The Mathematical Representation of Migration
Authors: Preston F. W. 
Published: May 2006

If sufficiently systematic numerical observations or trappings of birds or other animals on migration are made, and the accumulating totals plotted on probability paper on a time base or abscissa, they will frequently plot to a fairly good straight line. This means that the “distribution” of the observations in time is approximately Gaussian, and from the plot the peak of migration and the standard deviation of the timing can be ascertained with some accuracy. This can be done, for most passage—migrant species, for both spring and fall migrations, and so the distance or interval between peaks can be ascertained also. Theoretically the distribution cannot be exactly Gaussian, since the Gaussian curve would take an infinity of time to become complete. A more suitable theoretical form of graduating curve is the cosine—power distribution, whose parameters can be ascertained from the Gaussian parameters obtained from the probability—paper plots. The cosine—power law is a well—known statistical distribution, being the trigonometrical form of Karl Pearson's Type II statistical distribution curve, which is very similar to the Gaussian for high values of m, but extends only over a finite range (– p2 ° x ° p2). Different species of birds spend very different periods of time in the north, but in general those that go early return late, and the bisectrix of the interval between migration peaks tends to be fairly constant for nearly all species. It is found that for most species this bisectrix coincides quite closely with the peak—of—heat (or cold) and comes about a month after the peak—of—light (or darkness). Since it is relatively fixed in time, it forms a convenient zero of coordinates, and, in order so to use it, it is convenient to express the cosine—power curve as a sine—power curve (O ° x ° p). It is then readily possible to express in mathematical form the distribution of migration movement in time by adding the first harmonic (sin 2x), when the migrant spends in the north a period appreciably longer or shorter than 6 months. Real times appear to range from 4 to 8 months. Four species which comprise, at a given locality, a mixture of passage migrants and summer residents (or winter residents) it appears likely that the bisectrix is different for the two categories. This seems somewhat universally the case in Britain and frequently in North America. It may then become useful to give the phase angle of the bisectrix, being the angle or number of days by which the bisectrix lags the winter solstice (December 22). This angle may apparently by zero for some of the swallows, like the Purple Martin in the Pittsburgh area, or it may be 30@? (or 30 days) or more for more typical species. The phase angle appears to lag more for the most northerly—breeding local populations, and may increase somewhere around 1 degree (or one day of time) for each degree of latitude. There is some evidence that breeding birds occupy their summer habitat as soon as it is habitable, and leave as soon as they can–as soon as their breeding obligations are discharged. This time required for breeding may not be very different at different latitudes, but southerly zones become habitable earlier than the northern, and these two factors move the bisectrix accordingly. The Standard deviation of the time of migration for may species in eastern North America is of the general order of 5 to 10 days. It may be more in California and in milder climates generally. It is normally less on the spring migration than on the fall one: the birds are better synchronized in spring. In some cases it is probable that the fall migration is spread out, almost mad bimodal, by young and adults traveling at somewhat different times. In the spring, the two sexes, or breeding and nonbreeding birds, may move separately and so prolong the migration. The method or factor by which such close synchronism is achieved is a biological rather than a mathematical matter. In the ultimate analysis most migrations can be regarded as periodic fluctuations in the zonal or latitudinal distribution of life on the planet, and cannot be understood apart from it.