How to use this Galton–Watson extinction calculator

A Galton–Watson branching process tracks a population one generation at a time, where every individual in a generation produces a random number of offspring according to the same probability law. That makes the model useful for many different situations, from a chain reaction to a rumor to an epidemic-style spread or a family line, because the same reproduction rule is applied again and again. This calculator lets you choose a starting population, pick an offspring distribution, and see how the chance of extinction changes as generations pass. The point is not just to get a final answer, but to see how quickly a process becomes likely to die out and how that result changes when the offspring law becomes more or less generous.

The heart of the model is its probability generating function, written G(s). If the probability of producing k offspring is attached to the coefficient of s^k, then the extinction probability is the smallest solution to q = G(q) in the interval [0,1]. That fixed point is what the calculator iterates toward. Instead of treating extinction as an abstract theorem, the tool shows the recursion that drives it, so you can watch the probability rise generation by generation and compare the short-run trajectory with the eventual long-run limit.

The mean offspring count m = G’(1) tells you which branching regime you are in. When m < 1, the process is subcritical, so extinction is unavoidable even if the first few generations look lively. When m = 1, the process is critical, which means it can persist for a long time but still dies out with probability one. When m > 1, the process is supercritical and survival becomes possible, although it is not guaranteed because a highly variable offspring distribution can still collapse early. This is why the calculator reports both the mean and the variance instead of showing only the average number of offspring.

The calculator evaluates the chosen law in three connected ways. First, it computes the mean and variance so you can see the overall reproduction scale and how uneven that reproduction is from one lineage to another. Second, it iterates the generating function starting from q₀ = 0 and applying q_t+1 = G(q_t) until the estimate settles, which gives you a generation-by-generation extinction curve. Third, it applies the single-founder extinction probability to the initial population, because if one founder survives or dies independently of the others, the chance that the whole starting group disappears is the per-founder extinction probability raised to the power of the initial count. That means the same offspring law can look very different depending on whether you start with one ancestor or several founders.

Interpreting mean, variance, and criticality in Galton–Watson branching processes

In this branching-process calculator, the mean offspring count is the quickest way to tell whether the process is shrinking, holding steady, or growing in expectation. A value below one means the expected lineage size falls from generation to generation. A value near one means the process sits close to the knife edge between die-out and expansion. A value above one means the average lineage can grow, even though random fluctuations may still wipe out a particular run. Variance matters because the same mean can hide very different behavior: one reproduction law may be steady and tightly clustered, while another may spend a lot of probability mass on zero offspring and a smaller mass on very large counts. Those two laws can share the same mean and still feel very different in practice.

• Poisson(λ): useful when offspring counts are unbounded but cluster around a single average. In this model, the mean and variance are both λ, so the same parameter controls both the center and the spread of the offspring law.
• Binomial(n, p): useful when each individual has a fixed number of reproduction chances and each chance succeeds independently with probability p. The calculator caps the offspring count at n, which makes the distribution easier to interpret when there is a hard upper limit on reproduction.
• Custom discrete: useful when you already know the reproduction law from data, a hand-built toy model, or a scenario you want to approximate explicitly. Enter the probabilities as a discrete list and the calculator normalizes them so they sum to one.

The fixed-point equation makes the regime test easy to read. When the mean is below one, the only extinction root is q = 1, so eventual disappearance is certain. When the mean is above one, the calculator can find a root below one, and that root becomes smaller as the offspring law grows more fertile. The iterative method mirrors the way the process itself unfolds: each new estimate feeds the previous generation’s extinction probability back into the generating function, so you can see the probability climb instead of jumping straight to the answer. That is especially helpful when you want to compare similar distributions and see whether changes in variance or in the average offspring count matter more for extinction risk.

Worked example: a Poisson branching process with λ = 0.8

Suppose you start with five founders and choose the Poisson option with λ = 0.8. Because the mean offspring count is below one, the process is subcritical and extinction is certain in the long run. What the calculator shows, though, is how that certainty emerges gradually rather than instantly. For a single founder, the generation-by-generation extinction probability rises from about 0.4493 after one generation to about 0.6437 after two, 0.7520 after three, 0.8200 after four, 0.8660 after five, and 0.9628 by generation 10. That sequence is the visual signature of a process that is failing more and more often as the recursion keeps applying the same subcritical offspring law.

Once you have five starting individuals, the all-extinct probability is much larger because every founder has to disappear for the whole starting group to vanish. With the same Poisson law, the approximate chance that all five founders are extinct is about 1.83% by generation 1, 11.05% by generation 2, 23.92% by generation 3, 36.99% by generation 4, 48.71% by generation 5, and 82.77% by generation 10. That difference between the single-founder curve and the whole-group curve is the main reason the calculator reports both. In a branching process, one survivor can keep the lineage alive, so the starting population matters a great deal even when the offspring law stays the same.

Reading the table this way separates short-run survival from long-run fate. A supercritical process can still look fragile in the first few generations, especially when the variance is large, but its eventual extinction probability falls below one because some runs continue branching successfully. A critical process can keep going for a while too, but the probability of eventual die-out still reaches one. A subcritical process, by contrast, is headed toward extinction even if one or two generations happen to look unexpectedly healthy. If you change the starting population in the calculator, the single-founder probability does not change, but the probability that the whole initial group disappears can move very quickly because the model multiplies the founder-level outcome across everyone who starts the process.

Limitations and assumptions in Galton–Watson branching models

This calculator assumes that every individual in every generation follows the same offspring law and that those draws are independent. It does not include age structure, seasonal effects, spatial spread, immigration, migration, resource limits, or feedback where the population size changes the reproduction law. That keeps the model mathematically clean and makes the extinction recursion easy to interpret, but it also means the output should be read as a branching-process result rather than as a full simulation of a real population or outbreak. If you are working with a biological, social, or network process, the calculator is best understood as a compact way to test the consequences of one reproduction rule under idealized assumptions.

Custom probabilities are normalized so they sum to one, which is helpful when you are entering a rough empirical distribution or a toy example that only needs a few counts. Even so, it is still worth checking whether the counts you supply really match the process you want to study, because the calculator treats the normalized values as the exact reproduction law once you run it. The root search is numerical rather than symbolic, so distributions with a very flat generating function or a mean close to the critical threshold may converge more slowly, though the result is still a model-based estimate rather than a literal forecast. When you compare scenarios, the safest approach is to vary one parameter at a time and watch whether the process crosses from subcritical to critical or from critical to supercritical, because that boundary is usually where the extinction story changes most dramatically.

Formula: how Galton–Watson extinction is built from the offspring PGF

The calculator’s result comes directly from the branching-process inputs you enter: the initial population, the offspring distribution, and the generation horizon. There is no separate hidden model behind the scenes. Instead, the tool evaluates the chosen probability generating function, uses it to iterate the extinction probability, and then applies that probability to the number of founders you started with. In other words, the output is the model that your parameters define, which is why the same calculator can represent a very gentle reproduction law, a borderline critical case, or a strongly expanding process.