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 PresentationIntroduce 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:
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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 - )
12021281
1218
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
100200300400500600700800
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
Wim
Westera, Open Univ. of Netherlands
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 ModelProblem 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….
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