Modified aphid-ladybug dynamics In Exercise 22 we modeled populations of aphids and

Chapter 7, Problem 24

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Modified aphid-ladybug dynamics In Exercise 22 we modeled populations of aphids and ladybugs with a LotkaVolterra system. Suppose we modify those equations as follows: dA dt 2As1 2 0.0001Ad 2 0.01AL dL dt 20.5L 1 0.0001AL (a) In the absence of ladybugs, what does the model predict about the aphids? (b) Find an expression for dLydA. CAS (c) Use a computer algebra system to draw a direction field for the differential equation in part (b). Then use the direction field to sketch a phase portrait. What do the phase trajectories have in common? (d) Suppose that at time t 0 there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change. (e) Use part (d) to make rough sketches of the aphid and ladybug populations as functions of t. How are these two graphs related?

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