Statement
Populations have characteristic patterns of increase which are called population growth forms. For the purposes of compflrison we may designate two basic patterns, the I-shaped growth f01"m and the S-shaped or sigmoid growth f01m, which may be combined and/ or modified in various ways according to the peculiarities of different organisms and environments. In the J-shaped form density increases rapidly in exponential or compound interest fashion and then stops abruptly as environmental resistance be- comes effective more or less suddenly. This form may be repre- sented by the simple model:
-aN = rN with a definite limit On N at
In the sigmoid form, population growth is slow at first (estab- lishment or positive acceleration phase), becomes rapid (the logarithmic phase) as in the other type, but soon slows down gradually as the environmental resistance increases percentage
ORGANIZATION AT THE SPECIES POPULATION LEVEL: §8 183
wise (the negative acceleration phase), until a more or less equi- librium level is reached and maintained. This form may be repre- sented by the simple logistic model:
tiN = rN(K- N)
tit K
The upper level, beyond which no major increase can occur, as represented by the constant K, is the upper asymptote of the sigmoid curve and has been aptly called the carrying capacity.
In the J-form there may be no equilibrium level, but the limit on N represents the upper limit imposed by the environment. The two growth forms and certain variants are shown schematically in Figure 48.
Explanation
When a few individuals are introduced into or enter an unoccupied area (for example, at the beginning of a season) characteristic patterns of population increase have often been observed. When time is plotted on the horizontal axis and the number of individuals (or other measurement of density) on the vertical axis, the part of the growth curve which represents popu- lation increase often takes the form of an S or a
J ,
as shown in Figures 48, 49, 50 and 51. It is interesting to note that these two basic growth forms are similar to the two metabolic or growth types that have been described in the case of individual organisms (Bertalanffy, 1957). However, it is not known if there is a causal relationship between growth of individuals and growth of popu- lations; it is not safe at this point to do more than cal1 attention to the fact that there are some similarities in patterns.It will be noted that the equation given above as a simple model for the J-shaped form is the same as the exponential equa- tion discussed in the previous section, except that a limit is im, posed on N; that is, the relatively unrestricted growth is suddenly halted when the population runs out of some resource (such as food or space) or when frost or any other seasonal factor inter- venes. When the upper limit of N is reached, the density may remain at this level for a time, or, as is often the case, an immediate decline occurs, producing a "relaxation oscillation" pattern in density, as shown in Figure 48. This type of pattern, which Nicholson (1954) has called "density triggered" seems to be char-
184 BASIC ECOLOGICAL PRINCIPLES AND CONCEPTS: CH. 6
A B
LIMIT H----
\ \
TIM E .... . . )
8·1
A·I
A·2
Figure 48. Some aspects of population growth form, showing the J-shaped (exponential) (A) and the S-shaped (sigmoid) (B) forms and some variants. A-I and A-2 show oscillations which would be inherent in the J-shaped form. B-I, B:2 and B-3 show some possibilities (but by no means all) where there is a delay in denSity effect, which occurs when time elapses between production of young individuals and full influence of the individuals (the case in higher plants and animals). When nutrients or other requisites accumulate prior to population growth, an "overshoot" may occur as shown in A-2 and B-2. (This explainS why new ponds or lakes often provide better fishing than old onesl) (Curves adapted from Nicholson, 1954.)
acteristic of many populations in nature such as algal blooms, annual plants, some insects and perhaps lemmings on the tundra.
A type of growth form which is observed frequently follows an S-shaped or sigmOid curve. The Sigmoid curve is the result of greater and greater action of detrimental factors (environmental resistance) as the denSity of the population increases, in contrast to the previous model where environmental resistance was delayed until near the end of the increase. For this reason Nicholson ( 1954) has spoken of the sigmoid type as "density conditioned."
The Simplest case which can be conceived is the one in which
ORGANIZATION AT THE SPECIES POPULATION LEVEL: §8 185
detrimental factors are linearly proportional to the density. Such simple or "ideal" growth form is said to be logistic and conforms to the logistic equation which we have used as a basis for our model of the sigmoid pattern. The equation may be written as follows (three forms are shown above and the integrated form below) :
t,N (K- N) r ( N )
M = rN K Or = rN -
K
N! or = rN 1 - KN=- -- -K a- rt l+e
where t,N 1M is the rate of population growth (change in number in time), r the specific growth rate, as discussed in Section 6, N the population size (number), K the maximum population size possible, or "upper asymptote," e the base of natural logarithm and a equals r/K.
As will be noted, this is the same equation as the exponentia1 one written in the previous section with the addition of the ex- pression (K- N)/K or (rIK)N2• The latter expressions are two ways of indicating the environmental resistance created by the growing population itself, which brings about an increasing re- duction in the potential reproduction rate as population size approaches the carrying capacity. In word form, these equations simply mean:
populatIOn equals ( Ii 't d 'fi
increase un mt e Sp~CI C or
growth rate times
[
degree of real-]
~zntion of max- imum rate [
Rate of. ]
[~~:i~t:c!e~:!jblel
timesnumber~ in the minus [unrealized]
populatIOn) increase
It should now be emphasized that although the growth of a great variety of populations-representing microorganisms, plants, and animalS-including both laboratory and natural populations, have been shown to follow the sigmoid pattern, it does not follow necessarily that such populations increase according to the logistic equation. There are many mathematical equations which will pro- duce a sigmoid curve. Almost any equation in which the negative factors increase in some manner with denSity will produce sigmoid curves. Mere curve-fitting is to be avoided. One needs to have evidence that the factors in the equation are actually operating to control the population before an attempt is made to compare
186
tn
600<!
w >-450
u.. o
I-2300 ::>
o ~
<! 150
BASIC ECOLOGICAL PRINCIPLES AND CONCEPTS: CH. 6
a 2. 4 6 8 10 12 14 16 18 20 HOURS
Figure 49. Growth of yeast in a culture. A simple case of the sigmoid growth form in which environmental resistance (in this case, detrimental [actors produced by organisms themselves) is linearly proportional to the density. This type of population growth pattern is said to be logistic (because it corresponds to the logistic equation) and is characteristic of organisms with simple life histories. (From Allee et aI., 1949, after Pearl.)
actual data with a theoretical curve. The simple situation where environmental resistance increases linearly with density seems to hold for populations of organisms which have very simple life histories (for example, yeasts, see Fig. 49) and which are growing in a limited space (as in a culture). Populations often" overshoot"
the upper asymptote and undergo mild oscillations before settling down at the carrying capacity level (see Fig. 48). In populations of higher plants and animals, which have complicated life histories and long periods of individual development, there are likely to be delayed responses which greatly modify the growth fmID, produc- ing what Nicholson (1954) has called "tardy density conditioned"
pattems. In such cases a more concave growth CLU've may result (longer period required for natality to become effective) or definite oscillations may be produced. For a discussion of some of these possibilities and how they are related to the basic models reference may be made to Andrewartha and Birch (1954) and Nicholson's paper cited above.
As can well be imagined, data on population growth of field populations are few, incomplete, and hard to come by. One reason, aside from the difficulty of detelIDining numbers, is that many natural populations are in the "adult" stage when studied. The best opportunity to observe the fundamental growth fonn occurs when the population enters or is introduced into a new, unoccu-