Calculus Online: Lab 4 Solutions

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Question 1a

Which of the following plots could be the graph of a solution of the differential equation y'=-y³-y?

From the graph to the right of -y3-y we can tell that 0 is a stable equilibrium, which rules out choices b) and c). Choice a) is impossible since it passes through y=0 with non-zero slope (and y'(0)=-03-0=0). This leaves choices d) and e). But in choice e), at time t=0, y' is close to zero. As time increases y decreases but y' also becomes more negative. From the graph to the right we see that as y becomes less positive, y' increases. Thus the only choice remaining is d), which is the correct answer.

a)

b)

c)

d)

e)

Question 1b

For the values of a equal to -1.0, -0.6, -0.2, 0.2, 0.6 and 1.0, place appropriate boxes or squares in the plot on the right corresponding to each of the steady solutions of the differential equation for that parameter value.

The answer comes from changing the value of the parameter on the right. The equlibria happen where the graph on the right passes through the y axis. If it passes through with negative slope, it is a stable equilibrium. If it passes through with positive slope it is an unstable equilibrium.

Question 2a

There are two bifurcations that occur between R=0.1 and R=2.

Enter the value of R at which these bifurcations appear into the boxes below.

As R increases through .46, the number of steady solutions changes from 1 to 3. So this is a bifurcation point. As R increases through .58, the number of steady solutions changes from 3 to 1. This is the other bifurcation point.

Question 2b

We have seen in the experiment that if R is increased to 0.58 the budworm population explodes, and that it doesn't come down when R is lowered to 0.57. So the question is, what is the largest value of R that will bring the population back down again to, say, a value of b less than 2?

There is an easy way and a hard way to answer this question. You can go back to the simulator and experiment like you did to see the population explode. (This is the hard way). You can also think about what change made the population explode and see from the plot of g(b)-p(b) the value of R that will reverse this change.

Enter the value of R at which the the population is brought under control into the box below.

From the diagram above, we see that as R passes through .58 the smallest stable solution vanishes. Solutions for R>0.58 will tend to the high level stable solution.

Solutions starting near the high level stable solution will tend to it. So we need to get rid of that solution. This happens at the other point of bifurcation (this was a question about bifurcations, after all), R=0.46. This is the correct answer.