Ecology. Michael Begon
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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.
5.6.2 What type of competition?
But what type or range of competition is this model able to describe? We can answer this question by tracing the relationship between k values and log N (as in Section 5.3). Each generation, the potential number of individuals produced (i.e. the number that would be produced if there were no competition) is Nt R. The actual number produced (i.e. the number that survive the effects of competition) is Nt R /(1 + aNt ).
We have seen that:
Thus, in the present case:
(5.14)
or, simplifying:
(5.15)
Figure 5.21 shows a number of plots of k against log10 Nt 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 undercompensating and then compensates perfectly at higher values of Nt. The model is therefore limited in the type of competition it can produce. So far, we can only say that this type of competition leads to very tightly controlled regulation of populations.
5.6.3 Time lags
One simple modification that we can make is to relax the assumption that populations respond instantaneously to changes 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. For example, suppose that 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:
(5.16)
Equation 5.12 may be modified to:
(5.17)
time lags provoke population fluctuations
Now 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 behaviour of the modified model is as follows:
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 time lag has provoked the fluctuations in the model, and it can be assumed to have similar, destabilising effects on real populations.
5.6.4 Incorporating a range of competition
A further simple modification of Equation 5.12 allows us to incorporate a range of types of competition, as follows (Maynard Smith & Slatkin, 1973 ; Bellows, 1981):
We can see how this works from Figure 5.22, which plots k against log Nt, as in Figure 5.17, but now k is log10[1 + (aNt ) b ]. The slope of the curve, instead of approaching 1 as it did previously, now approaches the value taken by b in Equation 5.18. Thus, by the choice of appropriate values, the model can portray undercompensation (b <1), perfect compensation (b = 1), scramble‐like overcompensation (b > 1) or even density independence (b = 0). This model has the generality that Equation 5.12 lacks, with the value of b determining the type of density dependence that is being incorporated.
Figure 5.22 The intraspecific competition inherent in Equation 5.19. The final slope is equal to the value of b in the equation.
dynamic patterns: R and b
Equation 5.18 also shares with other good models an ability to throw fresh light on the real world. Analysing the population dynamics generated by the equation, we can draw guarded conclusions about the dynamics of natural populations. The mathematical method by which this and similar equations can be examined is described