> f:=x->r*(1-x/k); g:=x->x/(1+x^2);

> r:=.6:k:=10:

> plot({f(x),g(x)},x=0..15,p=0..0.6,color=blue);

> with(DEtools):
DEplot(diff(x(t),t)=r*x(t)*(1-x(t)/k)-x(t)^2/(1+x(t)^2),x(t),
t=0..20,[[x(0)=1],[x(0)=4],[x(0)=6],[x(0)=8],[x(0)=15]],x=0..16,stepsize=.05);

> r:=0.5:

> plot({f(x),g(x)},x=0..15,p=0..0.6,color=blue);

> DEplot(diff(x(t),t)=r*x(t)*(1-x(t)/k)-x(t)^2/(1+x(t)^2),x(t),
t=0..20,[[x(0)=0.1],[x(0)=1.8],[x(0)=4],[x(0)=6],[x(0)=8],[x(0)=15]],x=0..16,stepsize=.05);

> r:=0.4:

> plot({f(x),g(x)},x=0..15,p=0..0.6,color=blue);

> DEplot(diff(x(t),t)=r*x(t)*(1-x(t)/k)-x(t)^2/(1+x(t)^2),x(t),
t=0..20,[[x(0)=0.1],[x(0)=1.8],[x(0)=4],[x(0)=6],[x(0)=8],[x(0)=15]],x=0..16,stepsize=.05);

> r:=0.35:

> plot({f(x),g(x)},x=0..15,p=0..0.6,color=blue);

> DEplot(diff(x(t),t)=r*x(t)*(1-x(t)/k)-x(t)^2/(1+x(t)^2),x(t),
t=0..20,[[x(0)=0.1],[x(0)=1.8],[x(0)=4],[x(0)=6],[x(0)=8],[x(0)=15]],x=0..16,stepsize=.05);

>