A MATHEMATICAL THEORY OF RANDOM MIGRATION 



(8) Secondary Migration Problems. 



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Problem I. On one side of a straight line there is supposed to be a uniform 

 distribution of habitats ; on the other at starting no habitats. To investigate the 

 distribution in the unoccupied area after one migration. Each individual is supposed 

 to take n-fights to the new habitat. 



Let YYhe the straight line and a point at distance c from it on the unoccupied 

 side of it. Let N be the average density per unit of area on the occupied side. Then 

 after an n-flight migration, the contribution from P (co-ordinates r, ^) at will 

 be Nrhyhr <f> n (r 2 ), and integrating this all round a circle of radius r from A to C 

 within the occupied area, we have for the quantity F n (c) at 



f 00 /"cos -1 c/r 



F n (c) = 2NJ e j o <j> n {r*)rd x dr 

 = 2 JV I cos -1 c/r <f> n (r 2 ) r dr. 

 @ = -2NJ y n jp^ - = -2NJy n (c* + tf)dy (xliv). 



Hence 



dF„ 



dc 



The evaluation of this integral needs a further consideration of the w-functions. 

 By (xiv) 



-- - ^-er i ${j**} where e= ~ 2 ^- . 



4 



