1. LC Science Gang April 21, 2010 Doug Thomasey and Ashley Palmer

2.  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

3.  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.

4.  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

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

6.  Linear: y = mx + b  Quadratic:  Exponential:  Differential: www.szavay-blog.com

7.  Numerical values used in a model  Major influence on how model functions  Determined from observations (experiments) and assumptions about the system

8.  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:

9.  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 - )world record  PV: 14 ft – 693pts, 15 ft – 781pts (world record 20’2” - )world record  1500: 5:00 – 560pts, 4:30 – 745pts (world record 3:26.0 - ) 1202 1281 1218

10.  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.

11.  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

12.  An increase from 2.5 to 3m has a 50 pt increase  An increase from 6.5 to 7m has a 117 pt increase Wim Westera, Open Univ. of Netherlands

13.  World Record Holder in the Decathlon becomes

14.  Eventwise function  Piecewise function NEW WORLD RECORD!!! Doug Thomasey – 9,027 pts.

15.  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

16.  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

17.  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

18. μ =0.16μ =0.8

19.  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.

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

21.  Many factors can be solved algebraically  Equilibrium, reproduction numbers, etc.  Mathematical Software  Mathematica  Maple  Derive  Stella (Visual)

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

23.  μ =0.25  μ =0.75

24.  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

25.  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

26.  Inflows: 3 separate streams flowing into the lake Waterlick Brown Buffalo  Brown  Outflow: Spillway by the Dam

28.  Phosphorus loading  Najas guadalupensis  Herbivores

29.  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 slowmuse.wordpress.com

30.  Secchi Disk  Lowered into water, until it is no longer visible  After time, you are able to compare a reading to previous readings.  Turbidity

31.  Lake Tahoe  College Lake T. Shahady

33.  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

37.  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.