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. 1 Extinction of a branching process Reproduction is necessary for the survival of populations. Populations with high rates of reproduction will often avoid extinction. However, a population may have a high expected reproductive rate but may nevertheless go extinct with near certainty (Lewontin and Cohen, 1969). For example, populations with large variation in reproductive success can sometimes have a high probability of extinction, even if they have a high expected growth (Tuljapurkar and Orzack, 1980). Similarly, investors and gamblers can avoid Gambler’s Ruin through growth of capital. However, a gambler should not simply apply the strategy with the highest expected growth rate ∗corresponding author 1 ar X iv :1 20 9. 20 74 v3 [ qbi o. PE ] 1 6 M ay 2 01 3 as it may also have a high risk of ruin. For example, investors can use the Kelly ratio (Kelly, 1956) to maximize expected geometric growth of their capital but strict adherence to this ratio can be risky, and playing a more conservative strategy is often recommended (MacLean et al., 2010). To estimate the probability of Gambler’s Ruin, one can use approximations based on moments (Ethier and Khoshnevisan, 2002; Canjar, 2007; Hürlimann, 2005). Here we apply these approaches to estimate the probability of extinction in a branching process. The mathematics of Gambler’s Ruin is very similar to that of extinction in a branching process (Courtois et al., 2006). Both statistical models involve a random variable (payoff/offspring number), resulting in a random walk (change in capital/change in population size), and an absorbing state (ruin/extinction). Moreover, both processes are assumed to be Markovian, and finding the probability of ruin/extinction involves solving for the root of a convex function. Here we examine the random variable representing the number of offspring, and investigate how the moments of this random variable are related to the probability of extinction. We demonstrate an important relationship between these moments and extinction: odd moments favor survival and even moments favor extinction. The first moment of the offspring distribution, its mean, has the biggest influence on extinction. However, the first moment alone is not usually informative about extinction probabilities. In fact, strategies with arbitrarily large first moments can nevertheless go extinct with near certainty. Some of the “fittest” strategies can be highly unlikely to survive. Using the first few moments of the offspring distribution, one can obtain bounds on the ultimate probability of extinction (Courtois et al., 2006; Daley and Narayan, 1980). These bounds provide “best case” and “worst case” distributions. We present these bounds, termed s-convex extremal random variables, adapted from actuarial science and research on the gambler’s ruin problem (Denuit and Lefevre, 1997; Hürlimann, 2005; Courtois et al., 2006). We find the conditions under which these extremals provide non-trivial bounds. Using some simple examples, we demonstrate how these methods can be used to compare distributions using their moments. 2 Extinction in the Galton-Watson branching process To investigate biological extinction, we use a Galton-Watson branching process in which, at each discrete time interval, every individual generates i discrete offspring with probability pi, and zero offspring with p0. Without loss of generality we assume that an individual produces its offspring and then dies, so that each individual in a population is restricted to a single generation. The offspring number is a random variable, which we denote by X. Let n be the maximum value of X so that X takes values in the state space Dn = {0, 1, 2, ..., n} At any given time t, the size of a population (Zt) is the number of individuals in the branching process. We set Z0 ≡ 1 unless otherwise specified. The probability of extinction of a branching process is q ≡ limt→∞ P (Zt = 0|Z0 = 1). If the starting size of the population is