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(c) Here is a typical pair of differential equations that model the growth in population of two competing species x(t) and y(t):
x(t) = x — x2 — 0.5xy (4)
y(t) = y — y2 — 0.5xy.
The command dsolve can solve many pairs of ordinary differential equations — especially linear ones. But the mixture of quadratic terms in (4) makes it unsolvable symbolically, and so we need to use a numerical ODE solver as we did in the pendulum application. Using the commands in that application as a template, graph numerical212
Practice Set C: Developing Your MATLAB Skills 212
solution curves to the system (4) for initial data
x(0) = 0 : 1/12 : 13/12 y(0) = 0 : 1/12 : 13/12.
(Hint: Use axis to limit your view to the square 0 < x, y < 13/12.)
(d) The picture you drew is called a phase portrait of the system. Interpret it. Explain the long-term behavior of any population distribution that starts with only one species present. Relate it to part (b). What happens in the long term if both populations are present initially? Is there an initial population distribution that remains undisturbed? What is it? Relate those numbers to the model (4).
(e) Now replace 0.5 in the model by 2; that is, consider the new model
x(t) — x x 2xy (5)
y(t) — Ó - y2 - 2xy.
Draw the phase portrait. (Use the same initial data and viewing square.) Answer the same questions as in part (d). Do you see a special solution trajectory that emanates from near the origin and proceeds to the special fixed point? And another trajectory from the upper right to the fixed point? What happens to all population distributions that do not start on these trajectories? (f) Explain why model (4) is called "peaceful coexistence" and model (5) is called "doomsday." Now explain heuristically why the coefficient change from 0.5 to 2 converts coexistence into doomsday.
12. Build a SIMULINK model corresponding to the pendulum equation
x(t) — —0.5x(t) - 9.81 sin(x(t)) (6)
from The 360° Pendulum in Chapter 9. You will need the Trigonometric Function block from the Math library. Use your model to redraw some of the phase portraits.
13. As you know, Galileo and Newton discovered that all bodies near the earth's surface fall with the same acceleration g due to gravity, approximately 32.2 ft/sec2. However, real bodies are also subjected to forces due to air resistance. If we take both gravity and air resistance into account, a moving ball can be modeled by the differential equation
x — [0, —g] — ñ ||XXIl x. (7)
Here x, a function of the time t, is the vector giving the position of the ball (the first coordinate is measured horizontally, the second one vertically), x is the velocity vector of the ball, x is the acceleration of the ball, ||x||Practice Set C: Developing Your MATLAB Skills
is the magnitude of the velocity, that is, the speed, and c is a constant depending on the shape and mass of the ball and the density of the air. (We are neglecting the lift force that comes from the ball's rotation, which can also play a major role in some situations, for instance in analyzing the path of a curve ball, as well as forces due to wind currents.) For a baseball, the constant c turns out to be approximately 0.0017, assuming distances are measured in feet and time is measured in seconds. (See, for example, Chapter 18, "Balls and Strikes and Home Runs," in Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics, by Robert Banks, Princeton University Press, 1998.) Build a SIMULINK model corresponding to Equation (7), and use it to study the trajectory of a batted baseball. Here are a few hints. Represent x, x, and x as vector signals, joined by two Integrator blocks. The quantity x, according to (7), should be computed from a Sum block with two vector inputs. One should be a Constant block with the vector value [0, -32.2], representing gravity, and the other should represent the drag term on the right of Equation (7), computed from the value of x. You should be able to change one of the parameters to study what happens both with and without air resistance (the cases of c = 0.0017 and c = 0, respectively). Attach the output to an XY Graph block, with the parameters x-min = 0, y-min = 0, x-max = 500, y-max = 150, so that you can see the path of the ball out to a distance of 500 feet from home plate and up to a height of 150 feet.
(a) Let x(0) = [0, 4], x(0) = [80, 80]. (This corresponds to the ball starting at t = 0 from home plate, 4 feet off the ground, with the horizontal and vertical components of its velocity both equal to 80 ft/sec. This corresponds to a speed off the bat of about 77 mph, which is not unrealistic.) How far (approximately — you can read this off your XY Graph output) will the ball travel before it hits the ground, both with and without air resistance? About how long will it take the ball to hit the ground, and how fast will the ball be traveling at that time (again, both with and without air resistance)? (The last parts of the question are relevant for outfielders.)