Generations and dependence
Section of Galton–Watson branching, part of Stochastic processes
Generations and dependence
| Type | Section |
| Taught in | Galton–Watson branching |
| Part | Stochastic processes |
Under independent identical reproduction, Z_n is a Markov chain: the next generation depends on the past only through the current number of individuals. It is not a random walk. The law of Z_{n+1}-Z_n depends on Z_n, because more individuals mean more independent offspring draws. The conditional mean drift is (m-1)Z_n, which grows with the population. A density-dependent process would change that drift again, often by making a large population less productive per individual.
The extinction probability is the smallest root of f(s)=s in [0,1]. For a non-degenerate offspring law with p_0>0, that root is 1 when the mean is at most 1, and it is strictly smaller when the mean is greater than 1. The mean sets the threshold. The root uses the whole offspring law.
Used in
See also
Offspring probability generating function · Extinction criterion · Galton–Watson process · Mean and variance by generation · Extinction probability · Subcritical decay, and what the model leaves out · Other offspring laws · Total family size