The Past, Present, and Future of
Endangered Whale Populations:
An Introduction to Mathematical
Modeling in Ecology
Glenn Ledder
University of Nebraska-Lincoln
http://www.math.unl.edu/~gledder1
gledder@math.unl.edu
Supported by NSF grant DUE 0536508
Outline
1. Mathematical Modeling
A. What is a mathematical model?
B. The modeling process
2. A Resource Management Model
A.
B.
C.
D.
The general plan for the model
Details of growth and harvesting
Analysis of the model
Application to whale populations
(1A) Mathematical
Input Data
Math
Problem
Model
Output Data
Key Question:
What is the relationship between input
and output data?
Rankings in Sports
Game Data
Mathematical Ranking
Weight Factors Algorithm
Game Data: determined by circumstances
Weight Factors: chosen by design
Rankings in Sports
Game Data
Mathematical Ranking
Weight Factors Algorithm
Model Analysis:
For a given set of game data, how does the
ranking depend on the weight factors?
Endangered Species
Fixed
Parameters Mathematical Future
Model
Population
Control
Parameters
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 mathematical
object based on a real situation and
created in the hope that its mathematical
behavior resembles the real behavior.
Models and Modeling
A mathematical model is a mathematical
object based on a real situation and
created in the hope that its mathematical
behavior resembles the real behavior.
Mathematical modeling is the art/science
of creating, analyzing, validating, and
interpreting mathematical models.
(1B) Mathematical
Real
World
approximation
validation
Conceptual
Model
Modeling
derivation
analysis
Mathematical
Model
(1B) Mathematical
Real
World
approximation
validation
Conceptual
Model
Modeling
derivation
analysis
Mathematical
Model
A mathematical model represents a
simplified view of the real world.
(1B) Mathematical
Real
World
approximation
validation
Conceptual
Model
Modeling
derivation
analysis
Mathematical
Model
A mathematical model represents a
simplified view of the real world.
Models should not be used without
validation!
Example: Mars Rover
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
• Conceptual Model:
Newtonian physics
Mathematical
Model
Example: Mars Rover
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Conceptual Model:
Newtonian physics
• Validation by many experiments
Example: Mars Rover
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Conceptual Model:
Newtonian physics
• Validation by many experiments
• Result:
Safe landing
Example: Financial Markets
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Conceptual Model:
Financial and credit markets are independent
Financial institutions are all independent
Example: Financial Markets
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Conceptual Model:
Financial and credit markets are independent
Financial institutions are all independent
• Analysis:
Isolated failures and acceptable risk
Example: Financial Markets
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Conceptual Model:
Financial and credit markets are independent
Financial institutions are all independent
• Analysis:
Isolated failures and acceptable risk
• Validation??
Example: Financial Markets
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Conceptual Model:
Financial and credit markets are independent
Financial 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 on
http://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 on
http://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.
dX
 F ( X )  C G( X )
dT
Overall rate of increase = growth rate – consumption rate
(2B)
• Logistic growth
– Fixed environment capacity
 X
F ( X )  RX 1  
 K
Relative growth rate
R
F(X )
X
K
• Holling type 3 consumption
– Saturation and alternative resource
2
QX
G( X )  2
A X2
Q
0.75Q
G 0.5Q
0.25Q
0
0
A
2A
X
3A
4A
The Dimensional Model
dX
QX
 X
 RX 1    C 2
2
dT
A X
 K
2
Overall rate of increase = growth rate – consumption rate
The Dimensional Model
dX
QX
 X
 RX 1    C 2
2
dT
A X
 K
2
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
dX
QX
 X
 RX 1    C 2
2
dT
A X
 K
2
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
dX
QX
 X
 RX 1    C 2
2
dT
A X
 K
2
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.
X
t : RT , x :
A
Dimensionless Version
t
K
CQ
X  Ax , T  , k  , c 
R
A
RA
1  x 
dx
x 
 cx  1   
2
dt
c  k  1 x 
Dimensionless Version
t
K
CQ
X  Ax , T  , k  , c 
R
A
RA
1  x 
dx
x 
 cx  1   
2
dt
c  k  1 x 
k represents the environmental capacity.
c represents the number of consumers.
Dimensionless Version
t
K
CQ
X  Ax , T  , k  , c 
R
A
RA
1  x 
dx
x 
 cx  1   
2
dt
c  k  1 x 
k represents the environmental capacity.
c represents the number of consumers.
(Decreasing A increases both k and c.)
(2C)
1  x 
dx
x 
 c x  1   
2
dt
 c  k  1 x 
(2C)
1  x 
dx
x 
 c x  1   
2
dt
 c  k  1 x 
1 x
x
1   
2
c  k  1 x
The resource increases
(2C)
1  x 
dx
x 
 c x  1   
2
dt
 c  k  1 x 
1 x
x
1   
2
c  k  1 x
The resource increases
x
1 x
 1  
2
1 x
c k
The resource decreases
A “Textbook” Example
Line above curve:
Population increases
c=1
0.5
0.4
1 x
x
1   
2
c  k  1 x
0.3
y
0.2
0.1
0
0
2
4
6
v
8
10
A “Textbook” Example
Line above curve:
Population increases
c=1
0.5
0.4
1 x
x
1   
2
c  k  1 x
0.3
y
0.2
0.1
0
0
2
4
6
v
Low consumption –
high resource level
8
10
A “Textbook” Example
c=3
Curve above line:
Population decreases
0.5
0.4
x
1 x
 1  
2
1 x
c k
0.3
y
0.2
0.1
0
0
2
4
6
v
8
10
A “Textbook” Example
c=3
Curve above line:
Population decreases
0.5
0.4
x
1 x
 1  
2
1 x
c k
0.3
y
0.2
0.1
0
0
2
4
6
v
High consumption –
low resource level
8
10
A “Textbook” Example
c=2
0.5
0.4
0.3
y
0.2
0.1
0
0
2
4
6
8
v
Modest consumption –
two possible resource levels
10
A “Textbook” Example
Population stays low if x<2
(curve above line)
c=2
0.5
0.4
0.3
y
0.2
0.1
0
0
2
4
6
8
v
Modest consumption –
two possible resource levels
10
A “Textbook” Example
c=2
0.5
0.4
Population becomes large if x>2
(line above curve)
0.3
y
0.2
0.1
0
0
2
4
6
8
v
Modest consumption –
two possible resource levels
10
(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
Consumption increases to modest level.
x