Density dependence and the stabilization of animal numbers

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Abstract
Latto and Hassell (1987) disagree with the conclusion of Den Boer (1986), that the winter moth population at Wytham Wood, studied by Varley and Gradwell, was not regulated. They attempt to demonstrate regulation by means of a simulation model. In the present paper the validity of this model is tested step by step. The fixing of the initial and final densities, as practised by Den Boer and rejected by Latto and Hassell, did not prevent population explosions and extinctions, as was assumed by Latto and Hassell. It is shown that the deterministic formulation of the density dependence of pupal predation, as used by Latto and Hassell, deviates systematically from the field data. Replacing the values of the key-factor (k1) by random values drawn from a normal distribution (Latto and Hassell) affects the dynamics such that the ability of pupal predation to govern density is improved in the model. Changing mortalities other than the key-factor does not significantly influence the pattern of fluctuations nor the limits of density. Models should leave intact the essentials of the reality under study, while removing distracting elements (Levins 1968). As both the timing of the key-factor, and its correlation with pupal predation are essential features of the winter moth population at Wytham Wood between 1950 and 1968, the model of Latto and Hassell does not apply to this population. By simply changing log10 (eggs/female) it is shown that the power of the density dependence of pupal predation to govern possible trends in density of the winter moth population at Wytham Wood is weak. On the other hand, the model of Latto and Hassell gives insight into the conditions that might favour regulation of numbers. Although the model of Poethke and Kirchberg (1987) preserves more features of the pertinent winter moth population than that of Latto and Hassell (1987) it still deviates in one essential aspect: the succession in time of both the (coupled) mortalities and the deviations from the deterministic density dependence are taken at random. Therefore, also this model is still too far from the field population to be a sound base for the statistical speculation proposed by Poethke and Kirchberg.