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Consider the following system of differential equations:

[itex]\frac{dX}{dt} = 1 - X - XY - XZ, \\

\frac{dY}{dt} = aXY - Y,\\

\frac{dZ}{dt} = aXZ - Z.

[/itex]

Here, 'a' is a real positive constant and [itex] X,Y,Z \geq 0 . [/itex]

I can find all of the three steady states except for the one where all three exist, i.e. when [itex] \bar{X},\bar{Y},\bar{Z} \neq 0 . [/itex]

When solving for this steady state, I arrive at [itex] \bar{X} = \frac{1}{a} . [/itex]

Then, plugging this back into the first equation, I get the following

[itex] 0 = a -1 - \bar{Y} - \bar{Z}, [/itex]

or,

[itex] a-1 = \bar{Y}+\bar{Z}. [/itex]

Is there a way to solve for [itex]\bar{Y}[/itex] and [itex]\bar{Z}[/itex]?

-as a side note, I've explored this numerically with matlab and it looks like it depends on the initial conditions. Could it be a saddle?