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.