crowding affects all individuals equally, individuals are homogeneously
distributed and equilibrium conditions are independent of observational scale.
Assumption or Assumption or Published
Approximation approximation Authority
Violated ?
Smith 1963, Ayala 1969,
inverse linear density - nonlineari ties Gilpin and Ayala 1973, and Justice 1972
growth rate dependency nonlinearities Dorschner et al. 1987,
Hallam and Clark 1981
Davidson 1938, Jordon and
environmental independency disturbance drives Jacobs 1947, Ziswiler 1967,
(i.e. globally stable) instabilities Birch 1953, Park 1948,
Slobodkin 1964, May and
MacArthur 1972
Boyce and Daley 1980,
invariant equilibria nonlinearities Hutchinson 1958,
Nicholson 1954, 1975
dimensional stability W1Stable above one dimension Turchin 1993
Witteman et al. 1990
dynamics capturdd using instabilities are likely for Turchin 1993
first-order multi Witteman et al. 1 990
regulated by density Davidson 1938, Jordon and
feedback, (i.e. competition) disturbance moderated and Jacobs 1947, Ziswiler 1967,
social behaviour is not a driven, (biotic disturbance Birch 1953, Park 1948,
determining factor in includes social behaviour Slobodkin 1964, May and
behaviour MacArthur 1972
migration, emigration and spatially dependent Hallam and Clark 1981,
Ziswiler 1967, Gause 1935
deterministic probabilistic Park 1948, Aplet et al. 1988,
Turchin 1993, Davidson 1938
Boyce and Daley 1980,
competition intensity is equilibria and intrinsic Hutchinson 1958,
invariant growth rate are not Nicholson 1954, May 1975
invariant Dorschner et al. 1987,
Hallam and Clark 1981
historical time Hutchinson 1948
independent of genetic interactive outcomes
v ariation dependent of genetic Park (1948)
variation
Table 2.4 Summary table of the results of experimental testing of hypotheses (model constructs)
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Inverse linear density-dependency implies that the growth rate is density dependent even at the lowest densities. It may be more reasonable to suppose that there is a threshold density below which individuals do not interfere with each other (Pielou 1977).
An invariant growth rate assumes that there is a stable age distribution and that females in a sexually reproducing population always find mates, even when the density is low (Pielou 1977). The assumption of a stable age distribution rests upon Lotka (1922), who concluded that a population growing geometrically develops a stable age distribution.
There is also a number of assumptions regarding the instantaneous nature of regeneration, resource supply and the outcome of competitive interactions. Both regeneration and resource supply (reaction) time lags have been adapted for the logistic equation (Hutchinson 1948, Wangersky and Cunningham 1956,
Wangersky 1 978).
Finally, the Lotka-Volterra competition model assumes that the competition coefficients
( a
and /3) are invariant with respect to N1 and N2• It is unlikely this assumption is upheld in every case as is demonstrated by Istock(1977).
The competitive exclusion principle has been a major theoretical J
development of the Lotka-Volterra competition equations which has proven very difficult to substantiate. Furthermore, Krebs (1985) cites examples of exclusions which have occurred in the existence of a super-abundant food supply. May and MacArthur (1972) discovered that closely packed species coexisted in a stable environment. The same result could not be produced when threshold levels of environmental instability were introduced. MacArthur's
(1972) idea of limiting similarity and Strong's null models have also largely
failed to find conclusive evidence of interspecific competitive exclusion.
As noted earlier, Wiens (1977) contends that competition may be rare for some populations because of high environmental variability. Such populations exist well below any carrying capacity and are only affected by competition during times of infrequent food shortage. In essence, the same hypothesis was invoked by Hutchinson (1958) to account for the paradox of plankton. Both explanations are distinctly non-equilibrium in nature. At present, this appears to be the most plausible explanation, a position that has been argued by Strong
(1984, 1986).
