I have a homework problem in which I'm given an Ivanova reaction system
$$X+Y \longrightarrow 2Y$$
$$Y+Z \longrightarrow 2Z$$
$$Z+X \longrightarrow 2X$$
and I'm asked to write the mass-action ODEs with respective rate constants $k_{1}, k_{2}, k_{3}$.
This is a course on nonlinear dynamics, and the last time I took chemistry was in high school. Thus, I'm not sure how to write mass-action ODEs given an Ivanova reaction system, whatever that is. I've googled mass-action and Ivanova to try to find how to write the mass-action ODEs, but I'm still confused.
Here is my attempt after searching around on the internet:
$$\displaystyle\frac{1}{2} \frac{dY}{dt} = k_{1}XY$$
$$\displaystyle\frac{1}{2} \frac{dZ}{dt} = k_{2}YZ$$
$$\displaystyle\frac{1}{2} \frac{dX}{dt} = k_{3}XZ$$
If I could get some sort of verification and/or hints that would be much appreciated, and I feel like I can do the rest of what the homework asks.
The law of mass action says that the rate of reaction is proportional to the concentrations of reagents. This is how you can write your equation for $[X]$, concentration of $X$: $$ \dot{[X]}=k_3[Z][X]+k_3[Z][X]-k_3[Z][X]-k_1[X][Y], $$ where two first terms comes from $2X$ expression in one of your reactions, third term comes from $Z+X$ part, and the last term comes from $X+Y$ term in the first reaction. I also assumed that the rate constants are $k_1,k_2,k_3$ from top to bottom. Simplifying, you get $$ \dot{[X]}=k_3[Z][X]-k_1[X][Y], $$ and the rest should be easy.