Dyar's Rule and the Investment Principle: optimal moulting strategies if feeding rate is size-dependent and growth is discontinuous

We consider animals whose feeding rate depends on the size of structures that grow only by moulting (e.g. spiders'' legs). Our Investment Principle predicts optimum size increases at each moult; under simplifying assumptions these are a function of the scaling of feeding rate with size, the efficiency of moulting and the optimum size increase at the preceding moult. We show how to test this quantitatively, and make the qualitative prediction that size increases and instar durations change monotonically through development. Thus, this version of the model does not predict that proportional size increases necessarily remain constant, which is the pattern described by Dyar''s Rule. A literature survey shows that in nature size increases tend to decline and instar durations to increase, but exceptions to monotonicity occur frequently: we consider how relaxing certain assumptions of the model could explain this. Having specified various functions relating fitness to adult size and time of emergence, we calculate (using dynamic programming) the effect of manipulating food availability, time of hatching and size of the initial (or some intermediate) instar. The associated norms of reaction depend on the fitness function and differ from those when growth follows Dyar''s Rule or is continuous. We go on to consider optimization of the number of instars. The Investment Principle then predicts upper and lower limits to observed size increases and explains why increases usually change little or decline through development. This is thus a new adaptive explanation for Dyar''s Rule and for the most common deviation from the Rule.