With reference to the present discussion, competitive exclusion provides further uncertainty concerning the validity of Lotka-Volterra interspecific competition. If the non-equilibrium hypotheses of Hutchinson (1958) and Wiens
(1977) are correct, then the intraspecific competition mechanism of the logistic
equation is also invalidated in its initial form. While the details of the competition controversy will continue to be debated, it now seems quite clear,
that initial assumptions concerning the existence and intensity of competition are probably invalid in the case of some, if not most, natural populations. Furthermore, it now seems likely that competition intensity is closely related to the degree of environmental stochasticity present in a given habitat (Wiens
1977).
These conclusions acknowledge the apparent existence of equilibria in some cases. Exceptions to what appears to be a non-equilibrium rule may be explained by the following:
(i) It is assumed that intraspecific competition is the biotic mechanism
primarily responsible for the form of the logistic curve. The existence of intraspecific competition needs to be clearly demonstrated. Presumably, biotic or abiotic disturbance is also capable of driving short term logistic growth.
(ii) Apparent equilibrium behaviour must be demonstrated across a
range of different scales. In the absence of testing of this nature, it can always be claimed that apparent equilibrium behaviour is simply an artefact of temporal and spatial scale.
(iii) Apparent equilibrium behaviour can be driven by non-equilibrium
processes. It now seems evident that biotic and abiotic disturbance plays a major role in community organisation and represents what this author considers to be the greatest failin
g
of the logistic and Lotka-Volterra competition equations.Points (i) - (iii) above are supported by Facelli and Pickett (1990) who contend that good correspondence between a model and field data constitutes a poor test of a model, if its assumptions remain untested. Perhaps some of the great achievements of the logistic and Lotka-Volterra equations, are the discoveries that have resulted from attempts to test the assumptions of these models.
Having identified invalid assumptions, the difficulty then remains of trying to identify just which combination of neglected complications is causing the problem. Pielou (1977) maintains that the skill of a good modeller lies in being able to distinguish experimentally between important and unimportant causal variables. In a word of caution, Pielou suggests that good correspondence between predicted and observed events
cannot be taken to imply that the
model 's simplifying assumptions are reasonable in the sense that neglected
complications are indeed negligible in their effects
(Pielou 1977).The conclusion of the logistic Era in ecology appears to have been characterised by a realisation that many previously neglected complications, such as nonlinear density-dependency, growth rate dependency, environmental independency, invariant equilibria, dimensional stability, the adequacy of first order models, spatial independency, determinism, invariant competition
118
One possible answer to these deficiencies is to add greater complexity to the existing formulations of the logistic and Lotka-Volterra competition models. There are a number of difficulties in this approach. One problem, is in determining which simplifying assumptions are
too simple.
Pielou (1977) points out that obviouslyno model can allow for all conceivable complications; if it
did, it would (by definition) cease to be a model.
Furthermore, attempts to improve the realism of a model accomplish little, if they remove one objectionable assumption, only to replace it with a muchlarger number of other
assumptions; and any assumption in this context is merely a guess, perhaps a
wild one
(Pielou 1977).In adding greater complexity to a model, the degree and nature of a violated
assumption needs to be considered. Some simplifying assumptions demarcate paradigm boundaries. If this is the case, then the addition of greater complexity to an existing model amounts to a paradigm shift. Some of the assumptions and approximations of the logistic and Lotka-Volterra models demarcate the
equilibrium I non-equilibrium boundary in ecology. To replace these
assumptions with greater complexity· is effectively to construct a non
equilibrium model.
To illustrate this point, the equilibrium paradigm portrays stable equilibrium conditions1 as a tension maintained between biotic density feedback mechanisms and exogenous disturbance. In other words, the equilibrium paradigm accepts the existence of exogenous disturbance, but not as a regulatory or driving mechanism (Clements 1928).
If the addition of exogenous disturbance is required in the logistic or Lotka Volterra competition equations in order to predict the behaviour of a natural population or community, the product of this combination is essentially a non equilibrium model. The only exception to this condition would be if the level of the disturbance is low enough, to enable the density-feedback mechanism to dominate as the sole regulatory force.
Non-equilibrium behaviour is primarily driven by abiotic and biotic forms for disturbance. For this reason, the illustration used above generalises a range of non -equilibri urn scenarios.
DeAngelis