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The effect of environmental change on the rate of innovation and level of cultural complexity in a population is a theoretically understudied piece of an important puzzle at the heart of cultural evolution. Many mathematical models of cultural complexity have focused on the role of demographic factors such as population size or density. However, statistical studies often point to environmental variability as an important factor determining complexity in many cases. The aim of this study is to explore the interaction between environmental fluctuations and the rate of cultural innovation within a population and to examine the relationship between rates of innovation and the probability of maintaining a complex cultural repertoire in a changing environment. Two models are presented that draw on previous models used to examine rates of genetic mutation. The models show that, as in a genetic system, the stable rate of …