Question

Consider a branching process having \(\mu<1\). Show that if \(X_{0}=1\), then the expected number of individuals that ever exist in this population is given by \(1 /(1-\mu)\). What if \(X_{0}=n ?\)

Short answer

For a branching process with extinction probability \(\mu < 1\), if the initial population size is \(X_0 = 1\), the expected total number of individuals that ever exist in this population is given by \(\frac{1}{1-\mu}\). If the initial population size is \(X_0 = n\), the expected total number of individuals that ever exist in this population is given by \(n\frac{1}{1-\mu}\).

Step-by-step solution

Definition of branching process and extinction probability

A branching process is a stochastic process that models the growth of a population where individuals can reproduce independently. It is defined by a sequence of random variables \(\{X_n \}\), where \(X_n\) represents the number of individuals in generation n. The branching process starts with an initial population size \(X_0\). The extinction probability \(\mu\) can be understood as the probability that the population will eventually die out, meaning that there will be no individuals left in the population at some generation \(n\). If \(\mu < 1\), it indicates there is a non-zero probability that the population will survive indefinitely.

Calculate the expected number of individuals in each generation

Let \(E_n = E[X_n]\) be the expected number of individuals in generation n. We are given the extinction probability as \(\mu < 1\). Notice that the expected number of individuals in generation 1 is the initial population size, which is \(X_0 = 1\). So, \(E_1 = 1\). Now let's evaluate \(E_2\), which is the expected number of individuals in the second generation. Since each individual can reproduce independently, the expected number of offspring for each individual is simply \((1-\mu)\). So, \(E_2 = E_1 (1-\mu)\), as the expected number of individuals in the second generation is the product of the expected number of individuals in the first generation and the expected number of offspring for each individual.

Calculate the expected total number of individuals

The expected total number of individuals that ever exist in the population can be calculated as the sum of the expected number of individuals in each generation: $$ E_{total} = E_1 + E_2 + E_3 + \cdots $$ Using the recursion for the expected number of individuals in each generation: $$ E_n = E_{n-1}(1-\mu) $$ We can write down the expected total number of individuals as: $$ E_{total} = E_1 + E_1(1-\mu) + E_1(1-\mu)^2 + E_1(1-\mu)^3 + \cdots $$ This is a geometric series with a common ratio of \((1-\mu)\). Hence we can calculate the sum using the formula: $$ E_{total} = \frac{E_1}{1-(1-\mu)} = \frac{1}{1-(1-\mu)} = \frac{1}{\mu} $$ Since \(X_0 = 1\), the expected total number of individuals that ever exist in this population is given by: $$ \frac{1}{1-\mu} $$

Calculate the expected total number of individuals for initial population size \(X_0 = n\)

Now we want to compute the expected total number of individuals that ever exist in this population for an initial population size of \(X_0 = n\). Notice that the calculation will be the same as in Step 3, but now we have \(E_1 = n\) instead of 1. So the expected total number of individuals is: $$ E_{total} = E_1 \sum_{i=0}^{\infty} (1-\mu)^i = n \frac{1}{1-(1-\mu)} = n\frac{1}{\mu} $$ Thus, for initial population size \(X_0 = n\), the expected total number of individuals that ever exist in this population is given by: $$ n\frac{1}{1-\mu} $$

Key concepts

Extinction Probability

Extinction probability is a vital concept in the study of branching processes, which are mathematical models used to describe the dynamics of a population. In the context of branching processes, extinction probability is denoted as \(\mu\), and it represents the likelihood that the population will eventually cease to exist. When considering a population in which each individual reproduces independently, the future size and existence of that population depend on the offspring of each individual.

If the extinction probability \(\mu\) is less than one, there is a possibility for the population to continue indefinitely, as the probability of population survival—equivalently, the probability of non-extinction—is \(1-\mu\). For instance, if \(\mu = 0.5\), there's a 50% chance the population will eventually go extinct. However, this also means there's a 50% chance it will survive and potentially grow. It's crucial for students to understand that the lower the extinction probability, the higher the chance of population survival, which directly impacts the expected total number of individuals in the population over time.

Expected Number of Individuals

When studying population dynamics through branching processes, the expected number of individuals is an essential measure that tells us about the average population size we can predict in future generations. It is rooted in the idea of an 'average scenario' based on the probabilities of reproduction and extinction at play. It's not the number that will definitely appear in any given generation, but rather the number that we expect based on the rules and probabilities of the model.

Starting with a single individual, the expected number of individuals in the first generation \(E_1\) is 1. As long as the extinction probability \(\mu\) is less than 1, each subsequent generation has a positive expected size, calculated recursively by multiplying the expected number of individuals in the previous generation by \(1-\mu\), the probability of survival. When we calculate the expected total number of individuals over the entire existence of the population (assuming a start of one individual), we find this total to be \(\frac{1}{1-\mu}\), a result that holds as long as the assumption \(\mu<1\) remains true. For multiple initial individuals \(X_0=n\), the expected total number of individuals ever to exist in the population follows a similar pattern and is given by \(n\frac{1}{1-\mu}\). Understanding this concept allows students to grasp how initial conditions combine with the reproductive probabilities to dictate the long-term outcomes of populations.

Geometric Series

The geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This concept is intricately connected to branching processes because the total expected number of individuals in the population can be expressed as a geometric series.

In particular, when calculating the expected total number of individuals in a branching process, we sum across all generations. For a starting population of one individual and extinction probability \(\mu < 1\), the series becomes \(1 + (1-\mu) + (1-\mu)^2 + (1-\mu)^3 + \dots\). This series has a common ratio of \(1-\mu\) and sums up to the expression \(\frac{1}{1-(1-\mu)}\), which simplifies to \(\frac{1}{\mu}\).

It's important for students to recognize how the sum of an infinite geometric series provides key insights into long-term population predictions. By using the geometric series sum formula, \(\frac{a}{1-r}\), where \(a\) is the first term and \(r\) is the common ratio, they're able to reach a closed-form expression for the expected total population size. In the given exercise, when \(X_0=n\), the series starts with \(n\) instead of 1, and the resulting expression for the total expected number of individuals scales accordingly to \(n\frac{1}{1-\mu}\).