I'm running into a problem. This is mainly for reading over the summer and I'm working on getting through a dynamical systems book on my own. I've come across a system that I'm not too sure on the procedure.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Problem in a system of ODEs

**Physics Forums | Science Articles, Homework Help, Discussion**