In chapters 2 to 4 we discussed several models of extinction which make use of ideas drawn from the study of critical phenomena. The primary impetus for this approach was the observation of apparent power-law distributions in a variety of statistics drawn from the fossil record, as discussed in section 1.2; in other branches of science such power laws are often indicators of critical processes. However, there are also a number of other mechanisms by which power laws can arise, including random multiplicative processes (Montroll and Shlesinger 1982; Sornette and Cont 1997), extremal random processes (Sibani and Littlewood 1993), and random barrier-crossing dynamics (Sneppen 1995). Thus the existence of power-law distributions in the fossil data is not on its own sufficient to demonstrate the presence of critical phenomena in extinction processes. Critical models also assume that extinction is caused primarily by biotic effects such as competition and predation, an assumption which is in disagreement with the fossil record. As discussed in section 1.2.2.1, all the plausible causes for specific prehistoric extinctions are abiotic in nature. Therefore an obvious question to ask is whether it is possible to construct models in which extinction is caused by abiotic environmental factors, rather than by critical fluctuations arising out of biotic interactions, but which still give power-law distributions of the relevant quantities. Such models have been suggested by Newman (1996, 1997) and by Manrubia and Paczuski (1998). Interestingly, both of these models are the result of attempts at simplifying models based on critical phenomena. Newman's model is a simplification of the model of Newman and Roberts (see section 3.6), which included both biotic and abiotic effects; the simplification arises from the realization that the biotic part can be omitted without losing the power-law distributions. Manrubia and Paczuski's model was a simplification of the connection model of Solé and Manrubia (see section 4.1), but in fact all direct species-species interactions were dropped, leaving a model which one can regard as driven only by abiotic effects. We discuss these models in turn. The model proposed by Newman (1996, 1997) has a fixed number N of species which in the simplest case are noninteracting.
POWER-LAW DISTRIBUTIONS IN CIRCULATING MONEY: EFFECT OF PREFERENTIAL BEHAVIOR
