302 



POPULATIONS 



actual data. The following discussion leans 

 heavily on the book by Cause (1934) and 

 to some extent a paper by Thomas Park 

 (1939). 



Taking it for granted that the logistic 

 curve is a reliable picture of group increase, 

 what biological facts and suggestions can 

 be inferred from the study of the curve it- 

 self? Inherent in the curve are the follow- 

 ing properties of significance in an analysis 

 of population growth (Pearl and Reed, 

 1920; Pearl, 1924): 



1. The area (and/or volume) upon 

 which the population grows is a finite area 

 with definite limits, however large. 



2. The number of individuals (popula- 

 tion density) that can be supported in a 

 specified area is limited; in other words, the 

 asymptote of the curve approaches a finite 

 number, 



3. The lower asymptote of the curve is 

 zero; negative populations are unimagina- 

 ble. 



4. Population growth may be cyclical in 

 character with new logistic cycles additive. 

 Adaptive changes between the population 

 and its environment may initiate a new 

 cycle of growth superimposed on the other 

 one. For example, the agricultural stage of 

 human culture supported higher popula- 

 tion densities than did the pastoral stage. 



5. The general shape of the curve 

 shows, first, that populations have a slow 

 rate of growth; second, the rate increases 

 until it reaches a maximum (the inflection 

 point of the curve); and third, the rate be- 

 comes progressively less until the curve 

 stretches out nearly horizontally in close 

 approach to the upper asymptote. 



The differential equation from which the 

 logistic curve is derived is: 



dN , ^ (K-N) 

 _ =bN— ^^— , 



where b is the maximum potential rate of 

 reproduction for each organism in the 

 population; N is the total population size 

 at any moment of growth; t is time or age; 

 and K is the maximum population possible 

 under the obtaining ecological conditions. 

 Cause (1934) stated this equation in word 

 form as follows: 



The underlying rationale of the logistic 

 curve becomes clearer when an application 

 is made to an actual case. Cause has done 

 this for the growth of small laboratory 

 populations of Paramecium caudatum. In 

 his work five individual infusorians were 

 placed in 0.5 cc. of nutritive medium. The 

 experiment was repeated, and counts of or- 

 ganisms were taken at twenty-four hour 

 intervals for six days. When fitted to a 

 logistic curve, the actual observations cor- 

 respond closely with the curve itself. This 

 is seen in Figure 94, in which age of the 

 culture in days is plotted on the abscissa 

 and total population size on the ordinate. 

 This graph describes a population trend 

 (see next chapter). The feature of Cause's 

 study and logistic application is that some- 

 thing can be constructed about the popu- 

 lation growth factors by assuming that 

 the population actually grows in this fash- 

 ion and then calculating certain values from 

 the curve equation. As Cause puts it, we 

 are interested in the question: "What is 

 the potential rate of increase of Paramecium 

 under our conditions, and how does it be- 

 come reduced in the process of growth as 

 the environmental resistance increases?" 



From an inspection of Figure 94 it can 

 be seen that the maximal population pos- 

 sible under the respective conditions is 375 

 paramecia per 0.5 cc. of medium. This 

 value is called K. In fitting the curve, b, or 

 the rate of reproduction, is 2.309. This 

 means that in a twenty-four hour period 

 every individual protozoan has the capacity 

 to produce 2.309 others. Knowing these 

 two values. Cause computes certain other 

 expressions that have generalized popula- 

 tion importance. These are summarized in 

 Table 23. The first line shows the change in 

 number of organisms during the initial 

 four days of growth. The population in- 

 creases from 20.4 individuals to 137.2, to 

 319.0, to 369.0. If the ecological conditions 

 are not altered, N remains around 369. The 

 second line of the table expresses the po- 

 tential increase of the population per day, 

 or the number of ofiFspring that a given 

 population can potentially produce within 

 twenly-four hours. This is a geometric in- 

 crease and is not actually realized by the 

 population (p. 272). The unutilized oo- 



The rate of popu- 

 lation growth 



' The potential increase \ I The degree of realiza- , 



= ^ of the population per \ x \ *^°n ^^ *^^ potential ' 



unit of time I I increase i 



