David Schneider Memorial University,St. John’s, Canada

Scale, Scope, and Power Laws in Environmental Science. Part III.

Environmental Science 600017 September 2009

• Scale –Dependence: examples and definitionapplication (student examples)

• Space-Time Diagramsexamplesgroup projects in class

• Calculation across scalesexamples with backgroundexamples (worked in groups)

• Summary

Scale, Scope, and Power Laws in Environmental Science

1.0 10 100 1000

10

1.0

100

1000

10

1.0

100

1000

Kilometers

F

P

Z

= t1.17

Day

sD

ays

Fladenexperiment

Single ship dataP

Z

Fish stock surveys

F

(a)

(b)

H k V 2 3/

Euclidean ScalingRelates One Scope to Another

//V O

V O M ouse

M assM ouseM ass

2

2

2 32 3

V O

V O R at

A rea

A reaM ouse

L eng th

L eng thM ouse2

2

1 2

M ass

M assR at

V o lum e

V olum eM ouse

L eng th

L eng thM ouse

1 3

Respiration scales as Euclidean Area in Spherical Animals

Scaling Relations

Power Law

/VO

VO M ouse

M ass

M assM ouse2

2

2 3

Some algebra

Euclidean ScalingRelates One Scope to Another

//V O

V O M ouse

M assM ouseM ass

2

2

3 43 4

VO

V O R at

Supp lyV

Supp lyV M ou se

L

L M ouse2

2

1 3

V

V M ouse

Supp lyV

Supp lyV M ou se

L

L M ouse

L

L M ouse

4

Supply scales as delivery volume in a structured network

Scaling Relations

Power Law

/ /VO

V O M ouse

V

V M ouse

M ass

M assM ouse2

2

3 4 3 4

Some algebra

Euclidean ScalingRelates One Scope to Another

Respiration scales as Euclidean Area in Spherical Animals

/V O

V O M ouse

M ass

M assM ouse2

2

3 4

Weightmass (kg) Ratio

2/3 3/4mouse 0.035cat 4.5 129 25 38fat cat 15 429 57 94elephant 7500 214286 ?? ??elephant 7500 214286 3581 9960

Resp.Ratio

Weight grams of Caloriesmass (kg) Ratio food per day

2/3 3/4 per day @4Kcal/gmouse 0.035 4 16cat 4.5 129 25 38 153 611fat cat 15 429 57 94 377 1507elephant 7500 214286 3581 9960 39839 159355normal human 70 2000 159 299 ?? ??

70 2000 159 299 1196 4785Michael Phelps 87 2487 184 352 1409 5635

Resp.Ratio

Euclidean ScalingRelates One Scope to Another

Respiration scales as Euclidean Area in Spherical Animals

/

VO V O M ouseM ass

M assM ouse2 2

3 4

Species number scales as Area

          

N cich lids

cich lids

A L ake V ictoria

A L ake E dw ardoN o

2 0 04 0

6 9 4 8 4 2

2 1 5 0 2cich lid species km

kmcich lid species

,

,

5 3 2 3 2 . 0 .4 6 3

2 0 0

4 0

6 9 4 8 4

2 1 5 0

2

2

cich lid species km

kmcich lid species

,

,

0 .4 6 3

N spN sprefL ake

A rearefL akeA rea

0 4 6 3

0 4 6 3.

.Power Law

Allometric ScalingRelates One Scope to Another

Fractal ScalingRelates a Count to Unit Size

Steps along a coastline scale as step LengthDf

                     N L

L oN

- D f

o

1

4 0 0

1 0

0 1

g ian t step

baby

m

msteps

.

-1 .3

50 km Steps

200 km StepsDetail increases rapidly with increasingresolution along a complex coastline

The exponent Df quantifies the rate of change in detail

Power Laws in BiologyHave Limited Spatial Scopes

Spatial Scale (m)

10-6 100 106 1012

Literature search that excluded body size allometry and species-area relations

Nearly 200 found, but only 60 usable

Annual rate of publication increased exponentially.

Where spatial scope could be determined it was limited, usually less than 103.

Predicting Hydro Impact on Fish

Y ield k M E I M E I

ppmdep th

C atchA rea

kppm

dep th

1

2

4

1 4 16 64

Yie

ld

=

kg /

ha

MEI = ppm / m -1

8

a

We are scaling one ratio to another

Ryder’s Morphoedaphic Index is used to predict change in catch due to reservoir flooding

Completely empirical scaling

Predicting Hydro Impact on FishDimensional analysis showed that the MEI formula was driven by an artifact.

Dimensional analysis

2 dep th

A realake geom etry

1 ppm w ater clarity

C atchA rea

kppm

dep th

1 Water Clarity (ppm)

0 50 100 150 200 250

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

b

L

ake

G

eom

etry

Predicting Hydro Impact on FishDimensional analysis showed that the MEI formula was driven by an artifact.

C atch kdep th

A rea

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

-1

103

104

105

106

107

c

Lake Geometry

Tot

al C

atch

(kg

)

0 50 100 150 200 250

-1

103

104

105

106

107

d

Tot

al C

atch

(kg

)

Water Clarity

C atch k ppm 0

Predicting Hydro Impact on Fish

Improved formula: Process is flux across the surface

H k V 2 3/

Predicting Impact of Oil Spills

Test the prediction, via trained observers coordinated to emergency response.

Spill Size (Tonnes)101 102 103 104 105 106

Ca

rca

sses

101

102

103

104

105

Carcasses Tonnes

CI to

313 0223

95% 0024 0423

.

. .

Data from A.Burger (1993)

1000 barrel spillPredicted Count:

520 to 1760 carcasses

K ill V o l 1 0 3 1 3 0 2 2 3.

Power Laws: Phenomena, Impacts, and Action

• Examples– Coral Reefs– H1N1– meHg in fish

Coral reefsDynamics: Growth vs Loss via nutrification

Scale from lab measurements to lagoonAction: Reduce nutrient input into lagoon

Power Laws: Phenomena, Impacts, and Action

• Examples– Coral Reefs– H1N1– meHg in fish

H1N1 (Swine Flu)Dynamics: Ro = Reproductive Number = New/InfectedAction Ro > 4 locally (depends on crowding)

Ro < 1.4 in Mexico (public health response)

Power Laws: Phenomena, Impacts, and Action

• Examples– Coral Reefs– H1N1– meHg in fish

meHg in fishDynamics: Hg to meHg in anoxic environmentsAction

reduce local input (Minimata disease)

avoid mobilizing global input (Canada reservoirs)

Summary – Power LawsPhenomena, Impacts, Action

• Comparison of cases at different scales• Calculate predicted effects:

– From local to larger scale– From larger scale to local

• Research planning

• Calculate impacts

• Science based action

Scale, Scope, and Power Laws in Environmental Biology

Group project: In groups of 2 or at most 3, work through the problem sets provided.