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# Population Ecology Using Probability

### Branching Processes

### Probability Generating Functions

### Expectation

### Random Sums Formula

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Age 16 to 18

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Branching processes, or tree graphs, model the growth and eventual size of a population. If we know the probabilities of the number of offpsring produced at each generation, then we can determine the probability of ultimate extinction, or the eventual population size.

Consider a variable *X,* where $P(X=0)=p_0, P(X=1)=p_1, ...$

This is an integer valued variable with its mass function as a sequence. We set two conditions:

- All probabilities need to be positive $ p_k \geq 0 $
- Only one event can and must occur, so $p_0+p_1+...=\displaystyle\sum\limits_{k=0}^{\infty} p_k =1$

The *probability generating function* *G*, is an ordinary function in terms of *s:* $$G_X(s)=p_0+p_1 s+p_2 s^2+...$$ **Question:** What is the value of *G(s)* when $s=0$? And when $s=1$?

**Example:** Consider a random variable *Y* with the geometric distribution with parameter *p*.

Then $P(Y=k)=p(1-p)^{k-1}=pq^{k-1}$ for $k=0,1,...$.

So *Y* has PGF given by: $$\begin{align*} G_Y(s) & = \displaystyle \sum_{k=1}^{\infty} p q^{k-1} s^k \\ &= ps \displaystyle \sum_{k=0}^{\infty} (qs)^k \\ &= \frac {ps}{1-qs} \end{align*}$$

We can relate the PGF to the mean, or *expectation*. Recall that: $$E(X)=\bar x = \displaystyle \sum_{all x}^{ } xP(X=x)$$We can extend this definition to not just a variable, but to a function of a variable: $$E(g(X))=\bar{g}(x) = \displaystyle \sum_{all x}^{ } g(x) P(X=x)$$This definition reminds us of our PGF polynomial, with the important result: $$ G_X(s)=p_0+p_1
s+p_2 s^2+...=E(s^X)$$

Consider a population of meerkats, where each individual has a random number of offspring in the next generation. Using this information, we can determine the total expected number of offspring in future generations.

First let $N, X_1, X_2, ...$ be independent variables, with $X_1, X_2, ...$ all having the same probability generating function *G*. Think of these *X* as the individual meerkats in our population. This also means that our PGF is given by $G(s)=p_0+p_1s+p_2s^2+...$, where $p_0=P(\text{no offspring}), p_1=P(\text{one offspring}) , ...$

We are interested in finding the PGF of the sum $X_1+X_2+...+X_N$ $$\begin{align*} G_T(s) & = E[s^T] \\ &= \displaystyle \sum_{n=o}^{\infty} E\Big [s^T|N=n\Big ] P(N=n) \\ & = \displaystyle \sum_{n=o}^{\infty} G(s)^n P(N=n) \\ & = E[G(s)^n] \\ &= G_N \Big( G(s) \Big) \end{align*} $$**Example:** Elephants (in most cases) only have
one offspring at a time, with probability *p*, say. We can model the number of offspring using the Bernoulli distribution with parameter *p.*

Generation *n+1* consists of the offspring of generation *n*.

Let $Z_{n+1}= \displaystyle \sum_{j=1}^{Z_n} X_j$ , where $X_j$ is the number of offspring of the *j*th individual in generation *n.*

In the first generation: $G_{Z_1} (s)=G_X(s)=(1-p)+ps$

In the second generation: $G_{Z_2} (s)=G_{Z_1} \bigg(G_X (s) \bigg)=(1-p)+p\big((1-p)+ps\big)=(1-p^2)+p^2 s$

Continuing, we see that at the *n*th generation: $G_{Z_n} (s)=(1-p^n)+p^n s$

Now click here to find out about branching processes and how we can use probability to determine the likelihood of a population becoming extinct.

bioNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of the biological sciences, designed to help develop the mathematics required to get the most from your study of biology at A-level and university.

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