= Solution
The three jump vectors and their rates are $(1,0)$ at rate $\lambda$, $(-1,1)$ at rate $\gamma x_1$, and $(0,-1)$ at rate $\beta x_2$. Summing rate times jump gives the <Fokker-Planck drift vector>, while summing rate times the outer product of the jump gives the <Fokker-Planck diffusion matrix>:
$$
\boxed{A=\begin{pmatrix}\lambda-\gamma x_1\\\gamma x_1-\beta x_2\end{pmatrix},\qquad
B=\begin{pmatrix}\lambda+\gamma x_1&-\gamma x_1\\-\gamma x_1&\gamma x_1+\beta x_2\end{pmatrix}.}
$$
The maturation event simultaneously decreases larvae and increases adults; this accounts for the negative off-diagonal entries. Since $A$ is affine, its expectation is its value at the mean. Setting both mean derivatives to zero yields $\boxed{\langle x_1\rangle=\lambda/\gamma,\quad\langle x_2\rangle=\lambda/\beta}$.
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