A MATHEMATICAL THEORY OF RANDOM MIGRATION 29 



the range. The Rayleigh solution continues to give, sensible densities beyond 

 the range, although they may be sufficiently small to be neglected in practice. 



For rough purposes a first approximation to the infiltration curves may be found 

 from the Rayleigh solution, they will err on the side of safety if we are con- 

 sidering the effect of a clearance at a considerable distance from the boundary. 

 But with the aid of the tables of the w-functions and the ^-coefficients, it is 

 not difficult to obtain the actual form of the infiltration curves as I have done 

 in the present case. Diagram VII. compares the Rayleigh approximation and the 

 infiltration curve for n = 7. 



It will be seen that an infiltration curve of the first order gives not only the 

 density of the population after a first migration into cleared or unoccupied area 

 across a straight boundary, but also the diminution of density on the populated 

 side of the area, when we put c negative, i.e. it gives both the 'depopulation' 

 and ' repopulation .' The reduced density at the boundary is %N, and if we take 

 the point where the infiltration curve cuts the vertical through the boundary as 

 origin, we see that it is centrally symmetrical; or the loss of population at a 

 given distance from the boundary is exactly equal to the gain at the same 

 distance on the opposite side of the boundary. 



If we require an infiltration curve of the second order, we must now multiply 

 the ordinates of the curve of the' first order by (i) the average fertility of the 

 species, say ju,, and (ii) the survival rate A. If the environment be the same on 

 either side of the boundary, and neither ft nor A affected by the density of the 

 population, then /uA may be treated as a constant and the infiltration curves of 

 higher orders can be found with moderate ease for simple cases. We thus have 

 the distributions after two, three or more migrations accompanied by reproduction 

 and death. On the other hand both ju, and A may be functions of the density 

 of the population, and in this case the ordinates of the infiltration curves of the 

 second and higher orders can only be determined when the nature of /u, and A 

 is known. On the whole it is probable that the average fertility depending 

 on the mating frequency will be highest where the density is greatest, as mating 

 opportunities will then be most frequent, but in such cases the survival rate 

 A may be lower, as more enemies are likely to be present and the food supply 

 is also likely to be less, where the population is densest. Thus /aA as a whole 

 may not be very different on the depopulated and repopulated sides of the 

 boundary. We shall only consider in this memoir cases in which this product 

 is (i) supposed constant throughout, or (ii) constant for each migration season ; but 

 supposing uniform environment on both sides of the boundary, it is conceivable 

 that pA. will be correlated with the population density and this will modify the 

 basis of the distribution from which the second and later migrations start. 



