  Thus, as a prelude to modeling, Lande et al. argue that demographic stochasticity “refers to chance events of individual mortality and reproduction, which are usually conceived of as being independent among individuals” whereas environmental stochasticity “refers to temporal fluctuations in the probability of mortality and the reproductive rate of all individuals in a population in the same or similar fashion. The impact of environmental stochasticity is roughly the same for small and large  populations.”[27]  This is further elaborated: “Random variation in the expected fitness that is independent of population density constitutes environmental stochasticity. Random variation in individual fitness, coupled with sampling effects in a finite population, produces demographic  stochasticity.”[28]  Nevertheless, environmental fluctuations or even random catastrophes affect the size of a population only insofar as they affect reproduction and death rates, that is, by creating demographic fluctuations; at least in this sense, the second and third categories of stochasticity are not conceptually independent of the first. Moreover, Lande and others regard random catastrophes as extreme cases of environmental stochasticity. Consequently, it requires explicit mathematical models in which these distinctions are made exact through formal  definitions.[29]  Usually, models of demographic stochasticity are distinguished from models of environmental stochasticity using as a criterion whether the stochastic factor explicitly depends on the population size as a parameter. If it does, the model in question is one of demographic stochasticity; if it does not, it is one of environmental stochasticity. This choice captures the intuition mentioned earlier that the effect of the former depends on the population size whereas the effect of the latter does not.