STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 



47 



JVi, N 2 = number of individuals of the first and second species 

 in a mixed population at a given moment. 

 bi, b 2 = potential coefficients of increase in the number of indi- 

 viduals of the first and second species. 

 Ki, K 2 = maximal numbers of individuals of the first and second 

 species under the given conditions when separately 

 grown. 

 a, 13 = coefficients of the struggle for existence. 



The rate of growth of the number of individuals of the first species 

 in a mixed population is proportional to its potential rate (&iiVi), 

 which in every infinitesimal time interval is realized in greater or less 

 degree depending on the relative number of the still vacant places: 



— ^ — . An analogous relationship holds true for the 



second species. The growth of the first and second species is simul- 

 taneous. It can be expressed by the following system of simulta- 

 neous differential equations: 



Rate of growth 

 of the first spe- 

 cies in a mixed 

 population 



> = < 



Potential in- 

 crease of the 

 population of 

 the first species. 



>X 



Rate of growth 

 of the second 

 species in a 

 mixed popula- 

 tion 



Potential in- 

 crease of the 

 population of 

 the second spe- 

 cies 



X< 



Degree of reali- 

 zation of the po- 

 tential increase. 

 Depends on the 

 number of still 

 vacant places. 



Degree of reali- 

 zation of the po- 

 tential increase. 

 Depends on the 

 number of still 

 vacant places 



j ) 



• (ID 



Translating this into mathematical language we have: 

 dN 1 = lhNi Kt-(N 1 + aNd) 



dt 

 dN 2 



= b 2 N 2 



K 2 - (N t + ffli) 



(12) 



dt K 2 



The equations of the struggle for existence which we have written 



