A MATHEMATICAL THEOEY OF RANDOM MIGKATION 35 



that the advance of a slow scattering species* may be comparatively small. The 

 inference can accordingly be made that the existing boundaries of the geographical 

 distribution of certain forms of animal and plant life which are not marked by 

 natural barriers, and which do not correspond to obviously changing environ- 

 mental conditions, need not after all be associated with subtle physical differences 

 which have escaped the observation of the naturalist. The species may be pro- 

 gressing into an unoccupied area, but at a rate hardly observable in the time during 

 which accurate distribution observations are available. If this view be correct 

 we should expect such boundaries with no apparent environmental change in 

 the case of species for which we might reasonably predict a small n and I. 



Illustration II. I have endeavoured to apply the above theory to the im- 

 migration of mosquitoes into a cleared area. We will suppose in the present 

 treatment that the area bounded by a straight line (some attempt to allow 

 for curvature of the boundary will be considered later) has been cleared but is 

 not kept sterile to the species. I shall speak of a district as rendered sterile 

 to a species when it is made impossible for it to breed there, and kept sterile 

 when the breeding possibilities are persistently destroyed. The distinction is an 

 important one, especially in the mosquito case. For in the latter case all mos- 

 quitoes are immigrants, and in the former case we have not only immigrants, 

 but their produce. 



Major Ronald Ross, who has most kindly provided me with information as 

 to mosquito habits, makes the following remarks : 



(a) That the number of mosquitoes produced varies roughly (ceteris paribus) 

 as the extent of surface breeding area. 



(b) That the breeding area can be taken as consisting of numerous isolated 

 small pools or vessels of water scattered fairly uniformly over the country. 



(c) That the feeding places (houses, stables, birds, etc.) may be taken as 

 scattered pretty uniformly between the breeding pools. 



(d) That abundance or scarcity of food can scarcely influence the question 

 much. A single man or bird will yield enough food for many mosquitoes, and 

 if they starve it is not because the food is, not there, but because they cannot 

 reach it. They are therefore not likely to be drawn in general by special abun- 

 dance of food in any special direction. Wind tends to make mosquitoes " sit 

 tight," rather than allow themselves to be scattered. 



It would thus appear that on the average an " equi-swampous " condition of 

 the environment and random "flights" of the mosquito will not be very wide 

 of the truth. The difficulty is to form some estimate of n and I. On these 

 points again Major Ross came to my help, but naturally the statements he made 

 were with great reservation. 



* Of course any more quickly moving species that depends on this for food would have the same 



boundary, but in its case the boundary would be environmentally denned. 



5—2 



