Laser:population rate equation

[Clau]

1. The problem statement, all variables and given/known data
Simultaneous Laser Emission on Two Lines
Consider the multi-level system show in the figure where amplification can occur at frequencies \omega_{12} and \omega_{21}. The arrow "triplets" are meant to indicate the 3 Einstein absorption and emission processes that take place between levels 2 and 3. Write down the population rate equations for levels 1, 2 and 3.

The figure is attached.

My problem is: I don't know if I have to introduce the pumping power inside the population rate equations.

2. Relevant equations
Ni=number of particles in level i
Between levels 1 and 2 we have:
\downarrow \uparrow \downarrow
first down arrow: spontaneous trasition rate
N_{2}A_{21}
up arrow: absorption rate
N_{1}B_{12}W(\omega_{12})
second down arrow: stimulated emission rate
N_{2}A_{21}W(\omega_{12}


3. The attempt at a solution
For level 3 I have:
dN_{3}/dt = - N_{3}A_{31} + N_{1}B_{13}W(\omega_{13}) - N_{3}B_{31}W(\omega_{13})

Level 2:
dN_{2}/dt= - N_{2}A_{21} + N_{1}B_{12}W(\omega_{12}) - N_{2}B_{21}W(\omega_{12})

Level 1:
dN_{1}/dt = - dN_{3}/dt - dN_{2}/dt - N_{1}R{1}


Should I add P3 to level 3 and P2 to level 2?