Mathematica slope fields

Question

I would like to make a slope field. Here is the code

slopefield = 
 VectorPlot[{1, .005 * p*(10 - p) }, {t, -1.5, 20}, {p, -10, 16}, 
  Ticks -> None, AxesLabel -> {t, p},  Axes -> True, 
  VectorScale -> {Tiny, Automatic, None}, VectorPoints -> 15]

I solved the differential equations and plotted the curves manually. Three questions:

1. Is there an easier way to do it?
2. Ticks -> None doesn't seems to work. I still get labels for the tick marks.
3. I'd like to selectively label 2 tick marks.

Answer 1

I'm assuming here that the curves you mentioned are streamlines of the vector field. You can plot those automatically without having to solve any differential equations by using the options StreamPoints, for example to plot the stream lines going through the points

points =  Transpose@ArrayPad[{Range[-10, 16, 2]}, {{1, 0}, {0, 0}}]

(* ==> {{0, -10}, {0, -8}, {0, -6}, {0, -4}, {0, -2}, {0, 0}, {0, 2}, {0, 4}, 
        {0, 6}, {0, 8}, {0, 10}, {0, 12}, {0, 14}, {0, 16}} *)

you can do

slopefield = 
 VectorPlot[{1, .005*p*(10 - p)}, {t, -1.5, 20}, {p, -10, 16}, 
  FrameTicks -> None, AxesLabel -> {t, p}, Axes -> True, 
  VectorScale -> {Tiny, Automatic, None}, VectorPoints -> 15, 
  StreamPoints -> points,
  StreamStyle -> {Red, "Line"}]

Answer 2

To plot the vector field and the streamlines (curves) together, there are two other plot functions that are specialized for this purpose:

• StreamPlot
• LineIntegralConvolutionPlot

The difference to VectorPlot is that Mathematica can automatically pick a set of curves for you. You can specify the starting points for the curves, but you don't have to:

StreamPlot[{1, .005*p*(10 - p)}, {t, -1.5, 20}, {p, -10, 16}, 
 AxesLabel -> {t, p}, Axes -> True, 
 VectorScale -> {Tiny, Automatic, None}, VectorPoints -> 15,
 StreamStyle -> Red, FrameTicks -> {{5, 10}, {-10, 10}, {}, {}}]

Here, I've also added a FrameTicks specification that labels several special points on the horizontal and vertical axes.

The other alternative, which contains some additional visual information, is this:

LineIntegralConvolutionPlot[{1, .005*p*(10 - p)}, {t, -1.5, 
  20}, {p, -10, 16}, Ticks -> None, AxesLabel -> {t, p}, Axes -> True,
  VectorScale -> {Tiny, Automatic, None}, VectorPoints -> 15, 
 VectorStyle -> LightGray, ColorFunction -> ColorData["Rainbow"], 
 FrameTicks -> {{5, 10}, {-10, 10}, {}, {}}, Background -> Black, 
 BaseStyle -> White, FrameTicksStyle -> Yellow, ImageMargins -> 5]

The last plot can be customized by adding the actual streamlines with the StreamPoints option (see the documentation), but the colored background serves the same purpose. The idea is that the background pattern is physically like the pattern you'd get (e.g.) from grass seeds in an electric field, or irons filings in a magnetic field, etc. And of course the color encodes information about the field strength.

Answer 3

For example:

VectorPlot[{1, .005*p*(10 - p)}, {t, -1.5, 20}, {p, -10, 16}, 
          FrameTicks -> {
              Join[{{0, "ZERO", {0, .1}, Red}}, Table[{i, ""}, {i, -1.5, 20, 3}]], 
                                                Table[{i, ""}, {i, -10, 16, 3}]}, 
          AxesLabel -> {t, p}, 
          Frame     -> True, 
          VectorScale -> {Tiny, Automatic, None}, 
          VectorPoints -> 15]