The timing of life history events in the presence of soft disturbances

We study a model for the evolutionarily stable strategy (ESS) used by biological populations for choosing the time of life-history events, such as arrival from migration and breeding. In our model we account for both intra-species competition (early individuals have a competitive advantage) and a disturbance which strikes at a random time, killing a fraction 1-p of the population. Disturbances include spells of bad weather, such as freezing or heavily raining days. It has been shown by Iwasa and Levin 1995, that when the disturbance is so strong that it kills any individual present when it strikes (hard disturbance, p=0), then the ESS is a mixed strategy (individuals choose their arrival date in an interval of possible dates, according to a certain probability distribution). In this case, individuals wait for a certain time and afterwards start arriving (or breeding) every day. In this paper we explore a biologically more realistic situation whereby the disturbance kills only a fraction of the individuals (soft disturbance, p>0). We also remove some technical assumptions which Iwasa and Levin made on the distribution of the disturbance. We prove that the ESS is still a mixed choice of times, however with respect to the case of hard disturbance, a new phenomenon arises: whenever the disturbance is soft, if the competition is sufficiently strong, the waiting time disappears and a fraction of the population arrives at the earliest day possible, while the rest will arrive throughout the whole period during which the disturbance may occur. This means that under strong competition, the payoff of early arrival balances the increased risk of being killed by the disturbance. We study the behaviour of the ESS and of the average fitness of the population, depending on the parameters involved. We also investigate how the population may be affected by climate change: namely the occurrence of more extreme weather events, which may kill a larger fraction of the population, and time shifts of the distribution of the disturbance. We show how the ESS and the average fitness change under the new climate and discuss which is the impact of the new climate on a population that still follows the old strategy. In particular, we show that, at least under some conditions, extreme weather events imply a temporary decrease of the average fitness of the population due to an increased mortality. In addition, if the population adapts to the new climate, the population may have a larger fitness