Combining birth and death processes

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The birth-death process

Combining birth and death processes with birth and death rates \(\lambda\) and \(\mu\), respectively, we expect to have the following differential-difference equations for the birth-death process:

\[\begin{eqnarray}\frac{{d}p_0(t)}{{d}t} &=& \mu p_1(t),\\\frac{{d}p_n(t)}{{d}t} &=& \lambda(n-1)p_{n-1}(t) - (\lambda + \mu)np_n(t) + \mu(n+1)p_{n+1}(t),~~(n \geq 1).\end{eqnarray}\]

You should derive the above equations based on the following assumptions:

  1. Given a population with \(n\) individuals, the probability that an individual is born in the population during a short period \(\delta t\) is \(\lambda n \delta t + o(\delta t)\).
  2. Given a population with \(n\) individuals, the probability that an individual dies in the population is \(\mu n \delta t + o(\delta t)\).
  3. The probability that multiple individuals are born or die during \(\delta t\) is negligible. (The probability of one birth and one death during \(\delta t\) is also negligible.)
  4. Consequently, during \(\delta t\), the population can increase by 1 or decrease by 1 (if \(n > 0\)) or stay constant; these three events are disjoint.

Generating function equation

As usual, let's define the probability generating function
\[G(s,t) = \sum_{n=0}^{\infty}p_{n}(t)s^{n}. \tag{Eq:PGF}\]
We also assume the initial condition
\[p_n(0) = \delta_{n,n_0}.\]
Then, it is an exercise before breakfast to derive the following PDE from the above differential-difference equations:
\[\frac{\partial G(s,t)}{\partial t} = (\lambda s - \mu)(s - 1)\frac{\partial G(s,t)}{\partial s}.\]
We can solve this PDE by the same method as we used for the birth and death processes. But the solution depends on whether \(\lambda = \mu\) or not.

The case when \(\lambda \neq \mu\)

We have
\[G(s,t) = \left[\frac{\mu(1-s) - (\mu - \lambda s)e^{-(\lambda - \mu)t}}{\lambda(1-s) - (\mu - \lambda s)e^{-(\lambda - \mu)t}}\right]^{n_0}.\tag{Eq:LNM}\]
In principle, we can determine \(\{p_n(t)\}\) by expanding this PGF  into the power series of \(s\). But that is pretty complicated. We can still calculate the moments of the random variable \(N(t)\) from the PGF. For example, the expected population size is given as
\[\mathbb{E}[N(t)] = \frac{\partial G}{\partial s}(1,t) = n_0e^{(\lambda - \mu)t}.\]
Thus, the population size increases or decreases exponentially if \(\lambda > \mu\) or \(\lambda < \mu\), respectively.

The probability of extinction can also be obtained as (why?)
\[p_0(t) = G(0, t) = \left(\frac{\mu}{\lambda}\cdot\frac{1 - e^{-(\lambda - \mu)t}}{1 + e^{-(\lambda - \mu)t}}\right)^{n_0}.\]
Find the probability of ultimate extinction \(\lim_{t\to\infty}p_0(t)\).

The case when \(\lambda = \mu\)

In this case, we have
\[G(s,t) = \left[\frac{1+(\lambda t -1)(1-s)}{1 + \lambda t (1-s)}\right]^{n_0}.\]

Exercise. Note that this generating function cannot be obtained from (Eq:LNM) by simply setting \(\lambda = \mu\) as that would result in 0/0. What happens if we take the limit \(\mu \to \lambda\) in (Eq:LNM)? (Hint: Use L'Hopital's rule.)   □

The birth and death rates being equal means there are as many those who are born as those who die, on average. Thus, we expect the average population size to be constant:
\[\mathbb{E}(N(t)) \stackrel{?}{=} n_0.\]
You should check whether this conjecture is true or not.

Why are the generating functions in the form of \((\cdot)^{n_0}\)?

In the birth, death, and birth-and-death processes, the probability generating functions are all in the form of \((\cdot)^{n_0}\). Why is this so? Recall that the PGF can be written as
\[G(s,t) = \mathbb{E}[s^{N(t)}].\]
We may decompose the population as
\[N(t) = N_1(t) + N_2(t) + \cdots + N_{n_0}(t)\]
where each \(N_i(t), i = 1, 2, \cdots, n_0,\) represents the number of "descendants" of the individual \(i\) in the initial population of \(n_0\) individuals. Because each birth or death is independent of others, \(\{N_i(t)\}\) are independent of each other (and also identically distributed). Thus, \[\begin{eqnarray} G(s,t) &=& \mathbb{E}[s^{N(t)}]\\ &=&\mathbb{E}[s^{N_1(t)+\cdots + N_{n_0}(t)}]\\ &=&\mathbb{E}[s^{N_1(t)}]\cdot \cdots \cdot \mathbb{E}[s^{N_{n_0}(t)}] ~~(\text{independence})\\ &=&\{\mathbb{E}[s^{N_1(t)}]\}^{n_0} ~~(\text{identical distribution}). \end{eqnarray}\]

General population processes

So far, we have assumed that the birth and death rates are proportional to the population size. In general, those rates may be more complicated functions of population size (and time). Accordingly, we modify the differential equations to 
\[\begin{eqnarray} \frac{{d}p_0(t)}{{d}t} &=& -\lambda_0(t)p_0(t) + \mu_1p_1(t),\\ \frac{{d}p_n(t)}{{d}t} &=& \lambda_{n-1}(t)p_{n-1}(t) -[\lambda_n(t) + \mu_n(t)]p_n(t) + \mu_{n+1}p_{n+1}(t), ~~(n\geq 1). \end{eqnarray}\]
The models we have dealt with are
  • Birth process: \(\lambda_n = \lambda n\) and \(\mu_n = 0\) where \(\lambda\) is a constant.
  • Death process: \(\lambda_n = 0\) and \(\mu_n = \mu n\) where \(\mu\) is a constant.
  • Birth-and-death process: \(\lambda_n = \lambda n\) and \(\mu_n = \mu n\) where \(\lambda\) and \(\mu\) are constants.
Many other models are possible. For example, \(\lambda_n = \lambda\) (constant, not proportional to \(n\)) and \(\mu_n = \mu n\). In particular, \(\lambda_0 > 0\) allows individuals to be generated from void.

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