Partially Specified Ecological Models

Models are useful when they are compared with data. Whether this comparison should be qualitative or quantitative depends on circumstances, but in many cases some statistical comparison of model and data is useful and enhances objectivity. Unfortunately, ecological dynamic models tend to contain assumptions and simplifications which enhance tractability, promote insight, but spoil model fit and this can cause difficulties when adopting a statistical approach. Furthermore, the arcane numerical analysis required to fit dynamic models reliably presents an impediment to objective model testing by fitting. This paper presents methods for formulating and fitting partially specified models, which aim to achieve a measure of generality by avoiding some of the irrelevant incidental assumptions that are inevitable in more traditional approaches. This is done by allowing delay differential equation models, difference equation models and differential equation models to be constructed with part of their structure represented by unknown functions, while part of the structure may contain conventional model elements that contain only unknown parameters. An integrated practical methodology for using such models is presented along with several examples, which include use of models formulated using delay differential equations, discrete difference equations/matrix models, ordinary differential equations and partial differential equations. The methods also allow better estimation from ecological data by model fitting, since models can be formulated to include fewer unjustified assumptions than is usually the case if more traditional models are used, while still including as much structure as the modeller believes can be justified by biological knowledge: model structure improves precision, while fewer extraneous assumptions reduce unquantifiable bias.