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Extinction in a branching process: Why some of the fittest strategies cannot guarantee survival

The fitness of a biological strategy is typically measured by its expected reproductive rate, the first moment of its offspring distribution. However, strategies with high expected rates can also have high probabilities of extinction. A similar situation is found in gambling and investment, where strategies with a high expected payoff can also have a high risk of ruin. We take inspiration from the gambler's ruin problem to examine how extinction is related to population growth. Using moment theory we demonstrate how higher moments can impact the probability of extinction. We discuss how moments can be used to find bounds on the extinction probability, focusing on s-convex ordering of random variables, a method developed in actuarial science. This approach generates "best case" and "worst case" scenarios to provide upper and lower bounds on the probability of extinction. Our results demonstrate that even the most fit strategies can have high probabilities of extinction.