Math 221  Project with Maple  Due on Nov. 3, 2000

Name: Score:

Suggestions: To get started, go to my web site http://www.math.unl.edu/∼bdeng to get the ode sample file in Maple.

1 (a) Use Maple to find the exact solution to the logistic equation

\frac{dP (t)}{dt} = rP (t)(1 − \frac{P (t)}{K})

with r > 0 the intrinsic growth rate and K > 0 the carrying capacity.

(b) Plot the slope field for theequation with r = 0.5 and K = 5 together with solutions over the time interval [0, 20] with distinct initial conditions P (0) = 1, 2, 3, 4, 5, 6, 7, 8.

1 (a) Consider the equations of two competing species (Eq.(3) of page 493 of the handout supplement.)

\begin{aligned}\frac{dx}{dt} & = x(1 − x − y) \\ \frac{dy}{dt} & = y(0.75 − y − 0.5x). \end{aligned}

(i) Plot the direction field similar to Figure 9.4.1. of the supplement. (ii) Plot the direction field together with several solutions in one plot. Such a

plot is referred to as a phase portrait of the equations. With the exception of the direction field, this plot should look like Figure 9.4.2 of the supplement.

(iii) Plot the solutions curves x(t), y(t) verses the time t over the interval [0, 20] with initial conditions x(0) = 0.2, y(0) = 0.05. You need to put these two curves in one plot similar to Figure 9.5.3.

(b) Consider equations of two competing species (Eq.(21) of page 496 of the supplement.)

\begin{aligned}\frac{dx}{dt} & = x(1 − x − y) \\ \frac{dy}{dt} & = y(0.5 − 0.25y − 0.75x). \end{aligned}

Generate three plots exactly the same as those of (a)(i,ii,iii) above.

(c) Consider equations of one prey and one predator (Eq.(2) of page 505 of the supplement.)

\begin{aligned}\frac{dx}{dt} & = x(1 − 0.5y) \\ \frac{dy}{dt} & = y(−0.75 + 0.25x). \end{aligned}

Generate three plots exactly the same as those of (a)(i,ii,iii) above.

END