A Solution of a Lotka-Volterra Type DEQ

Needs["VisualDSolve`"];

\(\(PhasePlot[{\(r'\)[t] == .3  r[t] \((1 - r[t]/10)\) - .2\ r[t]\ f[t], 
        \(f'\)[t] == \(-.5\)\ f[t] + .9\ r[t] f[t]}, 
	{r[t], f[t]}, 

      	{t, 0, 100}, 
	{r, 0, 1.5}, 
	{f, 0, 4}, 
	
      InitialValues -> {1.2, 1.3}, 
	MainThickness -> .01, 
	
      MinorThickness -> .004, 
	PlotPoints -> 200, 
	
      ParametricPlotFunction -> ProjectionPlot3D, 
	BoxRatios -> {1, 1, 2}, 
      
	PlaneResolution -> 4, 
	
      AxesLabel -> {"rabbits (in hundreds)", "foxes", "time"}, 

      ViewPoint -> {1.300, \ \(-2.400\), \ 2.000}
	]; \)\)

\(\(rf = 
      PhasePlot[{\(r'\)[t] == .3  r[t] \((1 - r[t]/10)\) - .2\ r[t]\ f[t], 
          \(f'\)[t] == \(-.5\)\ f[t] + .9\ r[t] f[t]}, 
	{r[t], f[t]}, 

        	{t, 0, 100}, 
	{r, 0, 1.5}, 
	{f, 0, 4}, 
	
        InitialValues -> {1.2, 1.3}, 
	MainThickness -> .01, 
	
        MinorThickness -> .004, 
	PlotPoints -> 200, 
VectorField -> True, 
        FieldLogScale -> 10, FieldColor -> Hue, 
	BoxRatios -> {1, 1, 2}, 

        	PlaneResolution -> 4, 
	
        AxesLabel -> {"rabbits", "foxes", "time"}, 
	
        ViewPoint -> {2.344, \ \(-2.237\), \ 0.975}
	]; \)\)

\(\(rt = 
      SystemSolutionPlot[

        	{\(r'\)[t] == .3  r[t] \((1 - r[t]/10)\) - .2\ r[t]\ f[t], 
          \(f'\)[t] == \(-.5\)\ f[t] + .9\ r[t] f[t]}, 
	{r[t], f[t]}, 

        	{t, 0, 100}, 
AxesLabel -> {"time", "rabbits"}, 

        InitialValues -> {1.2, 1.3}, 
	PlotVariables -> {{r}}, 
		
        AxesOrigin -> {0, 0}, 
		PlotStyle -> Blue, 
		
        PlotRange -> {0, 1.2}]; \)\)

\(\(ft = 
      SystemSolutionPlot[

        	{\(r'\)[t] == .3  r[t] \((1 - r[t]/10)\) - .2\ r[t]\ f[t], 
          \(f'\)[t] == \(-.5\)\ f[t] + .9\ r[t] f[t]}, 
	{r[t], f[t]}, 

        	{t, 0, 100}, 
AxesLabel -> {"time", "foxes"}, 

        InitialValues -> {1.2, 1.3}, 
	PlotVariables -> {{f}}, 
		
        AxesOrigin -> {0, 0}, 
		PlotStyle -> Red]; \)\)

\(\(Show[
      Graphics[{
			Rectangle[{.4, 0}, {.4 + .8\ GoldenRatio, .8}, rt], 

          				Rectangle[{.4, \(-1\)}, {.4 + .8\ GoldenRatio, \(-.2\)}, 
            ft], 
				Rectangle[{\(-1\), \(-1\)}, {.2, .8}, rf]}]]; 
    \)\)

\(\(PhasePlot[{\(r'\)[t] == .3  r[t] - .2\ r[t]\ f[t], 
        \(f'\)[t] == \(-.5\)\ f[t] + .9\ r[t] f[t]}, 
	{r[t], f[t]}, 

      	{t, 0, 100}, 
	{r, 0, 1.5}, 
	{f, 0, 4}, 
	
      InitialValues -> {1.2, 1.3}, 
	MainThickness -> .01, 
	
      MinorThickness -> .004, 
	PlotPoints -> 200, 
	
      ParametricPlotFunction -> ProjectionPlot3D, 
	BoxRatios -> {1, 1, 2}, 
      
	PlaneResolution -> 4, 
	
      AxesLabel -> {"rabbits (in hundreds)", "foxes", "time"}, 

      ViewPoint -> {1.300, \ \(-2.400\), \ 2.000}
	]; \)\)

\(\(rf = 
      PhasePlot[{\(r'\)[t] == .3  r[t] - .2\ r[t]\ f[t], 
          \(f'\)[t] == \(-.5\)\ f[t] + .9\ r[t] f[t]}, 
	{r[t], f[t]}, 

        	{t, 0, 100}, 
	{r, 0, 1.5}, 
	{f, 0, 4}, 
	
        InitialValues -> {1.2, 1.3}, 
	MainThickness -> .01, 
	
        MinorThickness -> .004, 
	PlotPoints -> 200, 
VectorField -> True, 
        FieldLogScale -> 10, FieldColor -> Hue, 
	BoxRatios -> {1, 1, 2}, 

        	PlaneResolution -> 4, 
	
        AxesLabel -> {"rabbits", "foxes", "time"}, 
	
        ViewPoint -> {2.344, \ \(-2.237\), \ 0.975}
	]; \)\)

\(\(rt = 
      SystemSolutionPlot[

        	{\(r'\)[t] == .3  r[t] - .2\ r[t]\ f[t], 
          \(f'\)[t] == \(-.5\)\ f[t] + .9\ r[t] f[t]}, 
	{r[t], f[t]}, 

        	{t, 0, 100}, 
AxesLabel -> {"time", "rabbits"}, 

        InitialValues -> {1.2, 1.3}, 
	PlotVariables -> {{r}}, 
		
        AxesOrigin -> {0, 0}, 
		PlotStyle -> Blue, 
		
        PlotRange -> {0, 1.2}]; \)\)

\(\(ft = 
      SystemSolutionPlot[

        	{\(r'\)[t] == .3  r[t] - .2\ r[t]\ f[t], 
          \(f'\)[t] == \(-.5\)\ f[t] + .9\ r[t] f[t]}, 
	{r[t], f[t]}, 

        	{t, 0, 100}, 
AxesLabel -> {"time", "foxes"}, 

        InitialValues -> {1.2, 1.3}, 
	PlotVariables -> {{f}}, 
		
        AxesOrigin -> {0, 0}, 
		PlotStyle -> Red]; \)\)

\(\(Show[
      Graphics[{
			Rectangle[{.4, 0}, {.4 + .8\ GoldenRatio, .8}, rt], 

          				Rectangle[{.4, \(-1\)}, {.4 + .8\ GoldenRatio, \(-.2\)}, 
            ft], 
				Rectangle[{\(-1\), \(-1\)}, {.2, .8}, rf]}]]; 
    \)\)