Understanding Endangered Whale Populations through Mathematical Modeling

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Explore the past, present, and future of endangered whale populations through qualitative analysis and mathematical modeling. Delve into resource management models, input data analysis, species control parameters, and the importance of managing natural resources for the conservation of whales. Learn about the relationship between input and output data, future population dependence on control parameters, and the general biological resource model.

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The Past, Present, and Future of Endangered Whale Populations: Qualitative Analysis and Modeling Glenn Ledder University of Nebraska-Lincoln https://www.math.unl.edu/~gledder1 gledder@unl.edu

1. Mathematical Modeling 1. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations

Mathematical Model Input Data Math Problem Output Data Key Question: What is the relationship between input and output data?

Endangered Species Fixed Parameters Mathematical Model Future Population Control Parameters Model Analysis: For a given set of fixed parameters, how does the future population depend on the control parameters?

1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations

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?

1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations

General Biological Resource Model Let X be the biomass of resources. Let T be the time. Let G(X) be the resource growth rate. Let H(X) be the consumption per consumer. Let C be the (fixed) number of consumers. ?? ??= ?(?) ? ?(?) Overall rate of increase = growth rate consumption rate

1. Mathematical Modeling 1. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations

Growth Model Assume logistic growth ?? ??= ?? 1 ? ? ?(?) ? Overall rate of increase = growth rate consumption rate

Consumption Model Holling type 3 model Saturation and alternative resource Q 0.75Q ??2 ?2+ ?2 ?(?) = G 0.5Q 0.25Q 0 0 A 2A X 3A 4A

The Full Model ??2 ?2+ ?2 ?? ??= ?? 1 ? ? ? Overall rate of increase = growth rate consumption rate

1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations

Nondimensionalization ??2 ?2+ ?2 ?? ??= ?? 1 ? ? ? Overall rate of increase = growth rate consumption rate This model has 5 parameters a lot for analysis! Nondimensionalization reduces the number of parameters. 1. Replace X by x = X/Kand T by t = RT. 2. Define new dimensionless parameters.

Dimensionless Version ? =? 1 ? ?, ? = ??, ? = ???, ? = ??? 1 ? ?? ??= 1 ? ?? ? + ?2 p represents the processing-discovery ratio. c represents the number of consumers.

1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations

Qualitative Analysis ?? ??= ? ? ? ? + ?? ?? ? + ?2 ? ? = (? ?)(? + ??) represents growth ? ? = ?? represents consumption 1 ? ? + ?2> ?? The resource increases 1 ? ? + ?2< ?? The resource decreases

Example: p=0.01 The consumption function is ?? The growth function is (? ?)(? + ??)

Example: p=0.01, c=0.12 Curve above line 1 ? ? + ?2> ?? Growth > Consumption Population increases Low consumption high resource level

Example: p=0.01, c=0.36 Curve below line 1 ? ? + ?2< ?? Growth < Consumption Population decreases High consumption low resource level

Example: p=0.01, c=0.24 If ? ?< ? < ?.??: Growth > Consumption Population increases If ?.?? < ? <? ?: Growth < Consumption Population decreases Moderate consumption two resource levels

1. Mathematical Modeling 2. A Resource Management Model A. The general plan for the model B. Details of growth and harvesting C. Simplification D. Analysis of the model E. Application to whale populations

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 can be thought of as more consumers.

Stage 1 natural balance Low consumption High population

Stage 2 depletion Consumption increases Population decreases and then crashes

Stage 3 inadequate correction Consumption decreases Population increases, but only a little Consumption sustainable, but population can t recover

Stage 4 recovery Consumption decreases Eventually only a high population is stable We need population above the unstable equilibrium before we can increase consumption.