OT200 



< 



I 150 



Z '00 



o 



£ 50 



CD 



^ 



Instantaneous percentage 



Asymptote 



5.0 i 



-40 i 



( 

 [ 



Cumulative growth rote -13.0 ( 



..--Absolute growth rate -|2.0 g 

 I ■ 



10 I 

 0.0 I 



7.0 X 

 6 0° 



|FEB|MAR|APR|MAY| JUN| JUL|AUG|SEP| 



FIG. 10. 1 I Annual ecesis of the Invertebrate connmuni 

 herb and shrub strata of a deciduous forest. 

 223 + 266 + 218 + 273 + 272 

 5 



K 

 250-3.5 



:4.25; 



log e (20/3.5) 



250; 

 0.062; N = 



(4.25-0.062*) 

 l + e 



not fully expressed, even though its trend is present 

 inherently. 



The sigmoid curve shows that a population grows 

 slowly at first, then at an accelerating rate which is 

 at maximum at the point of inflection, after which the 

 population continues to increase but at a decelerating 

 rate, finally becoming stabilized at the upper asymp- 

 tote. Most growth curves are symmetrical, and the 

 point of inflection is one-half the value of the asymp- 

 tote. The lower concave part of the curve is called 

 the accelerating phase oj growth and the upper convex 

 part of the curve, the inhibiting phase of groivth. 



If the number of new individuals added during 

 each unit of time, absolute growth rate, is plotted 

 against time midway in each period, a bell-shaped 

 curve is obtained, the peak of this curve coinciding 

 with the point of inflection on the sigmoid curve. 

 However, the number of individuals involved in the 

 absolute growth rate varies with the length of the 

 time unit used, and the time unit of greatest signifi- 

 cance varies from one species to another. Compari- 

 sons of growth rate of different populations are diffi- 

 cult unless instantaneous growth rates are obtained. 



The instantaneous groivth rate is the rate of 

 growth at a point on a time scale and is usu- 

 ally expressed in terms of increase per individual or 

 unit biomass per unit of time. It cannot be measured, 

 but it can be calculated from the logistic curve by the 

 differential equation (Park 1939, Andrewartha and 

 Birch 1954) 



dN ,, K-N 

 dt k 



where jV is the size of the populations at any time t ; 

 dN /dt stands for the instantaneous rate of change 

 (dN) in the size of the populations during an interval 

 in time {dt) and hence may represent the growth rate 

 at any desired time on the growth curve ; r is the biotic 

 potential, innate capacity, or the intrinsic rate of in- 

 crease per individual per unit of time in an environ- 

 ment where there are no limiting factors : and K is the 

 maximum size of the population reached at the asymp- 

 tote. This equation means that the rate of growth 

 equals the potential rate of increase in the size of the 

 population (rN), multiplied by the fraction of the 

 maximum population size (carrying capacity) still re- 

 maining to be filled (K — N)/K. In its integrated 

 form, 



N = ^ 



1 + ^ C-rl) 



where the constant a is the natural logarithm of 

 (K — N)/N when t is zero. 



In order to solve the equation for the logistic 

 growth curve, it is necessary to determine the intrinsic 

 growth rate, r. In an environment without limiting 

 factors, population growth is logarithmic. The factor 

 r, the value of which varies with species, is the ex- 

 ponent that indicates this growth rate. The elephant, 

 for instance, has a very slow growth rate. It has been 

 estimated, however, that if all offspring survived and 

 in turn reproduced, a single pair could give rise to 

 19,000,000 elephants in 750 years. On the other hand, 

 a single stem mother of the common cabbage aphid 

 gives rise to an average of 41 young, and there may be 

 12 generations per year between March 31 and 

 August 15. If they all lived, the progeny resulting 

 would number 564,087,257,509,154,652 individuals in 

 only 4.5 months (Herrick 1926). It is of considerable 

 ecological value to determine both the maximum po- 

 tential rate at which a species could increase under 

 ideal conditions and the factors that prevent this in- 

 crease from being realized. 



The intrinsic rate of increase, r. has been defined 

 as the maximal rate of increase attained at any partic- 

 ular combination of temperature, moisture, quality of 

 food, and so on, when the quantity of food, space, and 

 other animals of the same kind are kept at an optimum 

 and other organisms of different kinds are excluded 

 from the experiment (Andrewartha and Birch 1954 

 p. ii). Such an ideal environment may be set up 

 under controlled experimental conditions. Actually, it 

 is sometimes approximated under natural conditions 

 during the very early stages of the accelerating phase 

 of growth. Under such conditions the value of r may 

 be approximated from the equation 



, log,. {Nr,/Nn) 



Thus if a population is doubled in a period of three 



60 Ecological processes and dynamics 



