A MATHEMATICAL THEOKY OF EANDOM MIGRATION 43 



Thus u f, (0, 0) = (^A) 9 (1 - -5156) N 



= (/xA) 9 -48iV. 



We see accordingly that if the fertility and the death-rate were the same in 

 the clearance and in the populated district outside, the density at the centre 

 of the cleared patch would at the end of the year be almost 50 per cent, of 

 that in uncleared country. It is thus obvious that clearance can be of small 

 use, unless it is followed by permanent preservation of sterility. Even if one 

 annual clearance were made it is very unlikely — if the actual values of the 

 constants are at all near those assumed — that the mosquitoes would not by the 

 9th or 10th breeding cycle within the year before the annual clearance was 

 repeated have reached a very substantial density even at the centre of the 

 patch. We have thus an additional argument in favour of rendering a district 

 not only sterile, but keeping it so. In such a case since v i and v„ i/f 2 , t/> 4 are 

 negative we shall have a density somewhat less than : 



^.(O, 0) = iV{l-4P (2-540) P (5'081)} = iV(l- "9889) about. 

 Thus: i^ 6 (0, 0) = , 01iV approximately. 



It follows that in the centre of such a rectangular patch, there would roughly 

 be only about 1 mosquito for every 100 in uncleared country. 



But while this shows that such a sterile patch would be a great improvement 

 for a denizen at the centre it is well to enquire what happens in such patches 

 some way from the centre. I accordingly add the following illustration. 



Illustration II. A square area of one mile side is cleared and kept permanently 

 sterile. What will be the density at the centre and a quarter of a mile from 

 the centre on the same assumption as before ? 



Here a- b = 880 yds. 



At the centre i ?1 = 77 2 = e 1 = e 2 = 2 , 54 and: 



x /, (0, 0) = N[l - 4 {P (2-54)} 2 ] = N{1 - (-9889) 2 } = -022iV ; 



or, we find one mosquito for every fifty in uncleared country. Taking our 



quarter of a mile directly towards one of the boundaries, we have h = 440, 



k = 0, and : 



^ = 1-27, t? 2 = 3-81, ei = e 2 = 2-54. 



Thus: 1 /,(440,0) = JV[l-{P (l-27)+P (3-81)}{2P (2-54)}J 



= N{1 -(-3980 + -4999) (-9889)}= -U2N. 



Thus at ^ mile from the centre (or from the edge) of the clearance, the 

 density is 11 per cent, of that in uncleared country. It may be doubted whether 

 this is a sufficient reduction, and, supposing the above assumptions to be any- 

 thing like roughly correct, it may be needful to render more than a square 

 mile permanently sterile to protect a patch of one square half-mile. 



6—2 



