Call Now to Set Up Tutoring:

(888) 888-0446

Create an account to track your scores
and create your own practice tests:

Create an Account Create a Practice Test

All Math Modeling Resources

20 Practice Tests Question of the Day Flashcards Learn by Concept

Want to review Math Modeling but don’t feel like sitting for a whole test at the moment? Varsity Tutors has you covered with thousands of different Math Modeling flashcards! Our Math Modeling flashcards allow you to practice with as few or as many questions as you like. Get some studying in now with our numerous Math Modeling flashcards.

Flashcards: 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

h=47

h=57

h=37

h=50

h=30

Report an Error Copyright Notice

Flashcards: Dynamic Models

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

Your answer:

h=57

Your answer:

h=37

Your answer:

h=50

Your answer:

h=30

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.

Flashcards: Dynamic Models

In a meadow there are two types of wild flowers growing, a green floor crawler and yellow daisies. The more desirable flower is the yellow daisy as they can be picked and sold for bouquets. The yellow daisies are also the more slow growing flower of the two. The green floor crawlers grow more rapidly and consume more of the meadow land. Yellow daisies compete with the green flower crawlers by growing taller and shading the crawlers new growth. The yellow daisies are also more resistant to certain types of bugs. Can these two types of flowers coexist on one portion of the meadow, or will one over run the other, driving it to extinction?

Both flower will coexist.

The steady state analysis cannot answer the question.

The Yellow Daisies will go extinct.

Both flowers will thrive.

The Green Floor Crawler will go extinct.

Report an Error Copyright Notice

Flashcards: Dynamic Models

Your answer:

Both flower will coexist.

Your answer:

The Yellow Daisies will go extinct.

Your answer:

Both flowers will thrive.

Your answer:

The Green Floor Crawler will go extinct.

Correct answer:

The steady state analysis cannot answer the question.

Explanation:

Since there is competition between the two differing species of flowers, the growth rate function is,

g(P)=rP-aP^2

where is the intrinsic growth rate and a<<r measures the strength of the limitations of resources.

Now identify the question as a dynamic model in steady state.

In other words,

$\\f_1(x_1,...,x_n) \\f_2(x_1,...,x_n)$

From here, identify what is known about the problem in mathematical terms.

$\\D=\text{Yellow Daisy Population} \\C=\text{Green Floor Crawler Population} \\g_D=\text{growth rate of yellow daisies} \\g_C=\text{growth rate of green floor crawler population} \\c_D=\text{loss due to competition for yellow daisies} \\c_C=\text{loss due to competition for green floor crawler}$

$\\g_D=r_1D-a_1D^2 \\g_C=r_2C-a_2C^2 \\c_D=b_1CD \\c_C=b_2CD \\D\geq0, C\geq0 \\r_1,r_2,a_1,a_2,b_1, b_2 \text{are positive reals}$

To answer the question determine whether $C\rightarrow0$ or $D\rightarrow0$ .

To formulate the model let

$\\x_1=D \\x_2=C \\\begin{Bmatrix} (x_1,x_2):x_1\geq0, x_2\geq0 \end{Bmatrix}$

where the steady state equations are as follows.

$\\\\r_1x_1-a_1x_1^2-b_1x_1x_2=0 \\r_2x_2-a_2x_2^2-b_2x_1x_2=0$

For the particular problem, the points on interests are at those where the two functions intersect.

$\\(0,0) \\\left( 0,\frac{r_2}{a_2}\right) \\\left(\frac{r_1}{a_1},0\right)$

and the intersection of

$\\a_1x_1+b_1x_2=r_1 \\b_2x_1+a_2x_2=r_2$

Using Cramer's rule to solve results in the following.

$\\x_1=\frac{r_1a_2-r_2b_1}{a_1a_2-b_1b_2} \\\\\\x_2=\frac{a1_r2-b_2r_1}{a_1a_2-b_1b_2}$

and when the question of coexistence is ask it is found that,

$\frac{r_2}{a_2}<\frac{r_1}{b_1}$ and $\frac{r_1}{a_1}<\frac{r_2}{b_2}$

Which after further examination learns to the conclusion that the steady state analysis cannot answer the question.

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)

View Tutors

Mariam
Certified Tutor

Yarmouk University, Bachelor of Science, Mathematics. Touro College, Master of Science, Teacher Education, Multiple Levels.

View Tutors

Reema
Certified Tutor

Saint Louis University-Main Campus, Bachelor of Science, Mathematics Teacher Education. Webster University, Master of Arts Te...

View Tutors

Ryanto
Certified Tutor

Excelsior College, Albany NY, Bachelor, Business with Math Minor. University of Houston, Master's/Graduate, Accounting.

Popular Subjects

English Tutors in Los Angeles, Algebra Tutors in San Diego, Algebra Tutors in Miami, SAT Tutors in Seattle, Math Tutors in San Diego, Statistics Tutors in San Francisco-Bay Area, ISEE Tutors in Miami, English Tutors in Phoenix, Calculus Tutors in Dallas Fort Worth, GRE Tutors in Philadelphia

Popular Courses & Classes

ISEE Courses & Classes in Philadelphia, GMAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Boston, MCAT Courses & Classes in Houston, GRE Courses & Classes in San Diego, SAT Courses & Classes in San Diego, ACT Courses & Classes in Chicago, SSAT Courses & Classes in Boston, GRE Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Los Angeles

Popular Test Prep

ACT Test Prep in Boston, GMAT Test Prep in Phoenix, SSAT Test Prep in Washington DC, ACT Test Prep in Chicago, MCAT Test Prep in Houston, GRE Test Prep in San Diego, ISEE Test Prep in San Diego, LSAT Test Prep in Los Angeles, GRE Test Prep in Boston, SAT Test Prep in Houston

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Create an account to track your scores
and create your own practice tests:

All Math Modeling Resources

Flashcards: Dynamic Models

Flashcards: Dynamic Models

Flashcards: Dynamic Models

Flashcards: Dynamic Models

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback:

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Create an account to track your scores and create your own practice tests:

All Math Modeling Resources

Flashcards: Dynamic Models

Flashcards: Dynamic Models

Flashcards: Dynamic Models

Flashcards: Dynamic Models

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Create an account to track your scores
and create your own practice tests: