Question

Branching Processes: Show that if μ=1 then probability of extinction is 1.

Answer 1

We will try to derive the given solution with the help of two approaches.

1. loud thinking.

2. Graphical approach

Thinking aloud

Think about the three possible scenario that can exist for the number of offsprings for each individual of a current generation.

If the mean number of offspring per individual( µ )is more than 1 (so on average, individuals replace themselves plus a bit extra), then the branching process is not guaranteed to die out — although it might do.

Why? Because with each passing generation the overall number is increasing and it is not converging /reducing .Hence it will not die out unless there is an external epidemic./disaster.

But what will happen if µ is equal to 1 or less than it.

if the mean number of offspring per individual µ is 1 or less, the process is guaranteed to become extinct (unless Y = 1 with probability 1).

This is simple to understand . Why ? because now we have a situation wherein the number of individuals with each passing generation is either stagnant of dwindling. This condition is such that it assures us that the overall population will eventually reduce to zero (extinct) .it is only a matter of number of generations based on the degrowth rate.

Graphical method

Let is the mean family size distribution .

Let {Z0 = 1, Z1, Z2, . . .} be a branching process with family size distribution Y . Let µ = E(Y ) be the mean family size distribution, and let γ be the probability of ultimate extinction.

Let G(s) be the PGF of family size Y

then

G(s)=E(s^Y ) (expected value of the size of the population in a given generation)

The function G(s) is an increasing function in the interval (0,1)

Graphically, µ the expected value of family size will be rate of growth and is thus the gradient /slope .

now look out for the gradient /slope of the curve G(s) (growth curve)

We shall check for the nature of the gradient of the curve G(s) at points =1 and below it.

We shall compare it with the gradient of µ =1 and check if they superimpose or not .

for

Look At the graph above . µ lies in the lower half of the region created by the line t=s in the first quadrant

What about the curve G(s) ?

In the range of (0,1) (zone of consideration -probability)

the curve will NEVER cross into the lower half of the region created by the line t=s in the first quadrant .

Why?

Because the curve G(s) is convex and converging in (0,1) and it meets the t axis again.

Thus the curve G(s) and µ do not intersect /meet .

However please visualize hte scenario when µ tends to 1.

Then t=s

and thus G(s)=s

What does this equation tell us?

this scenario /situation assures that is equal to 0.

Survival rate is 0 and hence there will be extinction.

and proability of extinction is 1.