the past, present, and future of endangered whale populations: an introduction to mathematical...
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![Page 1: The Past, Present, and Future of Endangered Whale Populations: An Introduction to Mathematical Modeling in Ecology Glenn Ledder University of Nebraska-Lincoln](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ced5503460f949ba903/html5/thumbnails/1.jpg)
The Past, Present, and Future of Endangered Whale Populations: An Introduction to Mathematical
Modeling in Ecology
Glenn LedderUniversity of Nebraska-Lincoln
http://www.math.unl.edu/[email protected]
Supported by NSF grant DUE 0536508
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Outline
1. Mathematical ModelingA. What is a mathematical model?B. The modeling process
2. A Resource Management ModelA. The general plan for the modelB. Details of growth and harvestingC. Analysis of the modelD. Application to whale populations
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(1A) Mathematical Model
MathProblem
Input Data Output Data
Key Question:
What is the relationship between input and output data?
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Rankings in Sports
MathematicalAlgorithm
Ranking
Game Data: determined by circumstances
Weight Factors: chosen by design
Game Data
Weight Factors
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Rankings in Sports
MathematicalAlgorithm
RankingGame Data
Model Analysis: For a given set of game data, how does the ranking depend on the weight factors?
Weight Factors
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Endangered Species
MathematicalModelControl
Parameters
Future Population
FixedParameters
Model Analysis: For a given set of fixed parameters, how does the future population depend on the control parameters?
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Models and Modeling
A mathematical model is a mathematicalobject based on a real situation andcreated in the hope that its mathematicalbehavior resembles the real behavior.
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Models and Modeling
A mathematical model is a mathematicalobject based on a real situation andcreated in the hope that its mathematicalbehavior resembles the real behavior.
Mathematical modeling is the art/science of creating, analyzing, validating, and interpreting mathematical models.
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(1B) Mathematical Modeling
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
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(1B) Mathematical Modeling
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
A mathematical model represents a simplified view of the real world.
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(1B) Mathematical Modeling
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
A mathematical model represents a simplified view of the real world.
Models should not be used without validation!
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Example: Mars Rover
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Newtonian physics
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Example: Mars Rover
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Newtonian physics
• Validation by many experiments
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Example: Mars Rover
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Newtonian physics
• Validation by many experiments• Result:
Safe landing
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Example: Financial Markets
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Financial and credit markets are independentFinancial institutions are all independent
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Example: Financial Markets
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Financial and credit markets are independentFinancial institutions are all independent
• Analysis:Isolated failures and acceptable risk
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Example: Financial Markets
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Financial and credit markets are independentFinancial institutions are all independent
• Analysis:Isolated failures and acceptable risk
• Validation??
![Page 18: The Past, Present, and Future of Endangered Whale Populations: An Introduction to Mathematical Modeling in Ecology Glenn Ledder University of Nebraska-Lincoln](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ced5503460f949ba903/html5/thumbnails/18.jpg)
Example: Financial Markets
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Financial and credit markets are independentFinancial institutions are all independent
• Analysis:Isolated failures and acceptable risk
• Validation?? • Result: Oops!!
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Forecasting the 2012 Election
Polls use conceptual models• What fraction of people in each age group vote?• Are cell phone users “different” from landline users?
and so on
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Forecasting the 2012 Election
Polls use conceptual models• What fraction of people in each age group vote?• Are cell phone users “different” from landline users?
and so onhttp://www.fivethirtyeight.com (NY Times?)• Uses data from most polls• Corrects for prior pollster results• Corrects for errors in pollster conceptual models
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Forecasting the 2012 Election
Polls use conceptual models• What fraction of people in each age group vote?• Are cell phone users “different” from landline users?
and so onhttp://www.fivethirtyeight.com (NY Times?)• Uses data from most polls• Corrects for prior pollster results• Corrects for errors in pollster conceptual models
Validation?? • Very accurate in 2008• Less accurate for 2012 primaries, but still pretty good
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(2) Resource Management
• Why have natural resources, such as whales or bison, been depleted so quickly?
• How can we restore natural resources?
• How should we manage natural resources?
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(2A) General Biological Resource Model
Let X be the biomass of resources.Let T be the time.Let C be the (fixed) number of consumers.Let F(X) be the resource growth rate.Let G(X) be the consumption per consumer.
)()( XGCXFdT
dX
Overall rate of increase = growth rate – consumption rate
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• Logistic growth– Fixed environment capacity
K
XRXXF 1)(
K
R
X
XF )(
Relative growth rate
(2B)
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• Holling type 3 consumption– Saturation and alternative resource
22
2
)(XA
QXXG
0 A 2A 3A 4A0
0.25Q
0.5Q
0.75Q
Q
X
G
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The Dimensional Model
22
2
1XA
QXC
K
XRX
dT
dX
Overall rate of increase = growth rate – consumption rate
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The Dimensional Model
22
2
1XA
QXC
K
XRX
dT
dX
Overall rate of increase = growth rate – consumption rate
This model has 4 parameters—a lot for analysis!
