Population Growth

Exponential:

Continuous addition of births and deaths at constant rates (b & d)

Such that r = b - ddNdt

=rN

Problem: no explicit prediction is madeSolution: isolate N terms on left, and integrate

Result of the integration:

Nt =N0ert Exponential growth

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N(t)

r=0.05

Exponential growth relationships

Exponential growth

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Slope of this curveIncreases with density

Slope of Curve on left

Density

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Slope of line = r

Nt =N0ert

dNdt

=rN

Exponential growth, log scaletime, t: N(t) log N

0 100 4.605170191 105.12711 4.655170192 110.517092 4.705170193 116.183424 4.755170194 122.140276 4.805170195 128.402542 4.855170196 134.985881 4.905170197 141.906755 4.955170198 149.18247 5.005170199 156.831219 5.05517019

10 164.872127 5.10517019

Linear increase of logvalues with time is a sign of exponential growth

lnNt =lnN0 +rt

Geometric GrowthTime is measured in discrete (contant) chunks

Simplest approach: Generations are the time unit

R0: Average number of offspring produced per individual,per lifetime-- Factor that a population will be multiplied by for each generation. Often called the Net Rate of Increase.

NT =N0R0T

Time is measured in generations in this equation.

Relationship between R0 and r

A population growing for one generation should show the same result using either of the following equations:

Continuous, where t=“generation time”)

Discrete, where T=1 generation

NT =N0R0TNt =N0e

rt

N1 =N0R01 =N0R0Nτ =N0e

rτ

If these give the same result, then

N0erτ =N0R0

R0 and r

N0erτ =N0R0

erτ =R0

rτ =lnR0

r =lnR0

τSo! Information about R and can lead us to r

Ways of finding R0 and

R0 = lxmxx∑

τ≈xlxmx

x∑

lxmxx∑

Cohort study

x nx total offspring0 431 21 442 10 323 2 24 0 0

Survivorship calculations

x nx total offspring lx0 43 11 21 44 0.488372092 10 32 0.232558143 2 2 0.046511634 0 0 0

Fecundity calculationsx nx total offspring lx mx0 43 1 01 21 44 0.48837209 2.09523812 10 32 0.23255814 3.23 2 2 0.04651163 14 0 0 0 0

mx

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Fecundity, mx

mx

Age-specific reproductionx nx total offspring lx mx lxmx0 43 1 0 01 21 44 0.48837209 2.0952381 1.023255812 10 32 0.23255814 3.2 0.744186053 2 2 0.04651163 1 0.046511634 0 0 0 0 0

Net Rate of Increase= 1.81395349

lxmx

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Offspring per initial individual, lxmx

lxmx

Ro=area under curve

Generation time

Generation Time,

lxmx

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lxmx

Ro=area under curve

Generation time

Approximate rx nx total offspring lx mx lxmx xlxmx0 43 1 0 0 01 21 44 0.48837209 2.0952381 1.02325581 1.023255812 10 32 0.23255814 3.2 0.74418605 1.488372093 2 2 0.04651163 1 0.04651163 0.139534884 0 0 0 0 0 0

Net Rate of Increase= 1.81395349 2.65116279Generation time= 1.46153846

r~ 0.21601909

Assumptions of exponential or geometric growth projections

Constant lx and mx schedules

This implies that reproduction and survival will not change with density

This also implies that any changes in physical or chemical environment have no influence on survival or reproduction

No important interactions with other species

if age-specific data are used, assume stable age distribution.

Suppose we let lx, mx and vary with density

Bottom line: let r (per capita growth rate) vary with N

dN/Ndt

N

r

K0

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Density-dependent growth

dN/Ndt

N

r

K

-r/K

Y = A + BX

dNNdt

=r −rKN

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Logistic equationdNdt

=rN 1−NK

⎡ ⎣

⎤ ⎦

Predictive form:

Nt =K

1+K −N0

N0

⎡

⎣ ⎢ ⎤

⎦ ⎥ e−rt

Human rates of change vs N

y = -0.0019x + 0.0254

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Projection based on Logistic model:

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Earlier US projection, similar approach:

Logistic ExamplesFull-loop (2x the bacteria)

Half-loop (half that on right)

Paramecium, 2 species, growing for 8 days at high <r> and low <l> resource levels. Scale has been stretched on right to be equivalent to that on the left

More logistic examples

Growth of a zooplankton crust-acean, Moina, at different temperatures

Growth of flour beetles in flower,In containers holding different amtsof flour

Drosophila studies

Evolution of K in DrosophilaPost-radiation

Control

Hybrid

Inbred

Results suggest that K responds to an increase in genetic variation,And that it changes gradually through time in response to selection.

Assumptions of Logistic Growth

Constant environment (r and K are constants)Linear response of per capita growth rate to densityEqual impact of all individuals on resourcesInstantaneous adjustment of population growth to change in NNo interactions with species other than those that are foodConstantly renewed supply of food in a constant quantity

Discrete Model for Limited Growth

Same assumptions, except population grows in bursts with each Generation-- built-in time lag

Models of this sort show the potential influence that a time lag can have on population change.

Nt+1 =Nt +rNt 1−Nt

K⎡ ⎣

⎤ ⎦

Simple model, complex behavior

Discrete Model

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R = 0.1, K = 1000

Simple model, complex behavior

Discrete Model

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R = 1.9, K = 1000Damped oscillation

Discrete Model

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Simple model, complex behavior

r= 2.2, K = 1000Limit cycle

Discrete Model

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Simple model, complex behavior

r= 2.5, K = 10004-point cycle

Discrete Model

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Simple model, complex behavior

r= 2.58, K = 10008-point cycle

Discrete Model

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Simple model, complex behavior

r= 2.7, K = 1000Erratic

Discrete Model

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Chaos

r= 3, K = 1000

Discrete Model

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Overshoot, Crash, Extinction

r= 3.000072, K = 1000

Discrete Model

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Concerns about Chaos

Biological populations don’t appear to have the growth capacity to generate chaos, but this shows the potential importance of time lags.

More complicated models can be even more sensitive

Some systems might be completely unpredictable

Evolution of Life HistoriesLife history features:

Rates of birth, death, population growthPatterns of reproduction and mortalityBehavior associated with reproductionEfficiency of resource use, and carrying capacity

Anything that affects population growth

Patterns

More patterns

Tradeoff