lotka: discontinuous evolution 53 



Let us expand the right hand member of (11) by Taylor's 

 theorem in the neighborhood of the point X„ F ro given by 



^ = R-e„b a ,Y CB = Q (13) 



dt 



^=riF m = (14) 



dt 



We thus obtain 



(B - ebY) = (B- e^b^Yj - \ d [ b ■ Y a I (X ■ - X J 



+ <M Y ^S)^ x ^ Y ~ Y ^ 

 +(U r --**^- Y 4- (15) 



or, in an obvious notation, and putting 



(X -IJ'=.u (16) 



(F-FJ = y (17) 



— = ax + @y + yx- + ixy '+ ey 2 + . • • (18' 



dt 



Similarly 



dl = a'x+p'y+y'x*+8'xy+eY+' • • (19) 



dt 



The solution of the system of differential equations (18) (19) 

 is 



x = A ie - ht + B ie - kt + A 2 e- 2ht + B 2 e- (h+k)t + C, e - 2kt 



+ yl3^ 3ht + 53e- l - h+k,t + CV- (h+2k,t + i)3e~ :3kt + • • • ■ .(20) 

 2 / = a i e- ht + & i e -kt +- . :' (21) 



The values of h, k, A i,A 2 , . . . a x , a 2 , .... can 

 be determined by substituting the solution in the original equation 

 and equating the coefficients of homologous terms. It is unnec- 



