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Problem Set 1 for 18.034: Logistic Model with Harvesting, Exercises of Differential Equations

This problem set includes four problems related to the logistic model with harvesting. Students are required to find equilibrium solutions, determine maximum values of harvesting rates, and find exact solutions to differential equations. The set also includes exercises from the textbook.

Typology: Exercises

2011/2012

Uploaded on 08/07/2012

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18.034 PROBLEM SET 1
Due date: Friday, February 13 in lecture. Late work will be accepted only with a medical note
or for another Institute-approved reason. You are strongly encouraged to work with others, but the
final write-up should be entirely your own and based on your own understanding.
Problem 1(20 points) The logistic model for a fish population with harvesting (p. 17) leads to
the following IVP:
y = ay cy2 H,
y(0) = y0
Here a and y0 are positive and c and H are nonnegative. The IVP is defined on the interval (0, ).
Also, the model is only valid as long as y(t) 0: If at any instant t1 (greater than 0) y(t1) equals
0, then the population is extinct, and the population will remain extinct for all t t1.
(a)(10 points) The equilibrium solutions are the solutions of the ODE (without the initial condi-
tion) for which y(t) = 0 for all t. Find inequalities among a, c, and H that determine when there
will be 2 equilibrium solutions, 1 equilibrium solution, or no equilibrium solutions.
(b)(10 points) Suppose that both a and c are positive. What is the maximum value of H for
which there is an equilibrium solution? If H is larger than this value, what is the long-term behavior
of any solution of the ODE?
Problem 2(20 points) After a change of variables, the logistic equation with harvesting reduces
to the following IVP (neglecting the extinction issue),
x = x2 + K,
x(0) = x0 > 0
where x = x(t) and where K is a constant. Suppose that K = b2 for some b > 0.
(a)(10 points) Formally rewrite the ODE as f(x)dx = g(t)dt and integrate to find an exact
solution. Express your answer in the form b x = h(t) for some expresion h(t). Don’t forget the
special case x0 = b.
(b)(10 points) At some instant t1, the value of x(t1) is very close to b. At that instant, the value
of b in the differential equation is abruptly increased to a larger value b1, and x(t) gradually moves
from the value b to the value b1. Assuming b1 b is small compared to b, approximately how much
time τ elapses before the difference b1 x(t1 + τ) is one half of the initial difference b1 b?
(c)(0 points not to be written up/handed in). Critical ecosystem double whammy. Interpret
your answer from (b). In particular, if the parameters a, c and H are near the critical value for
extinction, does the system respond more quickly or less quickly to a decrease in H than if the
parameters are far from the critical value?
Problem 3(5 points) Exercise 14, p. 49.
Problem 4(5 points) Exercise 20, p. 49.
1
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18.034 PROBLEM SET 1

Due date: Friday, February 13 in lecture. Late work will be accepted only with a medical note or for another Instituteapproved reason. You are strongly encouraged to work with others, but the final writeup should be entirely your own and based on your own understanding.

Problem 1(20 points) The logistic model for a fish population with harvesting (p. 17) leads to the following IVP: y�^ = ay − cy^2 − H, y(0) = y 0

Here a and y 0 are positive and c and H are nonnegative. The IVP is defined on the interval (0, ∞). Also, the model is only valid as long as y(t) ≥ 0: If at any instant t 1 (greater than 0) y(t 1 ) equals 0, then the population is extinct, and the population will remain extinct for all t ≥ t 1.

(a)(10 points) The equilibrium solutions are the solutions of the ODE (without the initial condi tion) for which y�(t) = 0 for all t. Find inequalities among a, c, and H that determine when there will be 2 equilibrium solutions, 1 equilibrium solution, or no equilibrium solutions.

(b)(10 points) Suppose that both a and c are positive. What is the maximum value of H for which there is an equilibrium solution? If H is larger than this value, what is the longterm behavior of any solution of the ODE?

Problem 2(20 points) After a change of variables, the logistic equation with harvesting reduces to the following IVP (neglecting the extinction issue),

x�^ = −x^2 + K, x(0) = x 0 > 0

where x = x(t) and where K is a constant. Suppose that K = b^2 for some b > 0.

(a)(10 points) Formally rewrite the ODE as f (x)dx = g(t)dt and integrate to find an exact solution. Express your answer in the form b − x = h(t) for some expresion h(t). Don’t forget the special case x 0 = b.

(b)(10 points) At some instant t 1 , the value of x(t 1 ) is very close to b. At that instant, the value of b in the differential equation is abruptly increased to a larger value b 1 , and x(t) gradually moves from the value b to the value b 1. Assuming b 1 − b is small compared to b, approximately how much time τ elapses before the difference b 1 − x(t 1 + τ ) is one half of the initial difference b 1 − b?

(c)(0 points – not to be written up/handed in). Critical ecosystem double whammy. Interpret your answer from (b). In particular, if the parameters a, c and H are near the critical value for extinction, does the system respond more quickly or less quickly to a decrease in H than if the parameters are far from the critical value?

Problem 3(5 points) Exercise 14, p. 49.

Problem 4(5 points) Exercise 20, p. 49. 1

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