8/4/2019 MATH 2930 - Worksheet 5 Solutions

http://slidepdf.com/reader/full/math-2930-worksheet-5-solutions 1/3

MA 2930, Feb 23, 2011

Worksheet 5 

1. Stability of critical points

For each of the following autonomous equations find its critical points, deter-mine if they are stable/unstable/semi-stable, and draw the phase line as well as qualitative sketches of representative solutions.

(1) dy/dt = y2(y − 1)(y − 2)(2) dy/dt = (1 + y2) arctan y(3) dy/dt = ey

2 − 2(4) dy/dt = sin2 y

(1) Critical points are the values of  y for which dy/dt = 0. Clearly theyare y = 0, 1 and 2 in this case.

To determine their stability we can either look at the sign of  dy/dt oneither side of them, or calculate f (y) and look at its sign at the critical point.Here the first method is easy to apply.

If  y < 0 (say y = −1), y > 0. If 0 < y < 1 (say y = 0.5), y > 0 aswell. So y = 0 is a semi-stable critical point: from below it attracts, butfrom above it repels.

If 1 < y < 2, y < 0. So y = 1 is stable. It attracts the flow from eitherside.

If  y > 2, y > 0. So y = 2 is unstable. The system flows away from it oneither side.

I’ll let you draw the phase line etc. because I don’t have a program fordrawing pictures!

(2) Critical points are where (1+y2) arctan y = 0, i.e., where arctan y = 0,i.e., y = 0.

It is clearly unstable because (1 + y2) > 0 always, whereas y < 0 if arctan y < 0 and y > 0 if arctan y > 0.

(3) Critical points are where ey2

= 2, i.e., y2

= ln 2, i.e., y = ±√ ln2.To determine stability let’s use f (y) = 2yey

2

.When y = −

√ ln2, f (y) = −2

√ ln 2eln 2 = −4

√ ln 2 < 0. So y(t) has a

minimum there, and therefore, y = −√ 

ln 2 is an unstable critical point.

1

8/4/2019 MATH 2930 - Worksheet 5 Solutions

http://slidepdf.com/reader/full/math-2930-worksheet-5-solutions 2/3

When y =√ 

ln2, f (y) = 2√ 

ln 2eln 2 = 4√ 

ln 2 < 0. So y(t) has a

maximum there, and therefore, y = √ ln 2 is an stable critical point.

(4) Critical points are where sin2 y = 0, i.e., sin y = 0, i.e., y = nπ wheren is any integer.

Since sin2 y is positive except at the critical points, it is clear that allthe critical points are semi-stable - attracting from below and repelling fromabove.

2. Population growth models

You have seen four models of population growth: exponential, logistic, thresh-

old and logistic-with-threshold. Make a comparison chart containing the dif- ferential equation, the phase line and a solutions sketch for each of them.

What would happen over time to the population in each of the four models above if the starting population is (a) very small, (b) very large? 

It should be easy for you to prepare the chart possibly after consultingthe book. I’ll only note that if the population starts out close to 0, it’ll growin the exponential and logistic models, but decline in the other two. If thestarting population is sufficiently large, it’ll grow further in the exponentialand threshold models and decline in the other two.

3. Sustainable harvesting

Consider the population  y of a species of tuna whose growth would obey the logistic model if it were left alone, but it is being harvested at a rate propor-tional to its population all the time. The resulting model is 

dy/dt = r(1− y/K )y − Ey

(a) What are equilibrium populations if (i) E > r, (ii) E < r? Are they stable or unstable? Draw the phase lines in each case.

(b) A sustainable yield  Y  is a rate at which fish can be caught indefinitely.It is the product of  E  and the stable equilibrium population  ye. Find  Y  and determine the value of  E  for which  Y  is maximum.

(a) To find the equilibrium populations we solve for critical points of the

2

8/4/2019 MATH 2930 - Worksheet 5 Solutions

http://slidepdf.com/reader/full/math-2930-worksheet-5-solutions 3/3