Logistic Curve 339 



limited by the degree of realization of maximal size. When the 

 growth of a self-limiting population is expressed graphically, a typi- 

 cal S-shaped curve is obtained known as the logistic curve, shown as 

 (B) in Fig. 9.11. The curve is truly logarithmic only at its begin- 

 ning, departs from the logarithmic increase as impeding factors be- 

 come effective, reaches an injiection point at which the acceleration of 

 growth becomes negative, and approaches an asymptote representing 

 the limiting size (K) of the population. 



If we now apply to our previous numerical example the more real- 

 istic condition that increase is progressively curtailed as the popula- 

 tion grows, we obtain some such growth values as those indicated in 

 the lower part of Table 18 with a maximal population size in this 

 case assumed as 50. Besides the population totals at the end of 

 each year (or other period) we are also interested in knowing the 

 numbers added to the population during each unit of time. With 

 unimpeded growth the annual increment becomes larger and larger 

 indefinitely; but with self-limited growth the annual increment passes 

 through a maximum at the time when the logistic curve reaches the 

 inflection point. In our example the numbers added per year in- 

 crease to a maximum of 20 during the third year and then drop off 

 to zero in the seventh year. A curve showing these changes in the 

 increments to the population and their- relation to the logistic curve 

 is presented in the lower portion of Fig. 9.11. The fact that the 

 population has approached its asymptote with annual increment ap- 

 proaching zero does not necessarily mean that little or no reproduction 

 is taking place in the population— it simply means that births are com- 

 pletely offset by deaths, natality is equaled by mortality, or A^= M. 

 Under these circumstances reproduction may continue to be high ac- 

 companied by high mortality, or reproduction may be low with low 

 mortality; as long as A and M are equal the size of the population will 

 not change. 



Early in the history of the population only small numbers are added 

 each year because the breeding stock is small. During the middle 

 period annual increments are large, but, as the population reaches its 

 maximum size, small annual increments again occur either because 

 breeding is sharply curtailed or because the young produced suffer 

 severe mortality. The reader should particularly notice that the 

 largest annual increment is not found when the population is at its 

 maximum, but occurs at the inflection point of the logistic curve, that 

 is, at the time when the population is growing most rapidly. 



Populations of a wide variety of organisms, ranging from bacteria 

 to whales, have been found to follow the logistic curve in their growth 



