George Washington University Differential Equation Problems Paper total of 6 questions
You have to show all your work.
One of the questions:
5. (All parts paper and pencil only. You may use Mathematica to confirm your solutions. However, you cannot use software as a substitute for computations.) The Lotka Volterra predator-prey model discussed in class assumes that in the absence of predators the number of prey grows exponentially. If we make the alternative assumption that the prey population grows logistically, the new system is
dx/dt = r /K (x(K ? x) )? ?xy
dy/dt = ?cy + ?xy
where x(t) is the population of the prey, y(t) is the population of predator r, K, ?, c and ? are positive and K > a b . (a) (5 points) Find all critical points. (b) (15 points) For each critical point, find the corresponding linear system. Find the eigenvalues of each linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable.
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