Abstract
Richard Levins has advocated the scientific merits of qualitative modeling throughout his career. He believed an excessive and uncritical focus on emulating the models used by physicists and maximizing quantitative precision was hindering biological theorizing in particular. Greater emphasis on qualitative properties of modeled systems would help counteract this tendency, and Levins subsequently developed one method of qualitative modeling, loop analysis, to study a wide variety of biological phenomena. Qualitative modeling has been criticized for being conceptually and methodologically problematic. As a clear example of a qualitative modeling method, loop analysis shows this criticism is indefensible. The method has, however, some serious limitations. This paper describes loop analysis, its limitations, and attempts to clarify the differences between quantitative and qualitative modeling, in content and objective. Loop analysis is but one of numerous types of qualitative analysis, so its limitations do not detract from the currently underappreciated and underdeveloped role qualitative modeling could have within science.
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Notes
More generally, a (n + m) × (n + m) matrix that also represents the effects of the m parameters on the variables could be constructed. Since variables do not affect parameters [(ii) above], the lower left quadrant of this matrix would be composed of zero entries. If the parameters do not influence one another, as is commonly assumed in scientific modeling, the lower right quadrant would also have zero entries. Only for simplicity of presentation, I have focused on relations between variables.
If the F i are partially differentiable, A may be represented by the familiar Jacobian matrix (see Section 3).
A function f is monotonic increasing if\({\left( {\forall x_1,x_2 }\right)\left( {\left( {x_1 < x_2 } \right)\to \left( {f(x_1 ) < f(x_2 )} \right)} \right)}\) . The inequality of the consequent is reversed for a monotonic decreasing function.
The following counterexample shows that a positive sign-directed interaction from x k to x j is not equivalent to a monotonic increasing interaction. Consider two increases in x k , from x 0 k to x 1 k and from x 0 k to x 2 k such that x 1 k < x 2 k . Since the interaction from x k to x j is positive sign directed, these increases in x k will induce increases in x j , to x 1 j and x 2 j , respectively. It may not be the case that x 1 j < x 2 j , however, as required if the interaction were increasing monotonic.
Neither hierarchy is intended to be exhaustive.
Wright (1921) probably first recognized this relationship between matrices and directed graphs, and consequently between determinants and loops (see below), in his development of path analysis: a method by which the effects of different factors inducing variation in a variable can sometimes be distinguished. Levins (1974) recognized this relationship independently of Mason (1953), who appreciated it in his analysis of the dynamics of electrical circuits, and Maybee’s (1966) similar recognition of the relationship in economics (see Bassett et al. 1968).
The definition of the Jacobian matrix in terms of partial derivatives requires a ij be defined in terms of how increases in x j affect x i .
See Justus (2005) for details.
Justus (in press) presents some problems with this view in the context of mathematical ecology.
Press perturbation analysis is equivalent to a method of qualitative analysis developed in the field of ‘comparative statics’ in economics (see Athey et al. 1998).
For a detailed discussion of this equation, see Puccia and Levins (1985, Ch. 3).
See Logofet (1993) for a review.
See Hale et al. (1999, Ch. 2) for a review.
Walley (1991, Ch. 5), for instance, argues that imprecise probabilities best represent uncertainty, lack of information, ambiguity, and other aspects of real-world agents’ belief states.
This is not the case for the essentially qualitative phenomena discussed above, such as agent beliefs and preferences.
Typically, scientists in one field appropriate models developed in another to make these simplifications. This practice involves a judgment that, despite their apparent differences, the systems being analyzed share a similar structure. It is also frequently motivated by the desire to import scientific rigor from a science with a secure theoretical foundation to a science without one. Most of the highly idealized models developed by Lotka, Volterra, and their contemporaries in the early development of mathematical ecology, for instance, originated in physics (Scudo 1971; Gasca 1996).
If adequate data are available to test them, predictions of veracity-idealized models can serve as a means towards developing improved models with fewer unrealistic features (Wimsatt 1987).
See Orzack (1990) for a clear discussion of examples of these problematic methods in biology.
See Brauer and Nohel (1969) for a description of this method.
See Justus (in press) for a detailed discussion of this issue.
There are some exceptions. In the special case that the perturbed parameter is strictly increasing or decreasing, subsequent changes in variables can be approximated by determining their change at successive values of the parameter (Levins 1974, 1975b), but only if the new system would establish an asymptotically stable point equilibrium at each new parameter value. The range of parameter values for which this holds depends upon how the dynamics of the system change as the parameter continues to increase or decrease. Flake (1980) has also shown that loop analysis can be extended using Laplace transforms to analyze the effects of some time-varying and periodic perturbations. This extension requires the precise mathematical form of these perturbations be specified, however.
It is unspecified whether this refers to the expected number of interaction coefficients appearing in the conditions required for local asymptotic stability, the number of loops in the model, or something else.
A survey of these methods is impossible here. See Hale et al. (1999).
See Hahn (1963) and for an exposition of this method.
With respect to a qualitative version of Lyapunov’s direct method, this may be the case. The construction of a Lyapunov function for a given model usually depends on the precise details of the model’s mathematical form.
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Acknowledgements
A Josephine De Kármán fellowship supported this research. Thanks to Alexander Moffett, Michael Weisberg, and an anonymous reviewer for helpful comments.
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Justus, J. Loop analysis and qualitative modeling: limitations and merits. Biol Philos 21, 647–666 (2006). https://doi.org/10.1007/s10539-006-9050-x
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DOI: https://doi.org/10.1007/s10539-006-9050-x