Example I: , , , , , .

>	phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/1-0.5*(y(t)/1)),               D(y)(t)=1*y(t)*(1-y(t)/1-.3*(x(t)/1))],     [x(t),y(t)],     t=0..10, [[x(0)=0.1,y(0)=0.1], [x(0)=1.1,y(0)=0.1], [x(0)=0.1,y(0)=1.1], [x(0)=1.1,y(0)=1.1]]);

This has a sink at the non-trivial equilibrium point .

See what happens when we lower to .  The equilibrium number of bluegills drops.

>	phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/0.8-0.5*(y(t)/1)),               D(y)(t)=1*y(t)*(1-y(t)/1-.3*(x(t)/0.8))],     [x(t),y(t)],     t=0..10, [[x(0)=0.1,y(0)=0.1], [x(0)=1.1,y(0)=0.1], [x(0)=0.1,y(0)=1.1], [x(0)=1.1,y(0)=1.1]]);

Qualitatively this is the same, only now the sink is at . Note that the equilibrium level of bluegill is now lower, but the equilibrium level of redear has stayed the same.  Perhaps the best way to see the change is to view both plots together. We will draw the arrows (color) of the second in blue and the solution curves of the second
(linecolor) in green.

>

plot1 := phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/1-0.5*(y(t)/1)),                         D(y)(t)=1*y(t)*(1-y(t)/1-.3*(x(t)/1))],         [x(t),y(t)],t=0..10,         [[x(0)=0.1,y(0)=0.1],[x(0)=1.1,y(0)=0.1],          [x(0)=0.1,y(0)=1.1],[x(0)=1.1,y(0)=1.1]]): plot2 := phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/0.8-0.5*(y(t)/1)),                         D(y)(t)=1*y(t)*(1-y(t)/1-.3*(x(t)/0.8))],         [x(t),y(t)],t=0..10,         [[x(0)=0.1,y(0)=0.1],[x(0)=1.1,y(0)=0.1],          [x(0)=0.1,y(0)=1.1],[x(0)=1.1,y(0)=1.1]],          linecolor=green,color=blue): display([plot1,plot2]);

Example II: , , , , , .

>	phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/1-0.3*(y(t)/1)),               D(y)(t)=1*y(t)*(1-y(t)/1-0.5*(x(t)/1))],     [x(t),y(t)],     t=0..10, [[x(0)=0.1,y(0)=0.1], [x(0)=1.1,y(0)=0.1], [x(0)=0.1,y(0)=1.1], [x(0)=1.1,y(0)=1.1]]);

The equilibrium is now at .

What happens when we lower to 0.8?

>	phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/0.8-0.3*(y(t)/1)),               D(y)(t)=1*y(t)*(1-y(t)/1-0.5*(x(t)/0.8))],     [x(t),y(t)],     t=0..10, [[x(0)=0.1,y(0)=0.1], [x(0)=1.1,y(0)=0.1], [x(0)=0.1,y(0)=1.1], [x(0)=1.1,y(0)=1.1]]);

Now the equilibrium has moved to .
Viewing the two pictures simultaneously:

>

plot1 := phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/1-0.3*(y(t)/1)),                         D(y)(t)=1*y(t)*(1-y(t)/1-0.5*(x(t)/1))],                      [x(t),y(t)],t=0..10,                      [[x(0)=0.1,y(0)=0.1],[x(0)=1.1,y(0)=0.1],                       [x(0)=0.1,y(0)=1.1],[x(0)=1.1,y(0)=1.1]]): plot2 := phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/0.8-0.3*(y(t)/1)),                         D(y)(t)=1*y(t)*(1-y(t)/1-0.5*(x(t)/0.8))],                       [x(t),y(t)],t=0..10,                       [[x(0)=0.1,y(0)=0.1],[x(0)=1.1,y(0)=0.1],                        [x(0)=0.1,y(0)=1.1],[x(0)=1.1,y(0)=1.1]],                       linecolor=green,color=blue): display([plot1,plot2]);

Example III: , , , , , .

Here the bluegill grow faster ( ), but are much more adversely effected by the other species.

>	phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/1-0.9*(y(t)/1)),               D(y)(t)=1*y(t)*(1-y(t)/1-0.1*(x(t)/1))],     [x(t),y(t)],     t=0..10, [[x(0)=0.1,y(0)=0.1], [x(0)=1.1,y(0)=0.1], [x(0)=0.1,y(0)=1.1], [x(0)=1.1,y(0)=1.1]]);

The equilibrium is now . Note that at time the curves have yet to converge on the equilibrium point.

Example IV: , , , , , .

Here is straight competition with the bluegills growing faster.

>	phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/1-1.0*(y(t)/1)),               D(y)(t)=1*y(t)*(1-y(t)/1-1.0*(x(t)/1))],     [x(t),y(t)],     t=0..10, [[x(0)=0.1,y(0)=0.1], [x(0)=1.1,y(0)=0.1], [x(0)=0.1,y(0)=1.1], [x(0)=1.1,y(0)=1.1]]);

Here there is a whole line of equilibria: .

Here is what it looks like if the two growth rates are the same.

>	phaseportrait([D(x)(t)=1*x(t)*(1-x(t)/1-1.0*(y(t)/1)),               D(y)(t)=1*y(t)*(1-y(t)/1-1.0*(x(t)/1))],     [x(t),y(t)],     t=0..10, [[x(0)=0.1,y(0)=0.1], [x(0)=1.1,y(0)=0.1], [x(0)=0.1,y(0)=1.1], [x(0)=1.1,y(0)=1.1]]);

The integal curves here are radial lines.

Here is the first of the two with the line of equilibria plotted in blue.

>

with(plots): plot1:=phaseportrait([D(x)(t)=2*x(t)*(1-x(t)/1-1.0*(y(t)/1)),               D(y)(t)=1*y(t)*(1-y(t)/1-1.0*(x(t)/1))],     [x(t),y(t)],     t=0..10, [[x(0)=0.1,y(0)=0.1], [x(0)=1.1,y(0)=0.1], [x(0)=0.1,y(0)=1.1], [x(0)=1.1,y(0)=1.1]]): plot2:=plot(1-x,x=0..1,color=blue): display([plot1,plot2]);