 973 9.264 0.977

348-362 22 0.022 0.022 1.000 -1.656 95 4.309 0.095

362- 0 0.000

Basic reproductive rateMean number of offspring produced by original ind. by the end of the cohort

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Iteroparity and the age-dependent reproduction

Static fecundity schedules for multiple breeding seasonsAge- or stage-related patterns...

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Matrices

• A matrix is a rectangular arrangement of data in the form m x n where m is the number of rows and n is the number of columns

• Capital letters are used for a matrix: A

• An element or entry is any datum of the matrix and lowercase is typically used

• The position of the element is denoted with subscripts ij where i is the row and j is the column.

– Example: a12,3 would be found in the 12th row, 3 third column

• A vector is a m x 1 or 1 x n matrix

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Leslie or Population ProjectionMatrix

In this exampleThere are three age classesFx is the fecundity of cohort xSx is the survival of cohort x to cohort x+1

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Population vector

In this example:Three age classesNx is the number in each age class

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Matrix multiplication

x nt=1 =

= nt=1

=

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Population Change

The resultant vector is the new population broken down by age structure

The new population is the sum of each of the age groups. In this case, Nt+1 = 112

Remember that λ is and the original population was or 100

0

1

N

Nt

so λ is 112/100 or 1.12

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Population Change (r)

The resultant vector is the new population broken down by age structure

The new population is the sum of each of the age groups. In this case, Nt+1 = 112

Remember that λ is and the original population was or 100

0

1

N

Nt

so λ is 112/100 or 1.12

λ over 1, pop is increasingln(λ) = r, so here r = 0.11r over 0, pop is increasingr is the intrinsic rate of natural increase

(increase in pop size per unit time)

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And? What do we do with this beast?

• A useful tool for seeing what happens to populations in the future

• Can incorporate stochasticity

– Gives a range instead a single value

• Can estimate the parameters that drive the results using a sensitivity analysis

– Important to figure out what to protect for endangered species or

– What to target for pest species

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Leslie Projection Matrices

• Used to see what would happen under different scenarios

• Can be expanded (gets ugly fast)

– Include spatial structure

– React to disease

– Competitors

– Etc

– Carrying capacity

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Carrying Capacity (K)

• Ideal populations size

– Everything needed is provided

– Populations increase exponentially

• Real populations

– Population has some limit set by the environment

– This upper limit is carrying capacity

– Highest density “allowed” by the environment

– Real populations may show exponential growth up to, over, or near the carrying capacity

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Carrying Capacity

• In most cases, exceedingly difficult to measure

– May be easy on a small scale where a limiting factor is easily monitored

• Rock structure for sessile organisms (density-dependent mortality vs density-dependent birth)

• Nesting sites for albatross

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Carrying Capacity

• In most cases, exceedingly difficult to measure

– May be easy on a small scale where a limiting factor is easily monitored

• Rock structure for sessile organisms

• Nesting sites for albatross

– Limiting factor can be biotic and abiotic

– Typically a mix of factors

– Ecology, being ecology, makes for complexities

• Species that limit some species are themselves limited by other species that are limited by other species that are limited by other species that are limited by other species that are limited by other species and so on

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Carrying Capacity

‘S’-shaped population increase, or sigmoidal curve

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Carrying capacity (K)

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Carrying capacity (K)

Much more likely to happen in real life

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Density Independent vs. Density Dependent Population Growth

dN by dt: speed at which a population increases in size N as time t progresses

r is the intrinsic rate of natural increase (increase in pop size per unit time)N pop sized density

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Density Independent vs. Density Dependent Growth

dN by dt: speed at which a population increases in size N as time t progresses

In absence of competition, or density independent.When density has no effect on pop, or pop growth.Which is unrealistic…..

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r and K

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r and K

Competition factor

When N rise to K, net increase fall to 0

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r and K selection

“r-selected “species

maximize“K-selected” species maximize

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r and K selection

When environment stable, K selectionhighly competitive, high survivorship, low reproductive outputFragile to rare disturbances (Tropics)When environment NOT stable, r selectionpoorly competitive, low survivorship, high reproductive outputEasily adapted to frequent disturbances (northern latitudes)

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r and K selection

Robert MacArthurE. O. Wilson

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R and K selection Characteristics

r-strategist K-strategist

Climate variable constant or predictable

Population size variable, recolonization at equilibrium

Competition variable, lax keen

Lifespan short, <1yr long, slower development

Size small larger

Reproduction much energy toward, rapid, large number of progeny

delayed reproduction, few offspring

Leads to high productivity efficiency

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Age Structure and Populations Age structured populations

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N t= N 0 ert

N t double= 2N0 2N0= N 0e

rtdouble

2= ertdoubleln 2 = rtdouble

tdouble=ln 2

r

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Species r (individuals/ individual*day)

Doubling time

T phage 300 3.3 minutes

E. coli 58.7 17 minutes

Rattus norvegicus 0.0148 46.8 days

Nothofagus fuscus 0.000075 25.3 years

Human Population Growth

The Human Population

• Current growth rate is ~1.3% per year

• ~2.1% per year in 1965-1970

• the absolute number of new people per year (~90 million) is at an all time high

• one billion in 1804

• Two billion in 1927

• Five billion in 1987

• Now – 6.57

• 9 billion in 2050 (revised down because of AIDS then revised back up because of new antivirals)

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Population Regulation

• Central charge of ecology

• Still largely unknown for most organisms

• Abiotic factors: weather (Andrewartha and Birch 1954)

• Biotic factors

• Interspecific and intraspecific interactions

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Bottom-up or top-down?

• Bottom up

– Nutrients

– Water

– Nesting sites

– Trophic levels below

• Top down

– Predators

– Parasites11/8/09

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population biology - wilkes university

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