models vary in the degree to which they have abandoned deterministic elements. The question may now be asked, to what extent does the behaviour of
a stochastically dominated population or community still reflect the operation of
deterministic processes?
Deterministic models have a long history in ecology, intricately bound with the assumptions of equilibrium theories of community organisation. Classical competition theory (Hutchinson 1959) is of particular interest to the present discussion. Classical competition models have traditionally been used to suggest
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that natural communities have a highly predictable structure, an argument that was dependent on at least five major assumptions and on some empirical data.
Firstly, it was assumed that life history characteristics could be summarised in a single parameter - the per capita growth rate. Secondly, it was assumed that deterministic equations could be used to model population growth in which environmental fluctuations were ignored. Thirdly, as a community grows, its structure was assumed to approach a stable age distribution. Fourthly, it was assumed that competition was the only important biological interaction
(Hutchinson 1959, Grant 1986). Finally, is was assumed that coexistence required
a stable equilibrium point.
The second, third, fourth and fifth of these assumptions are of special interest to the present discussion. The degree to which these assumptions are relaxed determines the extent to which the modified model retains predictive capability.
Community predictability is also closely associated with stability, a property shared by all forms of classical equilibrium theory (competition, predator-prey) to varying degrees. Competitive systems were traditionally considered to have global stability. A globally stable system would show little tendency to lose species, the community being able to recover from events that drive species to low densities. Communities would be assembled by immigration and because the system approaches equilibrium, historical changes in abundance could be considered unimportant in determining a future state.
Some ecologists have argued that these properties of global stability may
also be shared by models of non-equilibrium theories (Chesson and Case 1986).
For example, a system that demonstrates non-equilibrium behaviour at a local
scale, may have emergent equilibrium properties at a higher, landscape level of
scale. This hypothesis is currently an area of active research and will be discussed more fully later in this chapter.
In summary, the predictability of a community as proposed in classical stable equilibrium theory rests primarily upon the following seven assumptions:
(1) (2) (3) (4) (5)
community regulation by competition,
community structure approaches a stable age distribution, environmental fluctuations can be ignored,
coexistence requires stable equilibria,
a community has a stable species composition,
(6) communities recover from perturbations,
(7) historical events are not important for stable equilibria.
In the previous section, an important point was made concerning May's
(1974) observation of chaotic behaviour in the discrete logistic
Following May's observation, ecologists were led to realise that apparentlY
141 to deterministic chaos. This same observation may now be made concerning stochastic domination. Research efforts appear to indicate that elements of determinism or predictability are present in stochastic systems.
This proposition can be illustrated by reviewing published research in this area. Most of this work has ignored intrinsic stochasticity. Randomness is introduced by replacing one of the model's parameters by stochastic processes. The dynamics of the resulting model are then compared with those of its deterministic parent in an attempt to identify behavioural differences and properties of stability in the stochastic counterpart (Turelli 1978).
3 . 4 . 2 En viro n m e n tal Stochastic Domination Emp irical
R e s e a rc h
The addition of environmental stochasticity does not necessarily imply the existence of unpredictable, non-equilibrium behaviour. With strong internal biotic forces, the extent to which a community is affected by environmental stochasticity is dependent on factors like the frequency, magnitude and spatial extent of disturbance events. This assumes, of course, that competition is a valid regulatory process.
Armstrong and McGhee (1980) demonstrate that population limit cycles can actually facilitate coexistence for many species on a limiting resource - a conclusion that :invalidates one of the key predictions of classical theory. Classical competition theory requires that at least n limiting resources are
required for the coexistence of n species (Chesson and Case 1986).
In order to explain the paradox of plankton, Hutchinson (1961) turned this
axiom around and reasoned that the existence of more species of plankton than limiting resources must imply that the hypothesis of an equilibrium is wrong.
Hutchinson went on, to suggest that intermediate disturbance· may work to
moderate potential instabilities caused by strong competitive forces in the plankton community, that would normally result in competitive exclusion.
The results of Armstrong and McGhee (1980), appear to complement the conclusions of Hutchinson (1961). In the model of Armstrong and McGhee, the environment does not vary, fluctuations in population density and resource levels result from biotic instability (limit cycles) in the model's equilibrium point (Emlen 1984).
May and MacArthur (1972), introduced white noise into competition communities modelled by a variant of the deterministic logistic equation (29). The co-authors discovered that when the equation's parameters were held constant, as in the case of a uniform environment, stable equilibrium was the result. The co-authors then introduced environmental variability into the model in the form of Gaussian white noise and discovered that community stability was now dependent on a threshold level of white noise.
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Turelli (1978), pointed out that May's proposed scenario of competing deterministic and stochastic forces was quite sound. However, some of the assumptions implicit in May's use of white noise could only be upheld in very specific circumstances.
Another approach to modelling environmental fluctuations was used by
Hanson and Tuckwell (1978) who assumed that the logistic model described an
equilibrium state that was disturbed and continually reduced, by discrete disturbance occurring at return intervals, defined by Poisson processes.
where