MATH 416 Final: MATH414 BOYLE-M FALL2004 0101 FINAL EXAM

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turquoisegnu128

15 Feb 2019

School

University of Maryland

Department

Mathematics

Course

MATH 416

Professor

All

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Final exam math 414 fall 2004. For this test, you may use our text (brauer and nohel) and one page of notes. In your proofs and explanations, you do not need to (you can use both sides). reprove results you can cite in the text: judge each of the following statements to be true or false. You must justify your answers. (a) (15) there exists > 0 such that there is a unique function (t) from the interval ( , ) into r which satis es the conditions. 1(0) = 2(0) = 1 . (c) (10) there exists a homogeneous linear di erential equation for which the matrix (cid:18) et et. 2et et + 1(cid:19) is a fundamental matrix of solutions de ned on the entire real line: let y1, y2 denote functions from r to r. for the di erential equation y2(cid:19) (cid:18) y1.

Related Questions

A classical model in mathematical ecology is the LotkaâVolterra predator-prey model. Consider a simple ecosystem consisting of rabbits that have an infinite supply of food and foxes that prey on the rabbits for their food. This is modeled by a pair of nonlinear, first-order differential equations:

where t is time, r(t) is the number of rabbits, f(t) is the number of foxes, and Î± is a positive constant. If Î± = 0, the two populations do not interact, the rabbits do what rabbits do best, and the foxes die off from starvation. If Î± > 0, the foxes encounter the rabbits with a probability that is proportional to the product of their numbers. Such an encounter results in a decrease in the number of rabbits and (for less obvious reasons) an increase in the number of foxes. The solutions to this nonlinear system cannot be expressed in terms of other known functions; the equations must be solved numerically. It turns out that the solutions are always periodic, with a period that depends on the initial conditions. In other words, for any r(0) and f(0), there is a value t = t_p when both populations return to their original values. Consequently, for all t,

(a) Compute the solution with r₀ = 300, f₀ = 150, and Î± = 0.01. You should find that t_p is close to 5. Make two plots, one of r and f as functions of t and one a phase plane plot with r as one axis and f as the other.

(b) Compute and plot the solution with r₀ = 15, f₀ = 22, and Î± = 0.01. You should find that t_p is close to 6.62.

(c) Compute and plot the solution with r₀ = 102, f₀ = 198, and Î± = 0.01. Determine the period t_p either by trial and error or with an event handler.

(d) The point (r₀, f₀) = (1/Î±, 2/Î±) is a stable equilibrium point. If the populations have these initial values, they do not change. If the initial populations are close to these values, they do not change very much. Let u(t) = r(t)â1/Î± and v(t) = f(t) â 2/Î±. The functions u(t) and v(t) satisfy another nonlinear system of differential equations, but if the uv terms are ignored, the system becomes linear. What is this linear system? What is the period of its periodic solutions?

Hi, I need the full Matlab code and answers for all parts of the problem. Thanks!

MATH 416 Final: MATH414 BOYLE-M FALL2004 0101 FINAL EXAM

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