Nondimensionalization reduces the number of parameters.
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The Dimensional Model
22
2
1XA
QXC
K
XRX
dT
dX
Overall rate of increase = growth rate – consumption rate
This model has 4 parameters—a lot for analysis!
Nondimensionalization reduces the number of parameters.
X/A is a dimensionless population; RT is a dimensionless time.
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The Dimensional Model
22
2
1XA
QXC
K
XRX
dT
dX
Overall rate of increase = growth rate – consumption rate
This model has 4 parameters—a lot for analysis!
Nondimensionalization reduces the number of parameters.
X/A is a dimensionless population; RT is a dimensionless time.
A
XxRTt :,:
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Dimensionless Version
211
1
x
x
k
x
ccx
dt
dx
RA
CQc
A
Kk
R
tTAxX ,,,
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Dimensionless Version
211
1
x
x
k
x
ccx
dt
dx
RA
CQc
A
Kk
R
tTAxX ,,,
k represents the environmental capacity.c represents the number of consumers.
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Dimensionless Version
k represents the environmental capacity.c represents the number of consumers.(Decreasing A increases both k and c.)
211
1
x
x
k
x
ccx
dt
dx
RA
CQc
A
Kk
R
tTAxX ,,,
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211
1
x
x
k
x
cxc
dt
dx
(2C)
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211
1
x
x
k
x
cxc
dt
dx
211
1
x
x
k
x
c
The resource increases
(2C)
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211
1
x
x
k
x
cxc
dt
dx
211
1
x
x
k
x
c
k
x
cx
x1
1
1 2
The resource increases
The resource decreases
(2C)
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A “Textbook” Example
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
v
y
c = 1 Line above curve:Population increases
211
1
x
x
k
x
c
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A “Textbook” Example
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
v
y
c = 1
Low consumption – high resource level
Line above curve:Population increases
211
1
x
x
k
x
c
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A “Textbook” Example
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
v
y
c = 3
Curve above line:Population decreases
k
x
cx
x1
1
1 2
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A “Textbook” Example
High consumption – low resource level
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
v
y
c = 3
Curve above line:Population decreases
k
x
cx
x1
1
1 2
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A “Textbook” Example
Modest consumption – two possible resource levels
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
v
y
c = 2
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A “Textbook” Example
Modest consumption – two possible resource levels
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
v
y
c = 2Population stays low if x<2 (curve above line)
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A “Textbook” Example
Modest consumption – two possible resource levels
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
v
y
c = 2
Population becomes large if x>2(line above curve)
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(2D) Whale Conservation
• Can we use our general resource model for whale conservation?
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(2D) Whale Conservation
• Can we use our general resource model for whale conservation?
• Issues:– Model assumes fixed consumer population.
![Page 45: The Past, Present, and Future of Endangered Whale Populations: An Introduction to Mathematical Modeling in Ecology Glenn Ledder University of Nebraska-Lincoln](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ced5503460f949ba903/html5/thumbnails/45.jpg)
(2D) Whale Conservation
• Can we use our general resource model for whale conservation?
• Issues:– Model assumes fixed consumer population.
• We’ll look at distinct stages.
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(2D) Whale Conservation
• Can we use our general resource model for whale conservation?
• Issues:– Model assumes fixed consumer population.
• We’ll look at distinct stages.
– Model assumes harvesting with uniform technology.
![Page 47: The Past, Present, and Future of Endangered Whale Populations: An Introduction to Mathematical Modeling in Ecology Glenn Ledder University of Nebraska-Lincoln](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ced5503460f949ba903/html5/thumbnails/47.jpg)
(2D) Whale Conservation
• Can we use our general resource model for whale conservation?
• Issues:– Model assumes fixed consumer population.
• We’ll look at distinct stages.
– Model assumes harvesting with uniform technology.
• Advanced technology should strengthen the effects found in the model.
![Page 48: The Past, Present, and Future of Endangered Whale Populations: An Introduction to Mathematical Modeling in Ecology Glenn Ledder University of Nebraska-Lincoln](https://reader035.vdocument.in/reader035/viewer/2022062515/56649ced5503460f949ba903/html5/thumbnails/48.jpg)
Stage 1 – natural balance
x
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Stage 2 – depletion
Consumption increases to high level.
x
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Stage 3 – inadequate correction
Consumption decreases to modest level.
x
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Stage 4 – recovery
Consumption decreases to minimal level.
x
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Stage 5 – proper management
x
Consumption increases to modest level.