The Exponential Growth Model

It’s the Malthusian Model in the words of Newton …

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Time (years)

Po

pu

latio

n s

ize

N

N(t+1) = N(t)

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35

0 10 20 30 40 50

Time (years)

Po

pu

latio

n s

ize

N

The continuous-time exponential growth model:

rNdt

dN

Oneparameter: r = b - dthe instantaneous per capita population growth rate.

The continuous-time exponential growth model:

rNdt

dN

To use this equation to predict population size some time in the future, one needs to integrate it over time.

rNtdNd

Solving the exponential growth model by integration:

dtrN

dN 1. rearrange:

2. integrateboth sides:

t

t

t

t

dtrdNN

0 0

1

3. solveintegral: 00

lnln rtrtNN tt

Rules used:

cnx

adxax

cxdxx

nn

1

ln1

1

)(ln 0

0

ttrN

N

t

t

4. rearrange:

0

0

ttr

t

te

N

N 5. take the exponent:

0

0

ttrtt eNN

6. rearrange:

3. solveintegral: 00

lnln rtrtNN tt

Malthusian Growth Model

(discrete-time model)

Exponential Growth Model(continuous-time model)

One parameter = 1 + b – d

population growspopulation levelpopulation declines

One parameterr = b - d

r > 0 population growsr = 0 population levelr < 0 population declines

)()1( tNtN rNdt

dN 0

0

ttrtt eNN

0

0

ttrtt eNN

What is the numerical relationship between the and r of the two models?

)()1( tNtN

N(t) ………….. NN(t+1)…………N t

t0

The “1” in the Malthusian model needs to be specified:

= t - t0

rtt eNN00

rtt eNN00

)ln(

r

erRe-arrange:

Given this relationship:

)ln(

r

if the doubling time of a population is 10 days, what is the value of r?

1) How do we know if a population follows the exponential growth model?

2) Under conditions of “unchecked” growth, are there species-specific differences in how fast a population can grow?

Time

Nu

mb

ers

of

ind

ivid

uals

Is this growth exponential?

The exponential growth model:

)(lnln 00ttrNN tt

0

0

ttrtt eNN

y = b + s * x

Plotting ln N v time, should result in a straight line where the slope is r!

The exponential growth model:

)(lnln 00ttrNN tt

time

ln N

t0

intercept = ln(Nt0)

slope = r

Year after No of ln(animals)introduction animals

0 6 1.7921 20 2.9962 60 4.0943 300 5.7044 700 6.5515 1250 7.131

0.000

1.000

2.000

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6.000

7.000

8.000

0 1 2 3 4 5 6

Destruction Island pheasants:

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

0 1 2 3 4 5 6

Doubling time : = ln(2)/r = 0.69/1.11 = 0.62 years

y = 1.93 + 1.11x r2 = 0.985

Years after introduction

Ln N

Cheatgrass in the Western USA:

Doubling time : = ln(2)/r = 0.69/0.132 = 5.25 years

year year-1900 % final ln(% range)range

1900 0 0.02 -3.9121904 4 0.05 -2.9961914 14 0.18 -1.7151920 20 0.28 -1.2731928 28 1 0.000

-4.500

-4.000

-3.500

-3.000

-2.500

-2.000

-1.500

-1.000

-0.500

0.000

0 5 10 15 20 25 30

y = -3.72 + 0.132x r2 = 0.985

year million ln(N)people

1 200 5.2983171650 500 6.2146081850 1000 6.9077551930 2000 7.6009021975 4000 8.294051993 5500 8.6125031999 6000 8.699515

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0 500 1000 1500 2000Year AD

ln(N

)

Estimated worldwide human population size:

Humans grow at faster than exponential rates.

Adult bluefin tuna in the western North Atlantic

Nes

t box

occ

upan

cy (

%)

Smallwood et al. 2009, Journal of Raptor Research 43(4):274-282.

American Kestrel (Falco sparverius)

Loggerhead turtles on the coast of Turkey

Ilgaz et al. 2007. Biodivers Conserv 16:1027–1037

Loggerhead Turtle:

yearNo.

turtles ln (N)1993 120 4.791994 160 5.081995 80 4.381996 90 4.501997 120 4.791998 100 4.611999 110 4.702000 115 4.742001 80 4.382002 105 4.652003 60 4.09 Year

y = 90 - 0.043x r2 = 0.2971

Halving time : = ln(1/2)/r = -0.69/-0.043 = 16 years

When do we see exponential decline?

1. In general, when birth rates cannot keep up with death rates.

2. Exponential decline can occur :• when critical resources disappear, and no substitute is found,• when habitats disappear, and the species cannot find or move to

adequate habitat elsewhere (e.g. islands),• when a competitively superior species or a predator suddenly

appears or becomes more abundant.• when climate suddenly changes and the species cannot adapt,• By chance, in small, isolated populations.

3. Exponential decline can lead to extinction.

Excel Worksheets:

• Exponential Growth Analysis

Unchecked population growth rates vary across 5-6 orders of magnitude among organisms

Species Common r (individuals/ doubling timename (individual*day)

T phage Virus 300 3.3 minutesEscherichia coli Bacterium 58.7 17 minutesParamecium caudatum Protozoan 1.59 10.5 hoursHydra Hydra 0.34 2 daysTribolium castaneum Flour beetle 0.101 6.9 daysRattus norvegicus Brown rat 0.0148 46.8 daysBos taurus Domestic cow 0.001 1.9 yearsAvicennia marina Mangrove 0.00055 3.5 yearsNothofagus fusca Southern beech 0.000075 25.3 yearsHomo sapiens Human (last 25 years) 0.000049 40.5 years

James Brown, New Scientists (2004)

For all the diversity among the spcies of earth, some regular patterns arise:

1. Larger organisms consume more energy and grow faster, but

not exactly in proportion to their size.

A 200 kg elk, 10,000x heavier than a 20 g mouse, eats

only 1000x more food. Mouse metabolism is therefore

10x faster, but elk live 10x longer.

2. Smaller organisms have shorter time spans and reproduce

faster than larger organisms.

3. Smaller organisms achieve higher population densities.

Summary:

1. Two equivalent models express unchecked population growth: the Malthusian model, and the exponential model.

2. Both models have one parameter that describes a population’s potential for growth, when resources are unlimited.

3. The population growth rates of earth’s organisms vary over many orders of magnitude.

4. Smaller organisms acquire resources faster, live shorter lives, and have shorter doubling times.

5. The Human population explosion over the past 2000 years can be linked to dramatic increases in the average energy consumption of individuals.