On size and extinction: a random walk model predicts the body size of lowest risk for mammals

Question: Is the relationship between extinction risk and size in terrestrial mammals described by a peaked function or a monotonic function? Mathematical method: We develop a population viability analysis model where species take random walks at generational time steps. The model works like the classic gambler’s ruin problem where risky combinations of variance in growth rate, population density, and generation length are eliminated from an evolutionary game. Key assumptions: Our model ignores speciation. It assumes that the population growth rate at evolutionary time scales is zero. It assumes an unbiased random walk. Chronological time is adjusted for generation length, so that longer-lived species make fewer ‘gambles’ in the same period of time. Conclusions: Particular combinations of variance in growth rate and average population density yield an extinction function that predicts a size of lowest relative extinction risk for terrestrial mammals. This size is close to the mode of continental body size distributions (at about 0.1 kg). Generation length is a fundamental evolutionary time scale