Skip to main content
SpringerLink
Log in

Loop analysis and qualitative modeling: limitations and merits

  • Original paper
  • Published:
Biology and Philosophy Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. 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.

  2. If the F i are partially differentiable, A may be represented by the familiar Jacobian matrix (see Section 3).

  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.

  4. 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.

  5. Neither hierarchy is intended to be exhaustive.

  6. 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).

  7. 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 .

  8. See Justus (2005) for details.

  9. Justus (in press) presents some problems with this view in the context of mathematical ecology.

  10. 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).

  11. For a detailed discussion of this equation, see Puccia and Levins (1985, Ch. 3).

  12. See Logofet (1993) for a review.

  13. See Hale et al. (1999, Ch. 2) for a review.

  14. 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.

  15. Another, more controversial, example is whether interpersonal utility comparisons are possible, and if so, with what degree of precision they can be done (see Weirich 1984; Harsanyi 1976).

  16. This is not the case for the essentially qualitative phenomena discussed above, such as agent beliefs and preferences.

  17. 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).

  18. 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).

  19. See Orzack (1990) for a clear discussion of examples of these problematic methods in biology.

  20. See Brauer and Nohel (1969) for a description of this method.

  21. See Justus (in press) for a detailed discussion of this issue.

  22. 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.

  23. 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.

  24. A survey of these methods is impossible here. See Hale et al. (1999).

  25. See Hahn (1963) and for an exposition of this method.

  26. 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.

References

  • Athey S, Milgrom P, Roberts J (1998) Robust comparative statics. Monograph

  • Bassett L, Maybee J, Quirk J (1968) Qualitative economics and the scope of the correspondence principle. Econometrica 36:544–563

    Article  Google Scholar 

  • Boucher DH, James S, Keeler KH (1982) The ecology of mutualism. Annu Rev Ecol Syst 13:315–347

    Article  Google Scholar 

  • Brauer F, Nohel JA (1969) The qualitative theory of differential equations: an introduction. Dover Publications, New York

    Google Scholar 

  • Briand F, McCauley E (1978) Cybernetic mechanisms in lake plankton systems: how to control undesirable algae. Nature 273:228–230

    Article  Google Scholar 

  • Chesson PL, Case TJ (1986) Overview: nonequilibrium community theories: chance, variability, history, and coexistence. In: Diamond J, Case TJ (eds) Community ecology. Harper and Row, New York, pp 229–239

    Google Scholar 

  • Dambacher J, Luh H-K, Li H, Rossignol P (2003a) Qualitative stability and ambiguity in model ecosystems. Am Nat 161:876–888

    Article  Google Scholar 

  • Dambacher J, Li H, Rossignol P (2003b) Qualitative predictions in model ecosystems. Ecol Model 161:79–93

    Article  Google Scholar 

  • Desharnais RA, Costantino RF (1980) Genetic analysis of a population of tribolium. VII. Stability: response to genetic and demographic perturbations. Can J Genet Cytol 22:577–589

    Google Scholar 

  • Figuera J, Greco S, Ehrgott M (eds) (2005) Multiple criterion decision analysis: state of the art surveys. Springer, Berlin

    Google Scholar 

  • Flake RH (1980) Extension of Levins’ loop analysis to transient and periodic disturbances. Ecol Model 9:83–90

    Article  Google Scholar 

  • Gantmacher F (1960) Matrix theory, vol 2. Chelsea Publishing Company, New York

    Google Scholar 

  • Gasca AM (1996) Mathematical theories versus biological facts: a debate on mathematical population dynamics in the 1930s. Hist Stud Phys Biol Sci 26:347–403

    Google Scholar 

  • Hahn W (1963) Theory and application of Liapunov’s direct method. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Hale D, Lady G, Maybee J, Quirk J (1999) Nonparametric comparative statics and stability. Princeton University Press, Princeton

    Google Scholar 

  • Haray F, Norman RZ, Cartwright D (1965) Structural models: an introduction to the theory of directed graphs. Wiley, New York

    Google Scholar 

  • Harsanyi J (1976) Essays on ethics, social behavior, and scientific explanation. Reidel, Dordrecht

    Google Scholar 

  • Hastings A (1988) Food web theory and stability. Ecology 69:1665–1668

    Article  Google Scholar 

  • Justus J (2005) Qualitative scientific modeling and loop analysis. Philosophy of Science (in press)

  • Justus J (in press) Ecological and Lyapunov stability. Presented at the Biennial meeting of the Philosophy of Science Association. Vancouver

  • Lancaster K (1962) The scope of qualitative economics. Rev Econ Stud 29:99–123

    Article  Google Scholar 

  • Lane P (1986) Symmetry, change, perturbation, and observing mode in natural communities. Ecology 67:223–239

    Article  Google Scholar 

  • Lane P (1998) Assessing cumulative health effects in ecosystems. In: Rapport D, Contanza R, Epstein P, Gaudet C, Levins R (eds) Ecosystem health. Blackwell Science, Inc., Malden, pp 129–153

