the past, present, and future of endangered whale populations: an introduction to mathematical...
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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/~gledder1gledder@math.unl.edu
Supported by NSF grant DUE 0536508
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
(1A) Mathematical Model
MathProblem
Input Data Output Data
Key Question:
What is the relationship between input and output data?
Rankings in Sports
MathematicalAlgorithm
Ranking
Game Data: determined by circumstances
Weight Factors: chosen by design
Game Data
Weight Factors
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
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?
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.
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.
(1B) Mathematical Modeling
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
(1B) Mathematical Modeling
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
A mathematical model represents a simplified view of the real world.
(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!
Example: Mars Rover
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Newtonian physics
Example: Mars Rover
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Newtonian physics
• Validation by many experiments
Example: Mars Rover
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Newtonian physics
• Validation by many experiments• Result:
Safe landing
Example: Financial Markets
RealWorld
ConceptualModel
MathematicalModel
approximation derivation
analysisvalidation
• Conceptual Model:Financial and credit markets are independentFinancial institutions are all independent
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
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??
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!!
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
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
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
(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?
(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
• Logistic growth– Fixed environment capacity
K
XRXXF 1)(
K
R
X
XF )(
Relative growth rate
(2B)
• 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
The Dimensional Model
22
2
1XA
QXC
K
XRX
dT
dX
Overall rate of increase = growth rate – consumption rate
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.
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.
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 :,:
Dimensionless Version
211
1
x
x
k
x
ccx
dt
dx
RA
CQc
A
Kk
R
tTAxX ,,,
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.
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 ,,,
211
1
x
x
k
x
cxc
dt
dx
(2C)
211
1
x
x
k
x
cxc
dt
dx
211
1
x
x
k
x
c
The resource increases
(2C)
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)
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
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
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
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
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
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)
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)
(2D) Whale Conservation
• Can we use our general resource model for whale conservation?
(2D) Whale Conservation
• Can we use our general resource model for whale conservation?
• Issues:– Model assumes fixed consumer population.
(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.
(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.
(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.
Stage 1 – natural balance
x
Stage 2 – depletion
Consumption increases to high level.
x
Stage 3 – inadequate correction
Consumption decreases to modest level.
x
Stage 4 – recovery
Consumption decreases to minimal level.
x
Stage 5 – proper management
x
Consumption increases to modest level.
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