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Branching Process Simulator

Start with one individual, give each a random number of children, and repeat. The result is stark: the line either dies out or grows without limit — it never settles at a steady size. The threshold is exactly one child per individual on average, and the genuinely counter-intuitive part is what happens at that threshold. Replacing yourself exactly, on average, still means extinction with probability one.

What this generator does

Builds a distribution over having none, one or two children with the mean you choose, computes the exact probability the line eventually dies out, and simulates one run so the gap between a single outcome and the underlying probability is visible.

How to use this tool

  1. Set the mean number of children and how many generations to watch.
  2. Read the exact extinction probability before looking at the run.
  3. Generate several runs at the same setting — they differ enormously.
  4. Set the mean to exactly 100% and note that extinction is still certain.

Understanding the controls

Mean children each
As a percentage: 100% means each individual replaces itself on average, which is exactly the critical threshold. Below it extinction is certain; above it survival becomes possible.
Generations
How many generations to simulate, between 3 and 30. A surviving line grows very quickly, so long runs get large.
Seed
Any word reproduces exactly the same run.

Common use cases

  • Teaching branching processes and the critical threshold
  • Showing why surnames and family lines disappear
  • Demonstrating that a single simulation tells you very little
  • Generating population scenarios for a probability course
  • Setting the same run twice from a seed

How this generator works

The extinction probability is the smallest solution in the unit interval of an equation the process satisfies about itself: the chance the line dies out equals the chance it dies out through each possible number of children. Iterating that relation from zero converges to exactly that solution, and the check confirms the answer really is a fixed point by feeding it back in.

Randomness and fairness

The simulated run is random; the extinction probability is exact and comes from solving the equation rather than from counting simulations. A seed reproduces the same run exactly and is therefore explicitly not cryptographically secure. Without a seed the browser's cryptographically secure generator is used.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

  • Each individual has none, one or two children. Real offspring distributions have longer tails, which changes the numbers but not the threshold.
  • Generations do not overlap, and every individual reproduces independently with the same distribution.
  • The simulation stops at 5,000 individuals, since a surviving line grows without limit and would otherwise run away.
  • One run is shown. It illustrates the process; it does not estimate anything.
  • Nothing about a run survives the page closing, so copy the outcome table if you want to compare strategies later.

Privacy and your data

The process and the run are computed in your browser. Nothing about them or your seed is transmitted or kept.

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