A simple modification of Equation 5.12 of far more general importance was originally suggested by Maynard Smith and Slatkin (1973) and was discussed in detail by Bellows (1981). It alters the equation to:

The justification for this modification may be seen by examining some of the properties of the revised model. For example, Figure 5.21 shows plots of k against log Nt, analogous to those in Figure 5.20: k is now 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.

Another desirable quality that Equation 5.18 shares with other good the dynamic pattern? models is an ability to throw fresh it's R and b light on the real world. By sensible analysis of the population dynamics generated by the equation,

Chaos

Damped oscillations

Chaos

Damped oscillations

Monotonie damping

J_i_i_i_i_i i i 11_i_i_i_i_i i i 11_i_i_i_i_i i i 11

Chaos

Chaos

Stable limit cycles

Stable limit cycles

Damped oscillations

Monotonic damping

1000

Time

Figure 5.22 (a) The range of population fluctuations (themselves shown in (b)) generated by Equation 5.19 with various combinations of b and R inserted. (After May, 1975a; Bellows, 1981.)

it is possible to draw guarded conclusions about the dynamics of natural populations. The mathematical method by which this and similar equations may be examined is set out and discussed by May (1975a), but the results of the analysis (Figure 5.22) can be appreciated without dwelling on the analysis itself. Figure 5.22b shows the various patterns of population growth and dynamics that Equation 5.18 can generate. Figure 5.22a sets out the conditions under which each of these patterns occurs. Note first that the pattern of dynamics depends on two things: (i) b, the precise type of competition or density dependence; and (ii) R, the effective net reproductive rate (taking density-independent mortality into account). By contrast, a determines not the type of pattern, but only the level about which any fluctuations occur.

As Figure 5.22a shows, low values of b and/or R lead to populations that approach their equilibrium size without fluctuating at all ('monotonic damping'). This has already been hinted at in Figure 5.18. There, a population behaving in conformity with Equation 5.12 approached equilibrium directly, irrespective of the value of R. Equation 5.12 is a special case of Equation 5.18 in which b = 1 (perfect compensation); Figure 5.22a confirms that for b = 1, monotonic damping is the rule whatever the effective net reproductive rate.

As the values of b and/or R increase, the behavior of the population changes first to damped oscillations gradually approaching equilibrium, and then to 'stable limit cycles' in which the population fluctuates around an equilibrium level, revisiting the same two, four or even more points time and time again. Finally, with large values of b and R, the population fluctuates in an apparently irregular and chaotic fashion.

Thus, a model built around a density-dependent, supposedly regulatory process (intraspecific competition) can lead to a very wide range of population dynamics. If a model population has even a moderate fundamental net reproductive rate (and the ability to leave 100 (= R) offspring in the next generation in a competition-free environment is not unreasonable), and if it has a density-dependent reaction which even moderately overcompensates, then far from being stable, it may fluctuate widely in numbers without the action of any extrinsic factor. The biological significance of this is the strong suggestion that even in an environment that is wholly constant and predictable, the intrinsic qualities of a population and the individuals within it may, by themselves, give rise to population dynamics with large and perhaps even chaotic fluctuations. The consequences of intraspecific competition are clearly not limited to 'tightly controlled regulation'.

This leads us to two important conclusions. First, time lags, high reproductive rates and overcompensating density dependence are capable (either alone or in combination) of producing all types of fluctuations in population density, without invoking any extrinsic cause. Second, and equally important, this has been made apparent by the analysis of mathematical models.

In fact, the recognition that even simple ecological systems may contain the seeds of chaos has led to chaos itself becoming a topic of interest amongst ecologists (Schaffer & Kot, 1986; Hastings et al., 1993; Perry et al., 2000). A detailed exposition of the nature of chaos is not appropriate here, but a few key points should be understood.

