Simple mathematical models are suggested to represent morph-ratio clines in which the morphs are determined by pairs of alleles. 2. The models show that natural selection can bring about the spread of modifier genes influencing the slope and position of such clines. 3. If the selective effect of a modifier on the heterozygote is less than the arithmetic mean of its effects on the homozygotes, the cline will be made steeper. If its effect on the heterozygote is more than the arithmetic mean of its effects on the homozygotes, the cline will be flattened. An additive effect will leave the slope unchanged. 4. A modifier with opposite effects on different genotypes may spread in some parts of a cline but not in others. Thus it can produce one or more sharp local changes (steps) in the cline. The position of these discontinuities need not be related to any sharp changes in the environment, but will depend on the nature of the interaction between the modifier and the polymorphic locus. 5. Migration will not prevent the response to a modifier, but may influence its magnitude. 6. It is suggested that there is an evolutionary trend towards the steepening of morph-ratio clines, and towards the intensification of any steps that may occur. 7. These results provide an explanation of the "area effects" found in Cepaea and Partula. It is necessary to appeal neither to the existence of cryptic environmental discontinuities nor to the action of the founder principle. 8. The effects of natural selection on clines in metric characters are briefly considered.
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Bryan Clarke (1966) studied this question.