II. APPENDIX TO CHAPTER IV 



CALCULATION OF THE COEFFICIENTS OF MULTIPLICATION AND 

 OF THE EQUATIONS OF THE STRUGGLE FOR EXISTENCE 



We have had to deal frequently with the coefficients of geometric increase or 

 coefficients of multiplication as well as with the equations of competition. 

 For calculating the latter a very simple approximate integration by the method 

 of Runge-Kutta is necessary, with which, however, most biologists are not 

 familiar. Therefore we will briefly give here the technical details of the cal- 

 culations and examine a concrete example. 



/. We possess experimental data on the separate growth of the first and second 

 species. We wish to calculate the maximal masses, Ki and K 2 , and the coefficients 

 of multiplication, bi and b 2 . 



Example: The growth of Schizosaccharomyces kephir under anaerobic con- 

 ditions (1931). Placing the experimental data on the graph we determine 

 approximately the maximal mass K (Fig. 41). For calculation of the param- 

 eter b let us apply the method elaborated by Pearl and Reed. For every 



K - N 

 individual observation (N u N 2 , etc.) we take K — N, and then — — — and log 



K-N K-N 



— — — -. Placing the calculated log — — — against the corresponding values of 



time (t) we should obtain a straight line in the case of a symmetrical S-shaped 

 curve (Fig. 41, bottom). We draw this straight line approximately, approach- 

 ing the experimental observations. 



Some brief theoretical explanations are needed here. The differential equa- 



dN K — N 



tion of Pearl's logistic curve : = bN — — — after integration takes this 



if zr at 



form: N = . Hence it follows that e a ~ bt = — — — , and therefore 



1 + e a ~ bt N 



. K-N . , . K-N 



log — — — = (a — bt) loge. Taking natural logarithms we obtain: In — — — =■ 



P K- N 



a — bt. Now it is clear that if we place the Napierian logarithms of — — — 



against the absolute values of time (t) we must obtain a straight line whose 

 tangent will be the required coefficient of multiplication b, whilst a charac- 

 terizes the position of the zero ordinate. Their calculation is reduced to 

 determining the equation of the straight line we have drawn approximately. 

 For this purpose we measure the ordinates of two points of the straight line, 

 for instance t = 0, logio y = m, and t = 80, logio y = n. Passing on to the natural 

 logarithms, i.e. multiplying m by 2.3026 we obtain directly the coefficient a 

 (a — Ob = m X 2.3026). We can then easily calculate b in this way : a — 806 = 



149 



