Call Now to Set Up Tutoring:

(888) 888-0446

Math Modeling : Chaos and Fractals

Study concepts, example questions & explanations for Math Modeling

Create An Account

All Math Modeling Resources

20 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Dynamic Models

There are two similar species of monkey that inhabit the same forest. The intrinsic growth rate is 4% for species A and 7% for species B. The capacity of the forest is estimated at 125,000 specie A monkeys and 450,000 species B monkeys. Poaching these monkeys over the last 100 years has caused for a decline in each species. Species A has a population of 4,000 and species B, a population of 80,000. Use a discrete model to project the population growth with a time step of several years. How large a time step can be used to maintain a continuous time model with an equivalent qualitative behavior?

Assume $\alpha =10^{-8}$ , x_b=100

Possible Answers:

h=57

h=50

h=47

h=30

h=37

Correct answer:

h=47

Explanation:

Identify the known information and the dynamical system.

$\\A=\text{number of species A monkey} \\B=\text{number of species B monkey} \\g_A=\text{growth rate of species A per year} \\g_B=\text{growth rate of species B per year} \\c_A=\text{competition on species A} \\c_B=\text{competition on species B} \\\\g_A=0.04A(1-A/125,000) \\g_B=0.07B(1-B/450,000) \\c_A=c_B=\alpha AB \\A\geq 0, b\geq 0 \\\alpha \epsilon \mathbb{R}>0$

$\\\frac{dx_1}{dt}=f_1(x_1,...,x_n) \\\vdots \\\\\frac{dx_n}{dt}=f_n(x_1,...,x_n)$

The dynamical system equation is:

$\frac{dx}{dt}=F(x)$

From here solve by using the Euler Method for a discrete time dynamical system.

$\frac{\Delta x}{\Delta t}=F(x)$

Now formulate the model

$\\\frac{dx_1}{dt}=f_1(x_1,x_2)=0.04x_1\left(1-\frac{x_1}{125,000} \right )-\alpha x_1x_2 \\\\\frac{dx_2}{dt}=f_1(x_1,x_2)=0.07x_1\left(1-\frac{x_1}{450,000} \right )-\alpha x_1x_2$

$\\x_1=\text{species A monkey} \\x_2=\text{species B monkey}$

Now transform these equations into a set of difference equations.

$\\\Delta x_1=f_1(x_1,x_2)\Delta t \\\Delta x_2=f_2(x_1,x_2)\Delta t$

$\\\Delta x_1= \text{change in the species A monkey population} \\\Delta x_2= \text{change in the species B monkey population}$

Assume

$\\\alpha =10^{-8} \\x_1(0)=4,000 \\x_2(0)=80,000\\N=50$

where represents the number of iterations the computer will use when implementing the Euler Method.

The time step $h=\Delta t$ will need to be changed and the model reformulated to calculate how large the time step can be before it doesn't resemble a continuous time model.

Using computer implementation of the Euler Method on the population of species B monkeys the following graph is produced.

Screen shot 2016 05 05 at 6.06.50 am

It is seen that h=47 is the largest time step before the function begins to have chaos and not represent the behavior of a continuous system.

Report an Error

View Tutors

Manuel O.
Certified Tutor

Pedagogical Institute - Cuba, Bachelor of Science, Physics.

View Tutors

Heidi L.
Certified Tutor

Metropolitan State University, Bachelor in Arts, Language Arts Teacher Education. Hamline University, Certificate, Teaching E...

View Tutors

Richard
Certified Tutor

CUNY Brooklyn College, Bachelor of Science, Chemistry. CUNY Brooklyn College, Master of Arts, Experimental Psychology.

All Math Modeling Resources

20 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GMAT Tutors in San Francisco-Bay Area, SSAT Tutors in Los Angeles, Computer Science Tutors in San Francisco-Bay Area, Spanish Tutors in Washington DC, SAT Tutors in Los Angeles, GMAT Tutors in Boston, GMAT Tutors in Philadelphia, Computer Science Tutors in Los Angeles, Chemistry Tutors in Seattle, Calculus Tutors in Boston

Popular Courses & Classes

LSAT Courses & Classes in Denver, MCAT Courses & Classes in New York City, Spanish Courses & Classes in Atlanta, SAT Courses & Classes in Chicago, GRE Courses & Classes in Seattle, LSAT Courses & Classes in Phoenix, MCAT Courses & Classes in Washington DC, Spanish Courses & Classes in Philadelphia, GMAT Courses & Classes in Houston, GMAT Courses & Classes in Phoenix

Popular Test Prep

LSAT Test Prep in Miami, SAT Test Prep in Phoenix, ISEE Test Prep in Boston, ISEE Test Prep in Miami, GRE Test Prep in Dallas Fort Worth, GMAT Test Prep in Houston, GRE Test Prep in Houston, GRE Test Prep in Washington DC, ISEE Test Prep in New York City, ISEE Test Prep in Dallas Fort Worth

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Math Modeling : Chaos and Fractals

Study concepts, example questions & explanations for Math Modeling

All Math Modeling Resources

Example Questions

Example Question #1 : Dynamic Models

All Math Modeling Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: