152 THE STRUGGLE FOR EXISTENCE 



where the numerical values of the parameters are already known, and where it 

 is necessary to calculate the values Ni and 2V 2 corresponding to different 

 moments of time. As this system of differential equations cannot be solved at 

 present, we will use the method of approximate numerical integration of 

 Runge-Kutta. 



§1. Ordinates at origin. The system of differential equations does not in- 

 clude the special parameters characterizing the values N\ and N 2 with t = 

 (corresponding to parameter a of the logistic curve). We know these values 

 from the experiment, and they correspond to those quantities of the first and 

 second species which are introduced into a mixed population at the very be- 

 ginning. The same quantities are also introduced into the separate popula- 

 tions. Therefore, according to the logistic curves of separate growth we cal- 

 culate the value Ni with t = 0, and N 2 with t = 0. The quantities obtained are 

 taken for the initial ordinates of the system of differential equations. 



§2. Calculations. The meaning of the calculations is that knowing the 

 value of Ni and iV 2 at t = 0, we give to time a certain increment h and for this 

 new moment (t + h) we calculate according to the system of the equations of 

 competition the corresponding increments of the populations of the first and 

 second species: (Ni + k) and (iV 2 + I). These calculations are made with the 

 aid of the formulae of Runge-Kutta, which for our case take the form given in 

 Table 8. 



dN\ , dN 2 

 The initial values of the ordinates enable us to calculate — — and — — . 



at at 



Multiplying them by h we obtain ki, and h. Thereupon in the second line we 



calculate again and starting no longer from the initial ordinates but 



dt dt 



with ( Ni H — -J and ( N 2 + - ) . Multiplying the result by h we obtain k 2 and 



h, and continue to calculate until we obtain fci, k 2 , k 3 , k A andZi, l 2 , h, h. We then 

 pass on to the right side of the table. Here we calculate § (fci + fc 4 ) and then 

 k 2 + k 3 . Summing and taking § of the sum we obtain the required increment 

 of the first species equal to k. The increment of the second species is calcu- 

 lated in the same way. 



The ordinates thus calculated (Ni + k) and (N 2 + 1) can in their turn be con- 

 sidered as already known initial values, and following the same plan we can 

 calculate the new increments corresponding to the moment of time (t + 2h). 

 Continuing these calculations we will find the growth curves of the first and 

 second species in a mixed population. Table 9 presents a numerical example 

 of the calculation of such curves for Saccharomyces (No. 1) and Schizosaccharo- 

 myces (No. 2) in a mixed anaerobic culture according to the experiments of 1931. 



§3. Final values ofNi and N 2 . The process of calculating will bring us to the 



moment when and — will be near to zero, and we will approach the final 



dt dt 



values of Ni and N 2 (the experiments described in Chapter IV were made in 



conditions of limited resources of energy, where on the disappearance of the 



unutilized opportunity growth simply ceased). If in the approximation to 



