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In Section 4.7 we developed a simple model for species with discrete breeding seasons, in which the population size at time t, N_t, altered in size under the influence of a fundamental net reproductive rate, R. This model can be summarised in two equations:

(5.7)

and:

(5.8)

no competition: exponential growth

The model, however, describes a population in which there is no competition. R is constant, and if R > 1, the population will continue to increase in size indefinitely (‘exponential growth’, shown in Figure 5.20a).

Figure 5.20 Mathematical models of population increase. (a) In populations with discrete generations population increases with time: exponential increase (left) and sigmoidal increase (right). (b) The simplest, straight‐line way in which the inverse of generation increase (N_t /N_t+1) might rise with density (N_t ) offers a way of adding competition to exponential increase. For further explanation, see text.

incorporating competition

The first step is therefore to modify the equations by making the net reproductive rate subject to intraspecific competition. This is done in Figure 5.20b, which has three components. At point A, the population size is very small (N_t is virtually zero). Competition is therefore negligible, and the net reproductive rate is adequately defined by an unmodified R. Thus, Equation 5.7 is still appropriate, or, rearranging the equation:

(5.9)

At point B, by contrast, the population size (N_t ) is very much larger and there is a significant amount of intraspecific competition, such that the net reproductive rate has been so modified by competition that the population can collectively do no better than replace itself each generation, because ‘births’ equal ‘deaths’. In other words, N_t+1 is simply the same as N_t, and N_t /N_t+1 equals 1. The population size at which this occurs is, by definition, the carrying capacity, K (see Figure 5.13).

The third component of Figure 5.20b is the straight line joining point A to point B and extending beyond it. This describes the progressive modification of the net reproductive rate as population size increases; but its straightness issimply an assumption made for the sake of expediency, since all straight lines are of the simple form: y = (slope)x + (intercept). In Figure 5.21b, N_t /N_t+1 is measured on the y‐axis, N_t on the x‐axis, the intercept is 1/R and the slope, based on the segment between points A and B, is (1 − 1/R)/K. Thus:

(5.10)

or, rearranging:

(5.11)

Figure 5.21 The intraspecific competition inherent in Equation 5.13. The final slope of k against log₁₀ N_t is unity (exact compensation), irrespective of the starting density N₀ or the constant a (= (R − 1)/K).

a simple model of intraspecific competition

For further simplicity, (R − 1)/K may be denoted by a giving:

(5.12)

This is a model of population increase limited by intraspecific competition. Its essence lies in the fact that the unrealistically constant R in Equation 5.7 has been replaced by an actual net reproductive rate, R/(1 + aN_t ), which decreases as population size (Nt ) increases.

which comes first – a or K?

We, like many others, derived Equation 5.12 as if the behaviour of a population is jointly determined by R and K, the per capita rate of increase and the population’s carrying capacity – a is then simply a particular combination of these. An alternative point of view is that a is meaningful in its own right, measuring the per capita susceptibility to crowding: the larger the value of a, the greater the effect of density on the actual rate of increase in the population (Kuno, 1991). Now the behaviour of a population is seen as being jointly determined by two properties of the individuals within it – their intrinsic per capita rate of increase and their susceptibility to crowding, R and a. The carrying capacity of the population (K = (R − 1)/a) is then simply an outcome of these properties. The great advantage of this viewpoint is that it places individuals and populations in a more realistic biological perspective. Individuals come first: individual birth rates, death rates and susceptibilities to crowding are subject to natural selection and evolve. Populations simply follow: a population’s carrying capacity is just one of many features that reflect the values these individual properties take.

properties of the simplest model

The properties of the model in Equation 5.12 may be seen in Figure 5.20a (which shows a model population increasing in size over time in conformity with Equation 5.12) and Figure 5.20b (from which the model was derived). The population in Figure 5.20a describes an S‐shaped curve over time. As we saw earlier, this is a desirable quality of a model of intraspecific competition. But there are many other models that would also generate such a curve. The advantage of Equation 5.12 is its simplicity.

The behaviour of the model in the vicinity of the carrying capacity can best be seen by reference to Figure 5.20b. At population sizes that are less than K the population will increase in size; at population sizes that are greater than K the population size will decline; and at K itself the population neither increases nor decreases. The carrying capacity is therefore a stable equilibrium for the population, and the model exhibits the regulatory properties classically characteristic of intraspecific competition.