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Ecologists have fiercely debated for many decades whether populations are self-regulated by density-dependent biological mechanisms or are controlled by exogenous environmental forces. Here, a stochastic mechanistic model is used to show that the interaction of these two forces can explain observed large fluctuations in Dungeness crab ( Cancer magister ) numbers. Relatively small environmental perturbations interact with realistic nonlinear (density dependent) biological mechanisms, to produce dynamics that are similar to observations. This finding has implications throughout population biology, suggesting both that the study of deterministic density-dependent models is highly problematic and that stochastic models must include biologically relevant nonlinear mechanisms.

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1Step-wise refinement (SWR) asserts that complex programs can be derived from simple programs by progressively adding features. The length of a program specification is the number of features that the program has. Critical to the scalability of SWR are multi-dimensional models that separate orthogonal feature sets. Let n be the dimensionality of a model and k be the number of features along a dimension. We show program specifications that could be O(k n ) features long have short and easy-to-understand specifications of length O(kn) when multi-dimensional models are used. We present new examples of multidimensional models: a micro example of a product-line (whose programs are 30 lines of code) and isomorphic macro examples (whose programs exceed 30K lines of code). Our work provides strong evidence that SWR scales to synthesis of large systems.