My examiner on theoretical ecology, Marissa Baskett, suggested that I return to some basic literature on the why of mathematical modeling in ecology. Here are notes on three papers:
- Modeling purpose must be clearly defined, as this determines whether to focus on generality, realisms, and/or precision
- Complexity is as often obfuscating as illuminating
- When prediction is a desired goal, separating calibration from validation is essential
- Must test robustness of models against different mathematical assumptions, structures, and parameters
- Many of these assumptions relate to the need to model at a simpler level than reality
- Disclosure, clarity, and reproducibility in publication are required to justify model use for prediction and policy
- Model precision is irrelevant if the model fails to address a question of relevance.
Notes on Readings
- Levins (1966) - Modeling Strategies
- Aber (1997) - WhyDon’t We Believe the Models?
- May (2004) - Uses and Abuses of Mathematics in Biology
- Some Notes on My Approach
- Need to sacrifice at least one of generality, realism, and precision in a model, usually choosing as a results of goals of prediction, inference, and management
- All models have hidden assumptions and testing different structures is neccessary to establish robustness of results
- Models are simplifications of many-parameter systems, which introduces precision through loss of information and unknown relations between the finer-scale processes
- Models are not hypotheses or theories. A theory is supported by a group of related models that cover different phenomena and approaches.
- Population biology models complex systems, but truly accurate models of these process would have too many parameters, be intractable, computationally infeasible, and lack easily interpretable parameters
- One cannot maximize generality, realism, and precision simultaneously, as they aim towards different goals of understanding, prediction, and management. As such, three strategies:
Sacrifice generality, reducing parameters to those relevant for short-term prediction, solve numerically, and then test predictions in narrow situations. (e.g. fisheries models)
Sacrifice realism, expect many unrealistic assumptions will cancel each other out, and build in realism piece by piece to understand relevant complicatons (e.g., Lotka-Volterra)
Sacrifice precision. Focus on qualitative over quantitative and comparative results. Use abstract, general mathematical forms. (e.g. MacArthur and Levins)
- It’s important to remember which parts of the model are thought to be realistic, and which are not
- Test model robustness to changes in assumptions, keeping essential components
- The MacArthur and Levins (1962) paper on optimal fitness in heterogenous environments is an example of strategy 3.
- Three versions of the model have very different assumptions but arrive at similar results, contributing to its robustness.
- On the other hand, the proposition that increased rate of growth leads to lower populations size (as in the logistic and L-V models), is not robust to the addition of extraneous predation, or different growth and decline rates. Thus, it is non-robust
- The many parameters of a real system are generally abstracted to a few, but this does not mean that the underlying fine-grained parameters are independent or additive
- These may or may not have intuitive biological interpretations
- They may be formalizations of ecological concepts, such as measures of niche breadth, diversity, etc. These may not have “true” forms and should be tested for robustness
- They may represent variables found to be important in other studies
- The generalization of parameters leads to imprecision in three ways:
- Omit factors with small or rare efects
- Are only qualitative because the exact form of functions is unknown
- Information is lost in generalization
- Models are not hypotheses, in that they are not verifiable or falsifiable in a meaningful way
- Models are limited in scope and complexity
- A satisfactory theory is a cluster of inter-related models
- Multiplicity of models is inevitable and primarily due to a conflict of methods and goals.
Modeling papers aren’t usually held to a set of standards, and as such are usually doubted. They should contain the following:
- The structure of the model with both equations and a schematic, with a literature review justifying all the structure
- All parameters and their values, their method of selection.
- Even when the model is theoretical, empirically-justified parameters
- Validation against independent data. This is distinct from calibration, and this must be clear.
- Sensitivity to parameters, as well as comparison to other (simpler, null) model structures. This justifies the complexity of the model
- Prediction (though this isn’t always the point)
- Calibration process is important because it reveals over-fitting
“Failed” models that do not validate well are useful in illuminating areas for future research
- Mathematics plays different roles at different stages of the development of scientific knowledge. Statistical models are more common at observational stages, theoretical models at stages when patterns are being explained
- At the latter stages, theoretical models are meant to force clarity on thinking, and in many cases “less is more”. The relationship between inuitive understanding and the models should be preserved.
- Complex computer simulation may give us less understanding than analytical toy models. In the case of May’s work on HIV, a simple SIR model has outperformed a much more demographically detailed one.
- Confidence intervals and probibalistic values lend credence to results from aribitray assumptions/parameters, or ignore other important factors. In the case of Mad Cow disease, a model suggested banning T-bone stake because it could save a life, despite the fact that 100,000 cases would be due to other factors.
- Mathematical models in ecology only took off when authors such as Hutchison and MacArthur started asking testable, general questions in the line of physics, e.g., “How can similar species persist?”
- Most common, but difficult to idenity, abuses of mathematics “are situations where mathematical models are constructed with an excruciating abundance of detail in some aspects, whilst other important facets of the problem are misty or a vital parameter is uncertain”
- Interested in prediction and inference, but generality only as much as required for scenario-building.
- Best to mix model structure that improves prediction and allows for inference with non-parametric components (splines, S-maps)
- Choose structural components based on (a) inferential question, (b) management options, (c) data availability. e.g., include population structure when this is what managers can modify. Exclude genetic information because of inability to collect.
- Test nonparametric components against pure stochasticity
- Include observation as a model component
- Parse out sources of uncertainty when possible
- (More to come on this after reading the model selection literature provided by Andrew Latimer)
Aber, J. D. 1997. Why Don’t We Believe the Models?. Bulletin of the Ecological Society of America 78:232–233.
Levins, R. 1966. The strategy of model building in population biology. American Scientist 54:421–431.
May, R. M. 2004. Uses and abuses of mathematics in biology. Science 303:790–3.