Phase Portrait of Scalar ODE
by R Clark Robinson

Plot one trajectory

ClearSystemCache;

varnames := {t,x}
OdeScalar:= {x'[t] == x[t]*(1-x[t])}; 
 xmin := -0.5; xmax := 1.5; 
Tmin :=0;  Tmax  := 30;

SolnOde := NDSolve[ {OdeScalar, x[0]== 0.01},x,{t,Tmin,Tmax}]
Plot[Evaluate[x[t]/.SolnOde],{t,Tmin,Tmax},AxesLabel -> varnames,
PlotRange -> {xmin,xmax}, PlotStyle -> Purple ,Ticks -> {{0,10,20,30},{-0.5,0,0.5,1,1.5}}
]

Plot several trajectories

varnames := {t,x}
OdeScalar:= {x'[t] == x[t]*(1-x[t])}; 
odeIC := {0.1,0.2,0.3,0.4,1,1.3,1.5};  
xmin := -0.5; xmax := 1.5; 
Tmin :=0;  Tmax  := 7;

trajectory[t_,x0_]:=
Block[{x},
Evaluate[x[t]/.NDSolve[{OdeScalar,x[0]==x0},
x,{t,Tmin,Tmax}]]]
phaseplot =Plot[Evaluate[
Table[trajectory[t,iCond],{iCond,odeIC}] ] , 
{t,Tmin,Tmax},AxesLabel -> varnames,PlotRange -> {xmin,xmax}, Ticks -> {{0,7},{-0.5,0,0.5,1,1.5}}]

With Direction Field

odevectorfield := {1,x*(1-x)}
rangetvar := {t,Tmin,Tmax}
rangexvar := {x,xmin,xmax}
dirgraph=VectorPlot[{1,x*(1-x)},{t,Tmin,Tmax},{x,xmin,xmax},VectorScale  -> {0.04,1.5}]

Show[phaseplot,dirgraph]