In this case, we can assume both species have the same carrying capacity (rescaling one or the other). Let's simply suppose that both

Here is an *XPPAUT *file for the competitive system,
`compete.ode` :

# two species competing x'=a*x*(1-x-b*y) y'=c*y*(1-y-d*x) par a=1,c=1,b=.25,d=.25 @ xlo=-.1,ylo=-.1,xhi=1.1,yhi=1.1,xp=x,yp=y @ total=50 doneRun this program by typing

Now, lets increase the parameter *d* a little bit. This means that *x*
has a bigger effect on *y* than vice versa. Change *d* to 0.8, erase
the screen and redraw the nullclines. Verify that there is still a
stable coexistent solution but that it is biased toward *x* with the
value of *x* larger than that of *y*.

Increase *d* to 1.2. Describe the phaseplane, nullclines, and the
stable fixed points. Now the effect of *x* on *y* is greater than the
crowding effect of *y* on itself. Verify that the only stable
equilibrium is (1,0), that is, *x* wins the competition.

Lets make it fair again. Increase *b* to 1.2 and redraw the
phaseplane. What are the possible stable fixed points? Choose a bunch
of initial data and see if you can predict which fixed point the
solution will go to. This is an example of *bistability*, there
are two distinct stable fixed points, (1,0) and (0,1). Suppose
that
(*x*(0)=.3,*y*(0)=.2). Without simulating the system, which species
will emerge as the winner?