The basic reproductive ratio of a word, the maximum size of a lexicon

Language is about words and rules. While there is some discussion to what extent rules are learned or innate, it is clear that words have to be learned. Here I construct a mathematical framework for the population dynamics of language evolution with particular emphasis on how words are propagated over generations. I define the basic reproductive ratio of word, R, and show that R > 1 is required for words to be maintained in the lexicon of a language. Assuming that the frequency distribution of words follow Zipf's law, an upper limit is obtained for the number of words in a language that relies exclusively on oral transmission.