[Mahtematica] StreamDensityPlot

[Whole script]

f[m_] := RGBColor[m^alpha, m^alpha, m^alpha] 

StreamDensityPlot[{r*x - x^2/M - x*y/b/M, r*y - x*y/c/M - y^2/d/M}, {x, 0, xrange}, {y, 0, yrange},  

   ColorFunction -> f,    

   StreamPoints -> Coarse, StreamStyle -> Orange, StreamScale -> Large, 

   PlotRange -> {{0, xrange}, {0, yrange}}, 

   BaseStyle -> {FontFamily -> "Helvetica", 20}, 

   FrameLabel -> {"Abundance n", "Abundance m"},

   LabelStyle -> Directive[Black, 15], AspectRatio -> Automatic,   

   FrameTicks -> {{{0, 400, 800}, None}, {{0, 400, 800, 1200}, None}}];

[Explanation line by line]

//Define User Color Function 

f[m_] := RGBColor[m^alpha, m^alpha, m^alpha] 

StreamDensityPlot[

//Function for Coloring at Position (x,y), xrange, yrange   

{r*x - x^2/M - x*y/b/M, r*y - x*y/c/M - y^2/d/M}, {x, 0, xrange}, {y, 0, yrange},  

//Use the Color Function Defined as f

ColorFunction -> f,    

//Number of Stream Points, Color of line, and Scale of the stream

StreamPoints -> Coarse, StreamStyle -> Orange, StreamScale -> Large, 

//To Get Rid of Margin in Default Setting   

PlotRange -> {{0, xrange}, {0, yrange}}, 

//For Labelling as gnuplot style: Helvetica font, and Black color

BaseStyle -> {FontFamily -> "Helvetica", 20}, 

FrameLabel -> {"Abundance n", "Abundance m"},

LabelStyle -> Directive[Black, 15], AspectRatio -> Automatic,   

\\Ticks, {None} means no ticks at x2, and y2 axis

FrameTicks -> {{{0, 400, 800}, None}, {{0, 400, 800, 1200}, None}}];

[Example]

M = 1000; a = 1; b = 0.03; c = 0.71; d = 0.62; r = 0.5;

xrange = 1.2 r M ; yrange = .3 r M d;

plt1 = StreamDensityPlot[{r*x - x^2/M - x*y/b/M,  r*y - x*y/c/M - y^2/d/M}, {x, 0, xrange}, {y, 0, yrange}, 

   StreamStyle -> Orange, VectorScale -> Large,  ColorFunction -> "Rainbow", VectorStyle -> White, 

   StreamPoints -> Fine];

plt2 = ListPlot[{{(r M a c )/(a d - b c) (d - b), (r M b d )/( a d - b c) (a - c)}, {0, 0}, {r*M, 1}, {0, r*M*d}}, PlotStyle -> {PointSize[0.02], Yellow}, PlotRange -> {{0, xrange}, {0, yrange}}];

plt3 = Plot[{(d (c M r - x))/c, -((b M (-a r + M x))/a) }, {x, 0, 500}];

plt4 = ListLinePlot[path, PlotRange -> {{0, 600}, {0, 100}}, PlotStyle -> White];

fig = Show[plt1, plt2 , plt3, plt4, AxesLabel -> {Style[x, FontSize -> 10, Bold, Red],  Style[y, FontSize -> 10, Bold, Blue]}]