Branching Process - Extinction Problem

Extinction Problem

The ultimate extinction probability is given by

.

For any nontrivial cases(Trivial case refers to case where each member of the population has exactly one descendent. Then the extinction probability is 0.), the probability of ultimate extinction equals one if μ≤1 and strictly less than one if μ>1.

The process can be analyzed using the method of probability generating function. Let p₀, p₁, p₂...denote the probabilities of producing 0, 1, 2...offspring by each individual in each generation. Let d_m be the extinction probability by the mth generation. Obviously, d₀ =0. Since the probabilities for all paths that lead to 0 by the mth generation must be added up, the extinction probability is nondecreasing in generations. That is,

.

Therefore, d_m converges to a limit d, and d is the ultimate extinction probability. If there are j offspring in the first generation, then to die out by the mth generation, each of these lines must die out in m-1 generations. Since they proceed independently, the probability is (d_m-1)j. Thus,

.

The right-hand side of the equation is probability generating function. Let h(z) be the ordinary generating function for p_i:

.

Using the generating function, the previous equation becomes

.

Since d_m →d, d can be found by solving

.

This is also equivalent to find the intersection point(s) of lines y=z and y=h(z) for z≥0. y=z is a straight line. y=h(z) is an increasing (h'(z)=p₁+2p₂z+3p₃z2+...≥0)and convex (h″(z)=2p₂+6p₃z+12p₄z2+...≥0) function. There are at most two intersection points. Since (1,1) is always an intersect point for the two functions, there only exist three cases:

Case 1 has another intersect point at z<1.(See the red curve in the graph)

Case 2 has only one intersect point at z=1.(See the green curve in the graph)

Case 3 has another intersect point at z>1.(See the black curve in the graph)

In case 1, the ultimate extinction probability is strictly less than one. For case 2 and 3, the ultimate extinction probability equals to one.

By observing that h'(1)=p₁+2p₂+3p₃+...=μ is exactly the expected number of offspring a parent could produce, it can be concluded that for a branching process with generating function h(z) for the number of offspring of a given parent, if the mean number of offspring produced by a single parent is less than or equal to one, then the ultimate extinction probability is one. If the mean number of offspring produced by a single parent is greater than one, then the ultimate extinction probability is strictly less than one.