Invasibility, the chance of a population to grow from rarity and become established, plays a fundamental role in population genetics, ecology, epidemiology and evolution. For many decades, the mean growth rate of a species when it is rare has been employed as an invasion criterion. Recent studies show that the mean growth rate fails as a quantitative metric for invasibility, with its magnitude sometimes even increasing while the invasibility decreases. Here we provide two novel formulae, based on the diffusion approximation and a large‐deviations (Wentzel–Kramers–Brillouin) approach, for the chance of invasion given the mean growth and its variance. The first formula has the virtue of simplicity, while the second one holds over a wider parameter range. The efficacy of the formulae, including their accompanying data analysis technique, is demonstrated using synthetic time series generated from canonical models and parameterised with empirical data.

Invasibility, the chance of a population to grow from rarity and to establish a large-abundance colony, plays a fundamental role in population genetics, ecology, and evolution. For many decades, the mean growth rate when rare has been employed as an invasion criterion. Recent analyses have shown that this criterion fails as a quantitative metric for invasibility, with its magnitude sometimes even increasing while the invasibility decreases. Here we employ a new large-deviations (Wentzel-Kramers-Brillouin, WKB) approach and derive a novel and easy-to-use formula for the chance of invasion in terms of the mean growth rate and its variance. We also explain how to extract the required parameters from abundance time series. The efficacy of the formula, including its accompanying data analysis technique, is demonstrated using synthetic and empirically-calibrated time series from a few canonical models.

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