1 The term 'chaos' may itself be misleading if it is taken to imply a fluctuation with absolutely no discernable pattern. Chaotic dynamics do not consist of a sequence of random numbers. On the contrary, there are tests (although they are not always easy to put into practice) designed to distinguish chaotic from random and other types of fluctuations.

2 Fluctuations in chaotic ecological systems occur between definable upper and lower densities. Thus, in the model of intraspecific competition that we have discussed, the idea of 'regulation' has not been lost altogether, even in the chaotic region.

3 Unlike the behavior of truly regulated systems, however, two similar population trajectories in a chaotic system will not tend to converge on Cbe attracted to') the same equilibrium density or the same limit cycle (both of them 'simple' attractors). Rather, the behavior of a chaotic system is governed by a 'strange attractor'. Initially, very similar trajectories will diverge from one another, exponentially, over time: chaotic systems exhibit 'extreme sensitivity to initial conditions'.

4 Hence, the long-term future behavior of a chaotic system is effectively impossible to predict, and prediction becomes increasingly inaccurate as one moves further into the future. Even if we appear to have seen the system in a particular state before - and know precisely what happened subsequently last time - tiny (perhaps immeasurable) initial differences will be magnified progressively, and past experience will become of increasingly little value.

Ecology must aim to become a predictive science. Chaotic systems set us some of the sternest challenges in prediction. There has been an understandable interest, therefore, in the question 'How often, if ever, are ecological systems chaotic?' Attempts to answer this question, however, whilst illuminating, have certainly not been definitive.

Most recent attempts to detect chaos in ecological systems have been based on a mathematical advance known as Takens' theorem. This says, in the context of ecology, that even when a system comprises a number of interacting elements, its characteristics (whether it is chaotic, etc.) may be deduced from a time series of abundances of just one of those elements (e.g. one species). This is called 'reconstructing the attractor'. To be more specific: suppose, for example, that a system's behavior is determined by interactions between four elements (for simplicity, four species). First, one expresses the abundance of just one of those species at time t, Nt, as a function of the sequence of abundances at four successive previous time points: Nt-1, Nt-2, Nt-3, Nt-4 (the same number of 'lags' as there are elements in the original system). Then, the attractor of this lagged system of abundances is an accurate reconstruction of the attractor of the original system, which determines its characteristics.

In practice, this means taking a series of abundances of, say, one species and finding the 'best' model, in statistical terms, for predicting Nt as a function of lagged abundances, and then investigating this reconstructed attractor as a means of investigating the nature of the dynamics of the underlying system. Unfortunately, ecological time series (compared, say, to those of physics) are particularly short and particularly noisy. Thus, methods for identifying a 'best' model and applying Takens' theorem, and for identifying chaos in ecology generally, have been 'the focus of continuous methodological debate and refinement' (Bjornstad & Grenfell, 2001), one consequence of which is that any suggestion of a suitable method in a textbook such as this is almost certainly doomed to be outmoded by the time it is first read.

Notwithstanding these technical difficulties, however, and in spite of occasional demonstrations of apparent chaos in artificial laboratory environments (Costantino et al., 1997), a consensus view has grown that chaos is not a dominant pattern of dynamics in natural ecological systems. One trend, therefore, has been to seek to understand why chaos might not occur in nature, despite its being generated readily by ecological models. For example, Fussmann and Heber (2002) examined model populations embedded in food webs and found that as the webs took on more of the characteristics observed in nature (see Chapter 20) chaos became less likely.

Thus, the potential importance of chaos in ecological systems is clear. From a fundamental point of view, we need to appreciate that if we have a relatively simple system, it may nevertheless generate complex, chaotic dynamics; and that if we observe complex dynamics, the underlying explanation may nevertheless be simple. From an applied point of view, if ecology is to become a predictive and manipulative science, then we need to know the extent to which long-term prediction is threatened by one of the hallmarks of chaos - extreme sensitivity to initial conditions. The key practical question, however - 'how common is chaos?' - remains largely unanswered.

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