The model derived and discussed in Section 5.8 was appropriate for populations that have discrete breeding seasons and can therefore be described by equations growing in discrete steps, i.e. by 'difference' equations. Such models are not appropriate, key characteristics of chaotic dynamics

Takens' theorem: reconstructing the attractor how common - or important - is chaos?

r, the intrinsic rate of natural increase however, for those populations in which birth and death are continuous. These are best described by models of continuous growth, or 'differential' equations, which will be considered next.

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:

dN dt knJ

(5.19) increase in density (N) with time for models of continuous breeding. The equation giving sigmoidal increase is the logistic equation.

and:

(5.19) increase in density (N) with time for models of continuous breeding. The equation giving sigmoidal increase is the logistic equation.

A population increasing in size under the influence of Equation 5.20, with r > 0, is shown in Figure 5.23. 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 logeR. (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.

For the sake of realism, intraspecific competition must obviously be added the logistic equation to Equation 5.20. This can be achieved most simply by a method exactly equivalent to the one used in Figure 5.19, giving rise to:

dN dt

This is known as the logistic equation (coined by Verhulst, 1838), and a population increasing in size under its influence is shown in Figure 5.23.

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, whilst 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.

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