[answered] Math 1215 Third Project - Due December 5 2016 General Instr

Is there an answer key available for the MATH 1215 Blue Whale Population Growth project?

Math 1215 Third Project ? Due December 5 2016 General Instructions This project is to be typed or hand-written, in a neat and legible way. (Messy

 

work can cost you up to 5 points.) Projects consisting of more than one page need to be stapled - unstapled

 

work will not be graded. You need to give the question numbers with your answers, and show your work.

 

For the questions involving excel you need to include a printout of the tables. Projects may be done in

 

groups of up to two students and the students don?t need to be in the same section of the course. Make

 

sure that the name of each author, the section number and the Banner numbers are clearly written on the

 

first page of the project.

 

Projects are due at 5 PM on December 5, 2016, at your instructor?s office, but may be handed in in

 

class any time before that. Blue Whale Population Growth

 

When we describe the growth of a population with a differential equation, we can also do this in terms of

 

a per capita production function, and just as in the case of a discrete dynamical system, this needs to be

 

multiplied by the size of the population. So if the per capita production function of a population w(t) of

 

whales is ?(w), then the differential equation is

 

dw

 

= ?(w) ? w

 

dt

 

whales per year.

 

We will explore the dynamics of this equation for the blue whale population. Assume that the blue

 

whale population at the Southern Ocean Whale Sanctuary has per capita production function

 

?(w) = 0.001(?0.01w2 + 8w ? 700).

 

1. Give the differential equation (without initial condition) for w(t), the blue whale population at the

 

Southern Ocean Whale Sanctuary. All the questions below refer to this equation.

 

2. To get a first impression of what this differential equation, answer the following questions:

 

(a) If w = 50, will the population be increasing or decreasing?

 

(b) If w = 200, will the population be increasing or decreasing?

 

(c) If w = 400, will the population be increasing or decreasing?

 

(d) If w = 1000, will the population be increasing or decreasing?

 

3. Find the equilibrium points for this differential equation (i.e., those numbers of blue whales for which

 

the population would not change).

 

4. For each equilibrium point found in the previous problem, determine whether it is stable or unstable.

 

5. The Southern Ocean Whale Sanctuary has 50 blue whales.

 

(a) Will this population survive?

 

(b) If not, what is the minimum number of blue whales they will need to get from another place

 

and add to their population so that the resulting population will not be guaranteed to die out?

 

(c) If they just get the minimal number is this a safe situation for their blue whale population or

 

could the population be threatened by extinction in case there is a year with fewer births or

 

premature deaths?

 

1 6. Southern decides to ask for a transfer of 55 blue whales from the Indian Ocean Whale Sanctuary.

 

(a) If they can keep these whales indefinitely, what do you predict about the growth of the population? Will it grow or decline?

 

i. If it grows, will it grow indefinitely, or will it reach a maximum?

 

7. Assuming Southern gets the 55 blue whales:

 

(a) Write the iteration formula that corresponds to the Euler?s method approximation for w(t) with

 

? (ti+1 ) that involves W

 

? (ti ) as in the book or you may

 

?t = .2 (you may give a formula for W

 

give a formula for Wi+1 in terms of Wi as in the notes on our BBLearn website).

 

(b) Use Excel and the formula from part (a) to estimate the blue whale population after 5 years.

 

8. Indian agrees to lend Southern the 55 blue whales provided that an equal number eventually be

 

returned to them. Based on your Excel table from the previous problem, what is the earliest time

 

Southern can do this without putting their population in danger? 2

 

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