Consider the simple birth and death process:
![](https://book.transtutors.com/qimg/71ecc96e-2de2-42e1-b7c8-f722d794c0be.png)
in which G is a "gain." Assume that the reactions are catalyzed by enzymes and that the gain G can be tuned by changing the amounts of these enzymes. A deterministic ODE model for this system incorporating disturbances due to environmental perturbations is given by
![](https://book.transtutors.com/qimg/a9c4684a-ddcf-4087-ad35-979c5572c114.png)
Determine the Langevin equation for this birth and death process and compare its form to the deterministic one. Also, determine the frequency response of Z to noise for both the deterministic model and for the Langevin model. Does increasing the gain G have the same effect in both models? Explain.