Species arise and establish themselves over the geologic time scale. This process is manifested as a change in the relative frequency of occurrences of a given species in the global pool of species. Our main goal here is to model this rise and the eventual decline of microfossil species using a mixed-effects model where groups each have a characteristic occurrence trajectory (main effects) and each species belonging to those groups is allowed to deviate from the given group trajectory (random effects). Our model can be described as a “hat” with logistic forms in the periods of increase and decline. Using the estimated timings of rises and falls, we find that the lengths of the periods of rise are about as long as the lengths of the periods when species are above 50% of their estimated maximal occurrences. These latter periods are here termed periods of dominance, which are in turn about the same length as the species' periods of fall. The peak rates of the rises of microfossils are in general faster than their peak rates of falls. These quantified observations may have broad macroevolutionary and macroecological implications. Further, we hypothesize that species that have experienced and survived high levels of environmental volatility (specifically, periods of greater than average variation in temperature and productivity) during their formative periods should have longer periods of dominance. This is because subsequent environmental variations should not drive them to decline with ease. We find that higher estimated environmental volatility early in the life of a species positively correlates with lengths of periods of dominance, given that a species survives the initial stress of the environmental fluctuations. However, we find no evidence that the steepness of the rise of a species is affected by environmental volatility in the early phases of its life.

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