Problem 1: Death Process

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

a) Write the ODE for the mean population.

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Each individual dies independently at rate $\delta$. The population decreases proportionally to the current population: $$ \frac{d a}{dt} = - \delta a $$

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

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Let $P_n(t)$ be the probability that the population is $n$ at time $t$.

• From state $n$, any individual may die, so the transition rate $n \mapsto n-1$ is $n \cdot \delta$.
• From state $n+1$, we can transition to $n$ at rate $\delta \cdot (n+1)$.

CME:
$$ \frac{dP_n}{dt} = \delta \cdot (n+1) \cdot P_{n+1} - \delta \cdot n \cdot P_n $$

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

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Let $T_0(n)$ be the expected time to extinction from $n$ individuals.

• First event is a death with exponential waiting time mean $1/\delta n$.
• After the first death, the population is $n-1$.

Recurrence:
$$ T_0(n) = \frac{1}{\delta n} + T_0(n-1), \quad T(0) = 0 $$

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

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The extinction time is a sum of independent exponential waiting times with different rates: $1/\delta k$ for $k = n, n-1, \dots, 1$.

• Variance of an exponential with rate $\lambda$ is $1/\lambda^2$.
• Then sum the variances of each waiting time.

e) Argue that $\mathbb{P}(T_0(n) \leq t) = (1 - e^{-\delta t})^n$.

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The lifespan of each of the $n$ individuals of the starting population is an independent exponential random variable with rate $\delta$. Then use the cumulative distribution function of the exponential distribution.

f) Find the derivative $\frac{d}{dt}\mathbb{P}(T_0(n) \leq t)$.

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Use the chain rule.

g) 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 $T_A \leq T_B$.

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Use the law of total probability and the fact that the derivative from part f) is the probability density function of $T_0(n)$.

Problem 2: Birth and Death Process

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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.

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Each individual can give birth ($+\gamma$) or die ($-\delta$): $$ \frac{d a}{dt} = (\gamma - \delta) a $$

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

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Let $P_n(t)$ be the probability that the population is equal to $n$ at time $t$.

• Transitions: $n \mapsto n+1$ at rate $\gamma n$ and $n \mapsto n-1$ at rate $\delta n$.

CME for $n \geq 1$:
$$ \frac{dP_n}{dt} = \gamma (n-1) P_{n-1} + \delta (n+1) P_{n+1} - (\gamma+\delta) n P_n $$

c) Let $q_n$ be the probability of extinction when starting from an initial population of $n$. Find a recursion formula for $q_n$.

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Condition on the first event (birth or death): $$ q_n = \frac{\gamma}{\gamma+\delta} q_{n+1} + \frac{\delta}{\gamma+\delta} q_{n-1}, \quad q_0=1 $$

d) Find a solution of the recusion formula for $q_n$.

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Make the ansatz $q_n = r^n$ for some $r\in [0,1]$ and solve the resulting quadratic equation. Then distinguish the cases $\gamma > \delta$ and $\gamma \leq \delta$.

Intuition: when births are stronger than deaths, the population can "escape" extinction.

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