## Basic equations

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

and:

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 5.18). The first step is therefore to modify the equations by making the net reproductive rate subject to intraspecific competition. This is done in Figure 5.19, which has three components.

At point A, the population size is very small (Nt is virtually zero). Competition is therefore negligible, and the actual net reproductive rate is adequately defined by an unmodified R. Thus, Equation 5.7 is still appropriate, or, rearranging the equation: Figure 5.18 Mathematical models of population increase with time, in populations with discrete generations: exponential increase (left) and sigmoidal increase (right). Figure 5.19 The simplest, straight-line way in which the inverse of generation increase (N/Nt+1) might rise with density (Nt). For further explanation, see text.

At point B, by contrast, the population size (Nt) 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, Nt+1 is simply the same as Nt, and Nt/Nt+1 equals 1. The population size at which this occurs is, by definition, the carrying capacity, K (see Figure 5.7).

The third component of Figure 5.19 is the straight line joining point A to point B and extending beyond it. This describes the progressive modification of the actual net reproductive rate as population size increases; but its straightness is simply an

no competition: exponential growth incorporating competition assumption made for the sake of expediency, since all straight lines are of the simple form: y = (slope) x + (intercept). In Figure 5.19, Nt/Nt+1 is measured on the y-axis, Nt 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:

or, rearranging:

a hypothetical population increasing in size over time in conformity with the model). The population in Figure 5.18 describes an S-shaped curve over time. As we saw earlier, this is a desirable quality of a model of intraspecific competition. Note, however, that there are many other models that would also generate such a curve. The advantage of Equation 5.12 is its simplicity.

The behavior of the model in the vicinity of the carrying capacity can best be seen by reference to Figure 5.19. 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.

a simple model of intraspecific competition

For further simplicity, (R - 1)/K 5.8.2 What type of competition?

may be denoted by a giving:

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 + aNt), which decreases as population size (Nt) increases.

We, like many others, derived which comes first - Equation 5.12 as if the behavior of a popa or K? ulation 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 behavior 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.

The properties of the model in properties of the Equation 5.12 may be seen in Fig-

simplest model ure 5.19 (from which the model was derived) and Figure 5.18 (which shows

It is not yet clear, however, just exactly what type or range of competition this model is able to describe. This can be explored by tracing the relationship between k values and log N (as in Section 5.6). Each generation, the potential number of individuals produced (i.e. the number that would be produced if there were no competition) is NtR. The actual number produced (i.e. the number that survive the effects of competition) is NR/(1 + aNt).

Section 5.6 established that:

k = log (number produced) — log (number surviving). (5.13)

Thus, in the present case:

k = log NtR — log NtR/(1 + aN,), or, simplifying: k = log(1 + aNt).

Figure 5.20 shows a number of plots of k against log10Nt with a variety of values of a inserted into the model. In every case, the slope of the graph approaches and then attains a value of 1. In other words, the density dependence always begins by under-compensating and then compensates perfectly at higher values of Nt. The model is therefore limited in the type of competition that it can produce, and all we have been able to say so far is that this type of competition leads to very tightly controlled regulation of populations.

5.8.3 Time lags

One simple modification that we can make is to relax the assumption that populations respond instantaneously to changes

 ^ Final slope = 1 a = 0.5—y No = 5 / a = 0.05 " a = 0 .1 N0 = 10 1 1 1 1 1 1 1

Figure 5.20 The intraspecific competition inherent in Equation 5.13. The final slope of k against log10Nt is unity (exact compensation), irrespective of the starting density N0 or the constant a (= (R - 1)/K).

in their own density, i.e. that present density determines the amount of resource available to a population and this in turn determines the net reproductive rate within the population. Suppose instead that the amount of resource available is determined by the density one time interval previously. To take a specific example, the amount of grass in a field in spring (the resource available to cattle) might be determined by the level of grazing (and hence, the density of cattle) in the previous year. In such a case, the reproductive rate itself will be dependent on the density one time interval ago. Thus, since in Equations 5.7 and 5.12:

Nt+1 = Nt X reproductive rate, Equation 5.12 may be modified to:

time lags provoke population fluctuations

There is a time lag in the population's response to its own density, caused by a time lag in the response of its resources. The behavior of the modified model is as follows:

R < 1.33: direct approach to a stable equilibrium R > 1.33: damped oscillations towards that equilibrium.

In comparison, the original Equation 5.12, without a time lag, gave rise to a direct approach to its equilibrium for all values of R. The

 - / N0 = 10 / a = 0.1 / b = 5 _ / N0 = 10 a = 0.1 - b = 2 - N0 = 10 a = 0.1 b = 0.5 1 1 1 1 Figure 5.21 The intraspecific competition inherent in Equation 5.19. The final slope is equal to the value of b in the equation. time lag has provoked the fluctuations in the model, and it can be assumed to have similar, destabilizing effects on real populations. 