Modeling an Ecological

System

By: Alex(UW-La Crosse), Becca(KSU), Jessica(MU),

Justin(MU), and Victor(Bingrui)(UMASS Amherst)

California is currently experiencing a water crisis due to ● lack of annual rainfall ● overuse of groundwater from

aquifers● groundwater contamination ● fracking

Motivation: California Water Crisis

http://www.forbes.com/sites/williampentland/2014/02/04/as-water-supply-reaches-record-low-california-combats-drought-with-black-ops-weather-control-technology-from-vietnam-war/

Goals:

● To find a reasonable model that accurately depicts the relationship between farmers, plants, and water.

● Infer based on the data, how the water crisis may be affecting the system.

● Consider ways to solve the water crisis using mathematical and statistical techniques.

● Water has a seasonal component

● Plants depend on water

● Farmers are independent of water

● Farmers depend on plants

Our Story and Model Assumptions

Understanding the Variables

Farmer and water is not corrolated

Data Overview

LegendData1_lowData2Data3

Summary of Data

Original Data Set (Data1_Low)

1 = Water2 = Plants3 = Farmers

R2 Value: 0.9019

Second Data Set (Data2)

1 = Water2 = Plants3 = Farmers

Poor Models

R2 Value: 0.01242R2 Value: 0.02234

Bad Forms of Fit

● Linear Regression○ Doesn’t show flow of graph (highs and lows)

● Quadratic Regression○ Only represents frames of early data○ Does not accurately depict the data's trend well

(based on its continuous oscillating nature)

Using Fourier Series to Approx. Data

● Every continuous function can be written as a linear combination of sin and cos

● We used Fourier series, a combination of sines and cosines in order to fit the given data based on the amplitude and period of the curves.

● Advantage - derivative of the function has the same basis as the original function

R2 Value: 0.95

Model for Plants vs. Time

Model for Water vs. Time

R2 Value: 0.84

Model for Farmer vs. Time

R2 Value: 0.29

Best Fit Equations for Data

Graph of the Derivatives of the Best Fit Equations

Finding ODEs

● After using Fourier series to approximate the data of the water, plants, and farmers, we next determined coefficients to create differential equations which model the relationships between the three variables.

● We accomplished this using a computer program in R...

w’(t) = g (p, t)p’(t) = g (w, p, f)

f’(t) = g (p, f)

● Predator is dependent on a single prey ● Prey has an unlimited food supply*● There is no threat to the prey other than the predator

Lotka-Volterra Predator and Prey Model

x is the number of prey ;

y is the number of some predator ex: dx/dt = ax is the term from population dynamic -bxy is the death rate from interaction

Interrelationship

Results

Parameters: Estimate Std. Errora 4.967104 0.045907 b 2.069681 0.017873 c 2.925592 0.036546 d 4.565030 0.056859 e 2.570853 0.015306 f 0.200235 0.001184

dw=e*water*plant+0.04985*sin(1.745*t)+0.03364*cos(1.745*t)+f

dp = a*water*plant-b*plant*farmer

db = -c*farmer+d*plant*farmer

Residual standard error: 0.07783

Conclusions

● All variables are a function of time● There is a delayed change of plants as water

changes.● The farmers also have a delay in response.● We can’t change one without impacting the

others.

Limitations● Not able to capture all variables in

ecosystem● The assumptions are limiting

● Create plans to help deal with the water crisis that works best both financially and agriculturally

Future Research

Learning

From doing this project, we learned about the applications of math and statistics in real world environment. We learned how to take data and analyze it to form a differential equation that describes the system.

References● Aquifer. Digital image. Aquifer. Wikipedia, n.d. Web. 21

May 2015. <http://en.wikipedia.org/wiki/Aquifer>.● Foley, Kaye. "California's Water Crisis." Yahoo! News.

Yahoo!, n.d. Web. 21 May 2015. <http://news.yahoo.com/california-s-water-crisis-drought-katie-couric-explains-182006167.html>.

● Sternberg, Shlomo. "Lotka-Volterra." 19 Apr. 14. Lecture. <http://www.math.harvard.edu/library/sternberg/slides/11809LV.pdf>.

AcknowledgementsDr. Daniel Taylor-RodriguezDr. Kimberly KaufeldDr. Lea JenkinsThomas GehrmannNC State UniversitySAMSI

Thank You!

Questions?