Problem 7.3.4: A nonlinear system

ODE:  

x’  =  x (1.5 - 0.5 x - y)
w’ = y (0.75 - y -0.125 x)

vf[x_,y_]={x (1.5-0.5 x-y),y (0.75-y-0.125 x)}

df=VectorPlot[vf[x,y]/Norm[vf[x,y]],{x,0,7},{y,0,2},VectorStyle->Arrowheads[.03]]

Equilibria: (0,0), (3,0), (0,3/4), (2,1/2)

dfeq=Show[df,ListPlot[{{0,0},{3,0},{0,3/4},{2,1/2}},PlotStyle->Red]]

We know the behavior near the equilibrium points.  Let’s try to get some orbits.

soln[xzero_?NumericQ,yzero_?NumericQ]:=NDSolve[{dx'[t]==dx[t] (1.5-0.5 dx[t]-dy[t]),dy'[t]==dy[t] (0.75-dy[t]-0.125 dx[t]),dx[0]==xzero,dy[0]==yzero},{dx,dy},{t,0,10}]
xx[t_?NumericQ,xzero_?NumericQ,yzero_?NumericQ]:=dx[t]/.soln[xzero,yzero][[1]]
yy[t_?NumericQ,xzero_?NumericQ,yzero_?NumericQ]:=dy[t]/.soln[xzero,yzero][[1]]

orbit1=ParametricPlot[{xx[s,1,1],yy[s,1,1]},{s,0,10},PlotStyle->{Thick,Green}];

Show[dfeq,orbit1]

orbit2=ParametricPlot[{xx[s,5,1.5],yy[s,5,1.5]},{s,0,10},PlotStyle->{Thick,Green}];

Show[dfeq,orbit1,orbit2]

orbit3=ParametricPlot[{xx[s,5,0.5],yy[s,5,0.5]},{s,0,10},PlotStyle->{Thick,Green}];

Show[dfeq,orbit1,orbit2,orbit3]

orbit4=ParametricPlot[{xx[s,2,0.2],yy[s,2,0.2]},{s,0,10},PlotStyle->{Thick,Green}];

Show[dfeq,orbit1,orbit2,orbit3,orbit4,AspectRatio->Automatic]

orbit5=ParametricPlot[{xx[s,0.1,0.3],yy[s,0.1,0.3]},{s,0,10},PlotStyle->{Thick,Green}];

Show[dfeq,orbit1,orbit2,orbit3,orbit4,orbit5]

orbit6=ParametricPlot[{xx[s,6,0.2],yy[s,6,0.2]},{s,0,10},PlotStyle->{Thick,Magenta}];

Show[dfeq,orbit1,orbit2,orbit3,orbit4,orbit5,orbit6]

orbit7=ParametricPlot[{xx[s,6,0.02],yy[s,6,0.02]},{s,0,10},PlotStyle->{Thick,Magenta}];orbit8=ParametricPlot[{xx[s,0.5,0.02],yy[s,0.5,0.02]},{s,0,10},PlotStyle->{Thick,Magenta}];orbit9=ParametricPlot[{xx[s,2,0.02],yy[s,2,0.02]},{s,0,10},PlotStyle->{Thick,Magenta}];

Show[dfeq,orbit1,orbit2,orbit3,orbit4,orbit5,orbit6,orbit7,orbit8,orbit9]