The Clear Complexity of Lake Conditions Through Herbivores and Mathematical Models

LC Science Gang April 21, 2010

Doug Thomasey and Ashley Palmer

Goals of this Presentation

Introduce you to mathematical modeling

Learn how to construct a model

Apply the idea of a model to some real world phenomenon, Timber Lake in Campbell County

Mathematical Model

One or more equations used to describe the behavior of some system or phenomenon in mathematical terms.

Used in science, engineering, economics, and even psychology.

Where Models are used

Compound interest Liberal Arts Math

Carbon dating Pre-Calculus

Pendulum Trigonometry

Depth of water in a draining tank Calculus 1

Population growth/Spread of disease

Biology

Error Response Times Psychology

Dirt bike suspension systems Engineering

Suspension

Dr. J. R. White, UMass-Lowell (Spring 1997)

Some Types of Models

Linear: y = mx + b

Quadratic:

Exponential:

Differential:

www.szavay-blog.com

Parameters

Numerical values used in a model

Major influence on how model functions

Determined from observations (experiments) and assumptions about the system

Linear Model Parameters In a Linear Model (y=mx+b), you have

two ‘parameters’ m and b

When comparing temperature (°F °C) empirical evidence shows: 32°F= 0°C and 212°F= 100°C, so:

Scoring the DecathlonDay 1 – 100m, Long Jump, Shot Put, High Jump, 400mDay 2 – 110 hurdles, Discus, Pole Vault, Javelin, 1500m

Athletes are awarded points based on their PERFORMANCE in each of the 10 events

In a decathlon with 3 or more competitors, it is possible to win overall without winning any individual event.

Chasing points: 100m: 11.00 – 861pts, 12.00 – 651pts (world record 9.58 - )

PV: 14 ft – 693pts, 15 ft – 781pts (world record 20’2” - )

1500: 5:00 – 560pts, 4:30 – 745pts (world record 3:26.0 - )

1202

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What Type of Model is Used? Linear?

No▪ An increase from 65’ to 70’ is much harder to achieve

than and increase from 20’ to 25’, but yet the same increase in points would result.

20 25 30 35 40 45 50 55 60 65 700

100

200

300

400

500

600

700

800

Points for Shot Put

feet

Decathlon Model

x, y, and z are set parameter values that change for each event.

These parameter values are the basis for the model to function.

Running events

All other events

Long Jump Points Graph

An increase from 2.5 to 3m has a 50 pt increase An increase from 6.5 to 7m has a 117 pt increase

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The Beauty of Modeling

World Record Holder in the Decathlon becomes

How?

Eventwise function Piecewise function

NEW WORLD RECORD!!! Doug Thomasey – 9,027 pts.

Differential Equation Models

Definition: an equation containing the derivatives or differentials of one or more dependent variables with respect to one or more independent variables

Change continuously with time

Constructing a model

1) Identify the variables that are changing the system

2) Set reasonable assumptions about the system (parameter values depend on these assumptions)

3) Using any other empirical information, determine the DE or system of DE’s that would make up the model

Finding Parameter Values

Most of the time come through experiments or literature Example: Death rate of a population represented by

letter μ

If the death rate is low, then the value for μ will be small

If the death rate is high, then the value for μ will be high

Typically these values range from 0-1, based on a percentage

Visual Parameter Explanation

μ =0.16 μ =0.8

Smith Mountain Lake Model

Problem at hand: Striped Bass infected with parasite, Achtheres. We want to determine how long it would take for the entire population to become infected (if ever) given that 1 infected fish is introduced to the population. We will also assume that there is no fish reproduction, once a fish is infected, it will always be infected, and that we will stock 21 fish per acre per year into the lake.

Smith Mountain Lake Model

s: susceptible fish i: infected fish b: encounter rate d: death rate of susceptible v: death rate of infected fish

Results from SML

Many factors can be solved algebraically Equilibrium, reproduction numbers, etc.

Mathematical Software Mathematica Maple Derive Stella (Visual)

Results from SML

Y-axis: number of striped bass X-axis: months after one infected fish is introduced

Death Rate Parameter

μ =0.25 μ =0.75

Harder Model

10 Parameter Values? bx,η1,η2,η3,μ,ψ,β1,β2,α1,α2

Use a separate computer program to create lists of possibilities

www.math.hmc.edu

Timber Lake

Lacustrine system 65 acres in area, or 26 hec-3045.6

m² Soil type is mineral based and is

mainly saturated especially along the shore lines of the lake itself

Vegetation coverage: Typha, Najas guadalupensis, Algae

Unconsolidated bottom

Inflows: 3 separate streams flowing into the lake

Waterlick Brown Buffalo

Brown

Outflow: Spillway by the Dam

Formation of Timber Lake

Timber Lake

Concerns of Timber Lake

Phosphorus loading

Najas guadalupensis

Herbivores

Phosphorus and Clarity

Phosphorus is a nutrient required by all organisms for the basic processes of life.

Phosphorus is a natural element found in rocks, soils, and organic material. Abundant amount of Phosphorus▪ Creates Eutrophic conditions

Lack of Phosphorus▪ Decreasing zooplankton population and plant

life

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Testing Clarity

Secchi Disk

Lowered into water, until it is no longer visible

After time, you are able to compare a reading to previous readings.

Turbidity

Difference in Clarity

Lake Tahoe College Lake

T. Shahady

Phosphorus Cycle

Data Collected

Sampled March-December 2009 Hydrolab readings Zooplankton collection Phosphorus collection

Use Stella to Model the outlook for Phosphorus levels in Timber Lake Determine parameter values from data

collected

Stella (Easy)

Timber Lake Stella Model

Stella Graphs

Managing Phosphorus

BMPs are the most sufficient way to go about the preservation of the lake.

Keep a close eye on septic tanks Look into “lakescaping”- helps

reduce fertilizers that may run into the lake.

Become familiar with development within your watershed that might inhibit the activity that goes on within your lake.

Keeping a lake clear is more complex than most people think….