restart:Math 152-001 Lab 10/12/04 Michael Pelsmajer and Allen FlavellTry to finish this by the end of class, or Friday. The due date isTuesday at 5pm, so that in case you run into trouble, you have
Monday and Tuesday to get help. If you haven't already, save this file as "[your-name]-Feb24lab".
Save often! When you are done, revise your work so that it is easy for Allen to find and understand your solutions. (Also, for yourself, since you may need to come back to your earlier work.)To use the Maple commands for differential equations, first run the following command.with(DEtools):1. Whether or not we can solve a particular differential equation, we can use direction fields and Euler's Method to get some information about the solutions. Maple makes this much easier, for example:model1 := diff(y(x),x)=(exp(x)-2*y(x))/x;Here is how to make a direction field. DEplot( model1, y(x), x=-2...3, y=-2..3);To plot some solutions, define a list of initial values.initvals1 :=[[y(-2)=0.3],[y(0.1)=2], [y(-2)=0],[y(0.7)=-2]];
DEplot( model1, [y(x)], x=-2..3, y=-2..3, initvals1);The plot only has so much accuracy, and this makes trouble in two ways: a curve looks piecewise linear (it shouldn't be), and Maple can't compute values near the vertical asymptote. We can add (or lessen) the plot accuracy with "numpoints".DEplot( model1, [y(x)], x=-2..3, y=-2..3, [y(0.1)=2], numpoints=130);More importantly, the solutions are actually generated numerically, using methods like Euler's method. We can ask Maple to use Euler's method, and choose the step size. For example:DEplot( model1, [y(x)], x=-2..0, y=0..2, [[y(-2)=0.3]], method=classical[foreuler], stepsize=.4);Now your turn.Note that while the initial values y(-2)=0 and y(-2)=0.3 are close, but their behavior is quite different (one goes up, the other goes down).So solutions through y(-2)=c can have different behavior, depending on c.1.a.Using plots, find the exact value of c (accurate to three decimal places) where the behavior changes.
Show one plot with two solutions, with initial points with 0.01 of y=(-2)=c, with different global behavior.
2. Recall that this is the logistic equation that models a population with an environment that supports up to 1000 beasts.
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 A constant rate of decrease from the outside, say, by hunting 18 beasts/week, can be added to the model like this.
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2.a. Let k=0.09. Plot a directions field with some solutions for the first D.E., and then do the same for the second D.E.
Be sure to show solutions of different types.
Use a t-range of 0..40, and a P-range of -10..1100. 2.b. You should see an obvious qualitative difference between the types of possible solutions for the first D.E. and for the second D.E.
Describe it. Then discuss why it makes sense that the "-18" had this effect.
3. There is considerable evidence to suppose the theory that for some species, there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducting the factor ( 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 ). Thus, the modified logistic equation is given by the differential equation:
diff(P(t),t)= k*P(t)*(1-P(t)/K)*(1-m/P(t));
3.a. For the case where k=0.08, K=1000, m=200, draw a direction field from t=0 to t=50, with an approximate solution for each of the starting populations: 1300, 600, 250, and 190.