‘differential’ equations, which we consider next.
r, the intrinsic rate of natural increase
The net rate of increase of such a population will be denoted by dN/dt (referred to in speech as ‘dN by dt’). This represents the ‘speed’ at which a population increases in size, N, as time, t, progresses. The increase in size of the whole population is the sum of the contributions of the various individuals within it. Thus, the average rate of increase per individual, or the ‘per capita rate of increase’ is given by dN/dt(1/N). But we have already seen in Section 4.7 that in the absence of competition, this is the definition of the ‘intrinsic rate of natural increase’, r. Thus:
and:
A population increasing in size under the influence of Equation 5.20, with r > 0, is shown in Figure 5.25. Not surprisingly, there is unlimited, ‘exponential’ increase. In fact, Equation 5.20 is the continuous form of the exponential difference Equation 5.8, and as discussed in Section 4.7, r is simply loge R. (Mathematically adept readers will see that Equation 5.20 can be obtained by differentiating Equation 5.8.) R and r are clearly measures of the same commodity: ‘birth plus survival’ or ‘birth minus death’. The difference between R and r is merely a change of currency.
Figure 5.25 Exponential (solid line) and sigmoidal (dashed line) increase in density (N) with time for models of continuous breeding. The equation giving sigmoidal increase is the logistic equation.
the logistic equation
Intraspecific competition can be added to Equation 5.20 by a method exactly equivalent to the one used in Figure 5.18b, giving rise to:
(5.21)
This is known as the ‘logistic’ equation, and a population increasing in size under its influence is also shown in Figure 5.25.
The logistic equation is the continuous equivalent of Equation 5.12, and it therefore has all the essential characteristics of Equation 5.12, and all of its shortcomings. It describes a sigmoidal growth curve approaching a stable carrying capacity, but it is only one of many reasonable equations that do this. Its major advantage is its simplicity. Moreover, while it was possible to incorporate a range of competitive intensities into Equation 5.12, this is by no means easy with the logistic equation. The logistic is therefore doomed to be a model of perfectly compensating density dependence. Nevertheless, in spite of these limitations, the equation will be an integral component of models in Chapters 8 and 10, and it has played a central role in the development of ecology.
5.8 Individual differences: asymmetric competition
5.8.1 Size inequalities
Until now, we have focused on what happens to the whole population or the average individual within it. Different individuals, however, may respond to intraspecific competition in very different ways. For example, when larval salamanders (Ambystoma tigrinum nebulosum) were competed amongst one another in groups, the sizes of the largest surviving larvae were no different from those reared alone (P = 0.42) but the smallest larvae were much smaller (P < 0.0001) (Ziemba & Collins, 1999). Similarly, the overwinter survival of red deer calves, Cervus elaphus, in the resource‐limited population on the island of Rhum, Scotland, declined sharply as the population became more crowded, but those that were smallest at birth were by far the most likely to die (Clutton‐Brock et al., 1987). The effects of competition are far from being the same for every individual. Weak competitors may make only a small contribution to the next generation, or no contribution at all. Strong competitors may have their contribution only negligibly affected.
What effect does this have at the population level? Figure 5.26 shows the results of a classic experiment in which flax (Linum usitatissimum) was sown at three densities, and harvested at three stages of development, recording the weight of each plant individually. This made it possible to monitor the effects of increasing amounts of competition not only as a result of variations in initial density, but also as a result of plant growth (between the first and the last harvests). When intraspecific competition was at its least intense (at the lowest sowing density after only two weeks’ growth) the individual plant weights were distributed symmetrically about the mean. When competition was at its most intense, however, the distribution was strongly skewed to the left: there were many very small individuals and a few large ones. As the intensity of competition gradually increased, the degree of skewness increased as well. Something very similar seems to be happening with the data in Figure 5.27 for cod (Gadus morhua) living off the coast of Norway. At higher densities (and presumably greater intensities of competition) size decreased but the skewness in the distribution of sizes increased.
Figure 5.26 Intraspecific competition increases the skewing in the distribution of plant weights. Frequency distributions of individual plant weights in populations of flax (Linum usitatissimum), sown at three densities and harvested at three ages. The red bar is the mean weight.
Source: After Obeid et al. (1967).
