Algebraic determination of the evolutionary stable germination fraction

In this article, we consider a bet-hedging model with density dependence for germination behavior of dormant seeds. We show how to compute accurate values of the evolutionary stable germination fraction using a method based on expansion around the average seed bank size. The expansion is made up to the fourth order and is expressed in terms of the first four moments of the distribution of the year-to-year variation in the average yield per adult plant. This method leads to a simple algebraic equation, which can be solved without recurring to simulations of the seed bank dynamics. We first show the analytical derivation under general assumptions about the effect of density dependence. We then test our results with a numerical analysis where an explicit form of the density dependence has been taken.