Intraspecific variation may be continuous, or it may be quantized, if the number of structures present is always an integer. If there is some modal number of structures present in almost all individuals, variation is said to be modal. A developmental process is defined as one of ‘simple quantization’ if, first, it gives rise to an integral number of structures, and secondly, if the number of structures formed depends on the ratio between two continuous variables, for example the field size and the chemical wavelength in the model suggested by Turing (1952). Whether variation is quantized or modal will then depend on the accuracy with which these continuous variables are regulated. The larger the modal number, the more accurate must this regulation be. Data on the range of continuous variation within animal populations suggest that simple quantization cannot give rise to modal numbers greater than about 5 to 7. Yet modal numbers of 30 or more occur. Three processes which might account for this discrepancy are suggested, and evidence is presented to show that two of them occur. These are ‘multiplicative’ processes, involving successive processes of simple quantization, and ‘chemical counting’ processes, depending on qualitative differences between successively formed structures. The relevance of processes of quantization to the genesis of two-dimensional patterns is discussed.
Nonnegative multiplicative processes reaching stationarity in finite time