The Nature of Modeling and Modeling Nature

“The sciences do not try to explain, they hardly even try to interpret, they mainly make models… The justification of such a mathematical construct is solely and precisely that it is expected to work—that is, correctly to describe phenomena from a reasonably wide area.” John Von Neumann von Neumann “All models are wrong, but some are useful” Box Role Models on the Role of Models

Models What is modeling all about? Is it My feeling is no. Modeling is about: –abstraction –simplification –isomorphism (e.g., being able to envision fundamental similarities between different systems) This need not be mathematical. In a very real sense, we all approach our study systems through models, as we –generally work within frameworks of abstracted hypothetical mechanisms. –cannot possibly entertain all details of the system. –come to the system with an understanding of other systems, where similar processes may apply to the focal system. ?

A Broad Umbrella Verbal Graphical Statistical Computer-based Mathematical Competitive Exclusion Principle : “Complete competitors cannot coexist” (Hardin, 1960) travel time between patches t opt time spent in a patch (t) resources acquired from patch (R) R(t)R(t) Optimal Foraging Theory (Bull et al. 2004) Spatial Competition Hypothesis (Tilman 2004)

How much simplification? “Scale model” “Toy model” Detail-rich Specific in target Parameter values (or sensitivities to changes in these values) become important Predictions are narrow Empirical tests can be quantitative Highly abstracted General in target Relations between parameter values take precedence over their specific values Predictions are broad Empirical tests are often qualitative CONTINUUM

Why Toys? “…one of the main functions of an analogy or model is to suggest extensions of the theory by considering extensions of the analogy, since more is known about the analogy than is known about the subject matter of the theory itself.” Hesse “If you have a complex natural system you don't understand, and you model it by including all aspects you can think of, you just end up with two systems you can't understand!” Paola

So, what’s the point? Must a model make testable predictions in order to be valuable? Is Hardin’s competitive exclusion principle (and Newton’s laws of motion, Hubbell’s neutral theory, etc.) truly untestable? Forming a model is very much like creating a virtual world. –Claims made about this virtual world need to logically follow from assumptions (mathematics is a useful tool here) –This virtual world in essence becomes an experimental system (we ask what happens when we wiggle that parameter or fix that variable…) –One concern is whether our virtual world tells us useful things about the real world: Are the assumptions of the model satisfied or violated? Does the structure of the model reflect (aspects of) reality? Does the model suggest new empirical directions? One might suggest an iterative algorithm when it comes to modeling: The form of the virtual world is dependent on empirical findings and future empirical work is informed by this virtual world.

But why math? The major advantages of a mathematical model are: –The virtual world is very well-defined (e.g., Hardin’s C.E.P. verbal model is ambiguous) –The assumptions are (at least implicitly) made clear –Mathematical techniques address dynamics that we may not be able to intuit (e.g., feedback, network behavior, multiple spatial or temporal scales, etc.). Example (Buss & Jackson 1979) Buss & Jackson claimed that as A grows faster and faster, it will exclude B and C A mathematical model (Frean & Abraham, 2002) of an abstracted version of this system shows this plausible conclusion to be off the mark. These authors find that as A chases B faster, this liberates C with a net negative effect on A! One intuition (“faster growth means better competitive ability”) is supplanted by another (“the enemy of my enemy is my friend”). Mathematics helps tease such intuitions apart. A BC B C A

1.What do you think models are? 2.What role do models play in the context of science? 3.What role do models play in the context of ecology? 4.What role are models likely to play in your own research? 5.How central is accurate prediction to the worth of a model in your eyes? Are you convinced that models can play other roles (e.g., exploring possibilities, forming baselines for more complex systems, inspiring empirical/experimental directions, and providing explanations for phenomena)? 6.In the case of Hardin’s (1960) essay, if the competitive exclusion principle is taken to be a verbal model, what do you think its worth is? How does it relate to competition in laboratory or natural ecosystems? How do you react to Hardin’s statement that the “truth” of the principle cannot be established by empirical facts? Do you think this principle has something to offer those studying competition in the field? Are you convinced that the principle uncovers isomorphic behavior in a number of different systems (ranging from economics to genetics)? Questions