We bring together topics from previous chapters, seeking to account for variations in abundance.

Ecologists may emphasize stability or fluctuations. To resolve these contrasting perspectives, it is necessary to distinguish clearly between factors that determine and those that regulate abundance. In doing so, we review historical conflicts between the viewpoints of Nicholson and Andrewartha and Birch. We then outline the demographic, mechanistic and density approaches to the investigation of abundance.

Starting with the demographic approach, we explain key factor analysis, its uses, but also its shortcomings. We therefore also explain ^-contribution analysis, which overcomes some of a role for food?

the problems with key factor analysis, and in developing this explanation we describe and apply elasticity analysis.

The mechanistic approach relates the level or presence of a factor (amount of food, presence of predators) either to abundance itself or to the population growth rate. This may be simply correlational, but may alternatively involve the experimental perturbation of populations. We note that the introduction of a biological control agent is one particular example of this.

Correlations with density are not absent from other approaches, but the density approach focuses on density dependences in their own right. We explain how time series analyses seek to dissect density dependences, especially the relative strengths of direct and delayed density dependence when abundance at a given point in time is expressed as reflecting abundances at various times in the past ('time lags'). We show, too, how related analyses may be valuable in counting and then characterizing the lags in an optimal description of a time series, and also in evaluating the respective contributions of density-dependent and -independent processes (especially weather) in determining abundance.

Regular, multigeneration cycles have in many ways, and for many years, been the benchmark against which ecologists have tested their ability to understand the determination of abundance. We explain how cycles may be identified within time series and then examine three case studies in detail.

Red grouse cycles illustrate the difficulties of distinguishing between alternative explanations - parasites and kinship/territorial behavior - both of which have support.

Work on cycles in snowshoe hares illustrates the coming together of detailed time series analyses and results obtained by much more direct, experimental means. It also provides a very sobering reminder of the logistical and practical difficulties that need to be accepted and overcome in order to build explanations.

More effort has been expended in studying population cycles in microtine rodents (voles and lemmings) than in any other group of species. We describe geographic trends in cyclicity and the need for an explanation to account for these, and we note that any such explanation must acknowledge that the cycles are the result of a 'second-order' process: a combination of a direct and a delayed density-dependent process. We then examine, in turn, three sets of explanations differing in their source of delayed density dependence: (i) 'intrinsic' theories, including maternal effects;

(ii) the 'specialist predation hypothesis', supported by both mathematical models and field experiments, though both of these have also been subject either to criticism or contradictory evidence; and

(iii) theories focused on food, which also have their problems.

We conclude by acknowledging that none of the questions posed at the beginning of the chapter has simple answers.

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