Problem 1: Death Process

---

In this problem, we study a death process, which has the unique reaction $A \to \emptyset$ with rate constant $\delta$.

a) Write down an ODE describing the deterministic kinetics.

b) Write down the Chemical Master Equation for the stochastic kinetics. Qualitatively compare the two.

c) Find the expected value of the extinction time $T_0$ when starting from an initial population of $n$.

d) Find the variance of the extinction time $T_0$.

e) Consider two independent identical death processes, one for species $A$ starting at an initial population of $n_A$ and one for species $B$ starting at an initial population of $n_B$. Denoting by $T_A$ the extinction time of $A$ and by $T_B$ that of $B$, find (an estimation of) the probability that $T_A \leq T_B$.

Problem 2: Birth and Death Process

---

In this problem, we add a birth reaction to the death process from the previous problem, i.e., the reaction $A \to A + A$ with rate constant $\gamma$.

a) Write down an ODE describing the deterministic kinetics.

b) Write down the Chemical Master Equation for the stochastic kinetics. Qualitatively compare the two.

c) Find the probability of extinction when starting from an initial population of $n$.

d) For the case of almost-sure extinction, find the expected extinction time. Evaluate this formula for $n=1,2,3$.