sm311o_8d.mws,1-30-98,wdj

> restart;
with(DEtools):
with(plots):

The Lotka-Volterra Model is the system

> a:=1;b:=1;c:=1;d:=1;

> sys:=diff(x(t),t)=(a-b*y(t))*x(t),diff(y(t),t)=(c*x(t)-d)*y(t);
ic:=x(0)=1.2,y(0)=1;

> phaseportrait([sys],[x(t),y(t)],t=0..3,[[ic]],scene=[x(t),y(t)],stepsize=.05,x=0..2,y=0..2);

The differential equation governing the phase portrait in the x-y plane is

> de:=diff(y(x),x)=(c*x-d)*y(x)/((a-b*y(x))*x);

> soln:=dsolve({de,y(1.2)=1},y(x)):
y0:=s->subs(x=s,rhs(soln)): y0(x);

The phaseportrait command works better because the Lambert W function does not capture all of the curve in the phase portrait.

> plot(y0(x),x=.01..2,y=0..2);