Slope Fields for Systems in Maple

To plot the direction field of system of two first-order differential equations equations

dx/dt = f(x, y, t)
dy/dt
= g(x, y, t),

use the Maple command dfieldplot twice. Here is an example for the predator-prey system in Section 2.1 of Blanchard, Devaney & Hall, Differential Equations , 2nd Edition, ITP Brooks/Cole, 2002.

As for the slope field of a single first-order eqaution, execute the routine by placing the cursor at the end of each block of code in turn, and hitting the Enter key. The first command prints the differential equations whose direction field is plotted.

> with(DEtools) :
de1 := diff( R(t), t) = 2*R(t) - 1.2*R(t)*F(t) ;
de2 := diff( F(t), t) = -F + 0.9*R(t)*F(t) ;

> dfieldplot([de1, de2], [R, F], t = 0..10, R = 0..2, F = 0..3) ;

```Warning, F is present as both a dependent variable and a name. Inconsistent specification of the dependent variable is deprecated, and will be removed in the next release.
```

To plot solution curves in the phase plane, we can ask Maple to construct a numerical approximation to a solution x = R(t), y = F(t) , starting from some initial values. The next routine does that. First, it provides a number of initial-value points. Then it plots numerical approximations of solutions that pass through each of those points. Note that an initial-value point is specified as ( , , ), where the R - and F -values correspond to the initial time . Compare Figure 2.5, Section 2.1 of Blanchard, Devaney, and Hall. Note that Maple by default includes the direction field in the plot, even though no dfieldplot command is present.,

> inits := {[0, 1, 5/3], [0, 1, 0.5], [0, 1.5, 2.1], [0, 2, 2.5], [0, 3, 3.3], [0, 3.2, 3.8]};

>
DEplot( [de1, de2], [R, F], t = 0..10, inits, stepsize = 0.1, R = 0..5, F = 0..5, linecolor = BLACK) ;

>

```Warning, F is present as both a dependent variable and a name. Inconsistent specification of the dependent variable is deprecated, and will be removed in the next release.
```

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