Mathematical Contributions to the Theory of Evolution. 



XV. A MATHEMATICAL THEOEY OF RANDOM MIGRATION. By 

 Karl Pearson, F.R.S., with the assistance of John Blakeman, M.Sc. 



(1) Introductory. In dealing with any natural phenomenon, — especially one 

 of a vital nature, with all the complexity of living organisms in type and habit, — 

 the mathematician has to simplify the conditions until they reach the attenuated 

 character which lies within the power of his analysis*. The problem of migration 

 is one which is largely statistical, but it involves at the same time a close study 

 of geographical and geological conditions, and of food and shelter supply peculiar 

 to each species. Some years ago the late Professor Weldon started an extensive 

 study as to the distribution of various species and local races of land snails, but 

 he was struck by the absence in several cases of any definite change of environ- 

 ment at the boundaries of the distribution of a definite race. It occurred to me 

 in thinking over the matter that such boundaries, where they exist, may possibly 

 not be permanent. To take a purely hypothetical illustration : A species is pushed 

 back to a certain limit by a change of environmental conditions — say, an ice age. 

 Does it follow that if the environment again becomes favourable, that it will 

 rapidly occupy possible country ? What is the rate of infiltration of a species 

 into a possible habitat 1 It depends, of course, on a whole series of most complex 

 conditions, the rate of locomotion, the channels of communication, the distribution 

 of food areas and breeding grounds in the new country, and the connecting links 

 between all these. Every detail must be studied by the field naturalist in relation • 

 to each species. All the mathematician can do is to make an idealised system, 

 which may be dangerous, if applied dogmatically to any particular case, but which can 

 hardly fail to be suggestive, if it be treated within the limits of reasonable application. 

 The idealised system which I proposed to myself was of the following kind : 



(i) Breeding grounds and food supply are supposed to have an average uniform 

 distribution over the district under consideration. There is to be no special following 

 of river beds or forest tracks. 



* This is of course a perfectly familiar process to every mathematical physicist, but its unfamiliarity 

 leads the biologist to suspect or even discard mathematical reasoning, instead of testing the result 

 as the physicist does by experiment and observation. 



1—2 



