Extinction in agent-based and collective models of bet-hedging

Bet-hedging is a phenotype diversification strategy that combines a fast-growing vulnerable phenotype with a slow-growing resistant phenotype. In environments switching between favorable and unfavorable conditions, bet-hedging optimizes growth and reduces fluctuations over a long time, which is expected to reduce extinction risk. Here, we address directly how bet-hedging can reduce extinction probability in an agent-based model. An agent-based model is appropriate for studying extinction due to the low number of individuals close to extinction. We also show that the agent-based model converges to the collective model behavior for populations of $100$ individuals or more. However, the collective model provides relevant qualitative insight even for low populations. The collective model provides expressions for extinction that stress the relevance of the population number, showing that a factor four increase in the total population has a greater effect than a change of strategy from maximum growth to minimal extinction. This work provides further insight into the impact of finite population effects on the bet-hedging strategy's success.