ABSTRACT
An analysis is presented of the results to be expected from experiments on the homing of wild birds if the only factor operating is random search. It is found that this model reproduces the experimental results and predicts values for the parameters involved in the theory which are inherently plausible and which are in quantitative accord with experimental evidence. Attention is paid to the dependence of percentage return on distance of release, to the dependence of the average speed of return on this distance, and to the distribution in time of the returns. These three sets of data form a coherent picture within the framework of the hypothesis of random search. Certain types of migration are also briefly considered.
It is not suggested that this investigation proves that random search is indeed the mechanism by which the homing of wild birds is accomplished, but it is submitted that the large-scale experiments of the type considered here are not susceptible of the interpretation that a true navigational ability is involved.
It must be remarked immediately that this investigation concerns itself solely with the homing of wild, untrained birds ; the domestic pigeon or other birds which receive a special training are not considered.
The problem of random search is explicitly discussed in this paper in terms of bird movements, but the results may be applied to the ‘homing’ of any animate or inanimate object by a suitable choice of the parameters involved.
It is not suggested that, in practice, these flights are actually straight, they merely do not contain a major change of flight direction such as is supposed to terminate them.
A limit of another type is laid on L by the mathematical methods adopted in this work; when home is a small region enclosed by a continent L must not be many times as great as the linear dimensions of that region.
*This assumes that the whole coast is unknown. If a certain length of coast about home is presumed known then x′ refers more properly to the distance of the point of release from the nearest point of the known stretch. The results of this section on the average speed of homing then refer to the speed of return to the known stretch rather than to home itself, and a small correction to the observed speed of homing may be necessary on account of the flight from the beginning of known territory to home before comparison with the results of the theory based on random search. An example of such a correction is given in § 10.
See also the next footnote.
It should be remarked that the mathematical methods used in the treatment of these problems are approximate (see the Appendix, § 13) and that the approximation is the more accurate the greater the value of x′/L ; the sense of the approximation is to increase the average speed for small values of x′ by exaggerating the probability of return after short times. Another effect is to make R as quoted here rather smaller than a rigorous treatment would reveal. No sense is to be attached to Figs. 5 and 6 for values of t less than x′/v, a better curve would be given by setting the probability of return equal to zero below t = x′/v, taking a curve which starts from the abscissa at this point and fitting it to the given curve at a value of t two or three times as great as this.
A persistent phenomenon, which has impressed many observers, is the abnormally low speeds recorded by birds released in the immediate vicinity of home. A possible explanation, though one which, in view of other simple explanations, we would not press, is suggested by the present considerations. If, as the terrain gets more and more familiar, the bird searches ever more minutely—with an ever-decreasing L—then, by an extension of the above remarks, we should expect the average speed of return to fall, the nearer the point of release to home. Very familiar but not yet known territory would then behave almost as a reflecting barrier and progress through it would be very slow. But it seems to us more likely that the true explanation is based either on complete familiarity with the surroundings—the bird is then ‘in no hurry to return ‘—or on recollections of the trapping and transportation—the bird is then unwilling to return immediately to these unpleasant associations. Alternatively, it may be that the bird, having been trapped from the nest, does not realize, on release, that it is really its spell of duty at the nest and behaves as though it were its turn to seek food; it may not then return to the nest for the usual length of time, which may be days in some species.
If the whole coast is presumed to constitute known territory then the appropriate x′ will be the distance of the release point from the coast. Care must now be taken in interpreting the speed of homing as given by Fig. 3, since this will be the speed of homing to the coast only. An example of this kind is given in § 10.
Very few experiments are suitable for this type of analysis ; the requirements are that fairly large numbers of birds and large distances be involved.
Such observations are often made as a matter of course in investigations such as those cited above. Usually the initial scattering is random ; in some cases it is not, but as yet clear-cut evidence on initial orientation towards home in wild birds has not been forthcoming though it has been demonstrated by Matthews (1951) in the domestic pigeon.
From 
, to retrace one’s steps, and δtaσπoρá, dispersion. I am grateful to Prof. A. J. Beattie and Mr H. J. Lloyd-Jones for the suggestion of these words.
See note on previous page.
Although the flight characteristics do not enter the result of this calculation, the value of L enters the allowable latitudes of 600–700 miles mentioned above ; these are for an L of about 100 miles, but they change only as the square root of L; see the Appendix, § 17.
If it is indeed permissible to assume, as we do here, an orientation of the whole flight diagram towards home with an accuracy of about 20° either way, then most of the birds will, in fact, encounter some portion of the Hawaiian archipelago after time τ only, and so the effort required is only twice that appropriate to flight in a straight line. A rather sharper flight pattern could derive from crude solar navigation and with it successful homing to the archipelago with a still smaller effort. It is the writer’s belief that this is probably the case for the Pacific golden plover.
Very few golden plovers have ever been observed in flight over the ocean and none has ever been seen resting on the sea; this is not a strong argument against their so resting, as a flying bird is a very much more conspicuous object than a resting bird, and it is probable that resting would occur at night. There can be little doubt that the golden plover is capable of alighting on and taking off from open water when it is remembered that such unlikely species as the sand-martin (Stewart, 1950) and common buzzard (Stanes, 1950) appear to be capable of this feat.
This is not strictly the correct boundary condition; a more appropriate one would be that of boundary radiation into a vacuum (see Marshak (1947), for example), but the error introduced is small in our case.
