The Past, Present, and Future of Endangered Whale Populations: An Introduction to Mathematical Modeling in Ecology Glenn Ledder University of Nebraska-Lincoln Supported by NSF grant DUE

Outline 1.Mathematical Modeling A.What is a mathematical model? B.The modeling process 2.A Resource Management Model A.The general plan for the model B.Details of growth and harvesting C.Analysis of the model D.Application to whale populations

(1A) Mathematical Model Math Problem Input DataOutput Data Key Question: What is the relationship between input and output data?

Rankings in Sports Mathematical Algorithm Ranking Game Data: determined by circumstances Weight Factors: chosen by design Game Data Weight Factors

Rankings in Sports Mathematical Algorithm Ranking Game Data Model Analysis: For a given set of game data, how does the ranking depend on the weight factors? Weight Factors

Endangered Species Mathematical Model Control Parameters Future Population Fixed 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 Modeling Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation

(1B) Mathematical Modeling Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation A mathematical model represents a simplified view of the real world.

(1B) Mathematical Modeling Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation A mathematical model represents a simplified view of the real world. Models should not be used without validation!

Example: Mars Rover Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation Conceptual Model: Newtonian physics

Example: Mars Rover Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation Conceptual Model: Newtonian physics Validation by many experiments

Example: Mars Rover Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation Conceptual Model: Newtonian physics Validation by many experiments Result: Safe landing

Example: Financial Markets Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation Conceptual Model: Financial and credit markets are independent Financial institutions are all independent

Example: Financial Markets Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation Conceptual Model: Financial and credit markets are independent Financial institutions are all independent Analysis: Isolated failures and acceptable risk

Example: Financial Markets Real World Conceptual Model Mathematical Model approximationderivation analysisvalidation 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 Conceptual Model Mathematical Model approximationderivation analysisvalidation 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 (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 (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. Overall rate of increase = growth rate – consumption rate

Logistic growth – Fixed environment capacity K R Relative growth rate (2B)

Holling type 3 consumption – Saturation and alternative resource

The Dimensional Model Overall rate of increase = growth rate – consumption rate

The Dimensional Model Overall rate of increase = growth rate – consumption rate This model has 4 parameters—a lot for analysis! Nondimensionalization reduces the number ofparameters.

The Dimensional Model Overall rate of increase = growth rate – consumption rate This model has 4 parameters—a lot for analysis! Nondimensionalization reduces the number ofparameters. X/A is a dimensionless population; RT is a dimensionless time.

The Dimensional Model Overall rate of increase = growth rate – consumption rate This model has 4 parameters—a lot for analysis! Nondimensionalization reduces the number ofparameters. X/A is a dimensionless population; RT is a dimensionless time.

Dimensionless Version

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.)

(2C)

The resource increases (2C)

The resource increases The resource decreases (2C)

A “Textbook” Example Line above curve: Population increases

A “Textbook” Example Low consumption – high resource level Line above curve: Population increases

A “Textbook” Example Curve above line: Population decreases

A “Textbook” Example High consumption – low resource level Curve above line: Population decreases

A “Textbook” Example Modest consumption – two possible resource levels

A “Textbook” Example Modest consumption – two possible resource levels Population stays low if x<2 (curve above line)

A “Textbook” Example Modest consumption – two possible resource levels 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

Stage 2 – depletion Consumption increases to high level.

Stage 3 – inadequate correction Consumption decreases to modest level.

Stage 4 – recovery Consumption decreases to minimal level.

Stage 5 – proper management Consumption increases to modest level.