# Section 5 - The Equilibrium

## What are the equilibria of this system?

To find the equilibria of the system, set the system equal to 0 and solve.

c') 0 = ba - gc - dc
a') 0 = gc - da - uay
y') 0 = - ny + uay

There is always an equilibria at {0,0,0}. At that point, all of the animals are dead and obviously will no longer reproduce -- and so will remain dead (at 0).

To solve for the other equilibria, then, assume that neither a, c nor y = 0

y' goes to:

ny = uay
n = ua
a = n/u [which is necessarily >0 ; all parts of it are positive]

c' goes to:
ba = gc + dc
bn/u = c (g + d)
```        bn
c =  ---------  [which is necessarily >0 ]
u (g + d)
```
a' goes to:
gc - da = uay
(gc - da)/(ua) = y
```
bg - d(g + d)
y =  ----------------- > 0
u (g + d)
```
(which is necessarily > 0 because bg - d(g + d) > 0 by definition in section 2)

We can check these on the phase portraits simply by pointing the mouse at the approximate center of the sink on each plot and plugging in the values of {b,g,d,u,n} into the equations for the equilibrium. The center of the sink on the a/c plot should be approximately [1,2/3]; the center on the a/y portrait should be approximately [1,1/3]; and the center on the c/y portrait should be approximately [2/3,1/3].

Note that the other equilibrim (at {0,0,0}) is always a saddle.

So, this system always stabilizes, and it stabilizes at a realistic place -- i.e. the coordinates of the equilibrium are always positive and so all populations are alive -- when bg - d(g + d) > 0, as was set in section 2 to make sure that x didn't automatically die out.

If one allows bg - d(g + d) = 0, then there will be a birfurcation at those parameters; y will eventually die out (y-coordinate of the second equilibrium (the sink) is 0) and x will go to a certain point, as it would with no predation (see Section two)

If one allows bg - d(g + d) < 0, y will be negative at the equilibrium and hence that equilibrium will be irrelevant: y will simply die out. This is because, as seen in section two, x will die out. Then, since the only positive term in y' is related to the prey, y' will always be negative and predators will die out -- in realistic terms, they have no food, so they die.