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 xpp compete.ode. With the current parameter settings, b,d are small. Draw the nullclines. Notice that there are 4 equilibria, (0,0), (1,0),(0,1), and (0.8,0.8). Using whatever methods you'd like check the stability of these fixed points. Verify that the coexistent equilibrium point is the only stable one. (You can check the stability by clicking on and selecting one of the 4 intersections of the nullclines. Answer No to all the questions. Alternatively, you can just pick initial conditions nearby the equilibria.) Which equilibria represent the cases in which one or the other species ``wins'' the competition and wipes out the other? Are these stable?
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?