    Google Scholar 

  • Lane P, Levins R (1977) The dynamics of aquatic systems II. The effects of nutrient enrichment on model plankton communities. Limnol Oceanogr 22:454–471

    Article  Google Scholar 

  • Levine SH (1976) Competitive interactions in ecosystems. Am Nat 110:903–910

    Article  Google Scholar 

  • Levins R (1966) The strategy of model building in population biology. Am Sci 54:421–431

    Google Scholar 

  • Levins R (1968a) Evolution in changing environments. Princeton University Press, Princeton

    Google Scholar 

  • Levins R (1968b) Ecological engineering: theory and technology. Q Rev Biol 43:301–305

    Article  Google Scholar 

  • Levins R (1970) Complex systems. In: Waddington CH (ed) Towards a theoretical biology, vol 3. Aldine Publishing, Chicago, pp 73–88

    Google Scholar 

  • Levins R (1974) Qualitative analysis of partially specified systems. Ann N Y Acad Sci 231:123–138

    Article  Google Scholar 

  • Levins R (1975a) Problems of signed digraphs in ecological theory. In: Levin SA (ed) Ecosystem analysis and prediction. Society for Industrial and Applied Mathematics, Philadelphia, pp 264–277

    Google Scholar 

  • Levins R (1975b) Evolution in communities near equilibrium. In: Cody M, Diamond J (eds) Ecology and evolution of communities. Belknap Press, Cambridge, pp 16–51

    Google Scholar 

  • Levins R (1998) Qualitative mathematics for understanding, prediction, and intervention in complex ecosystems. In: Rapport D, Contanza R, Epstein P, Gaudet C, Levins R (eds) Ecosystem health. Blackwell Science, Inc., Malden, pp 178–204

    Google Scholar 

  • Levins R, Schultz BB (1996) Effects of density dependence, feedback and environmental sensitivity on correlations among predators, prey and plant resources: models and practical implications. J Anim Ecol 65:802–812

    Article  Google Scholar 

  • Logofet D (1993), Matrices and graphs: stability problems in mathematical ecology. CRC Press, Ann Arbor

    Google Scholar 

  • Lyapunov A ([1892], 1992) The general problem of the stability of motion. London: Taylor and Francis

  • Mason SJ (1953) Feedback theory: some properties of signal flow graphs. Proc Inst Radio Eng 41:1144–1156

    Google Scholar 

  • Maybee J (1966) Remarks on the theory of cycles in matrices. Mathematics Department, Purdue University. Manuscript

  • Orzack S (1990) The comparitive biology of second sex ratio evolution within a natural population of a parasitic wasp, Nasonia vetripennis. Genetics 124:385–396

    Google Scholar 

  • Orzack S, Sober E (1993) A critical assessment of Levins’s the strategy of model building in population biology (1966). Q Rev Biol 68:533–546

    Article  Google Scholar 

  • Puccia C, Levins R (1985) Qualitative modeling of complex systems. Harvard University Press, Cambridge

    Google Scholar 

  • Puccia C, Levins R (1991) Qualitative modeling in ecology: loop analysis, signed digraphs, and time averaging. In: Fishwick P, Luker P (eds) Qualitative simulation modeling and analysis. Springer-Verlag, New York, pp 119–143

    Google Scholar 

  • Quirk J, Ruppert R (1965) Qualitative economics and the stability of equilibrium. Rev Econ Stud 32:311–326

    Article  Google Scholar 

  • Roberts FS (1971) Signed digraphs and the growing demand for energy. Environ Plan 3:395–410

    Article  Google Scholar 

  • Roberts FS, Brown TA (1975) Signed digraphs and the energy crisis. Am Math Mon 82:577–594

    Article  Google Scholar 

  • Samuelson P (1947) Foundations of economic analysis. Harvard University Press, Cambridge

    Google Scholar 

  • Schmitz OJ (1997) Press perturbations and the predictability of ecological interactions in a food web. Ecology 78:55–69

    Article  Google Scholar 

  • Scudo F (1971) Vito volterra and theoretical ecology. Theor Popul Biol 2:1–23

    Article  Google Scholar 

  • Vandermeer J (1980) Indirect mutualism: variations on a theme by Stephen Levine. Am Nat 116:441–448

    Article  Google Scholar 

  • Walley P (1991) Statistical reasoning with imprecise probabilities. Chapman and Hall, New York

    Google Scholar 

  • Weirich P (1984) Interpersonal utility in principles of social choice. Erkenntnis 21:295–317

    Article  Google Scholar 

  • Weisberg M (2004) Qualitative modeling and chemical explanation. Philos Sci 71:1071–1081

    Article  Google Scholar 

  • Wiens JA (1984) On understanding a non-equilibrium world: myth and reality in community patterns and processes. In: Strong Jr DR, Simberloff D, Abele LG, Thistle AB (eds) Ecological communities: conceptual issues and the evidence. Princeton University Press, Princeton, pp 439–457

    Google Scholar 

  • Wimsatt W (1970) Some problems with the concept of ‘Feedback’. In: PSA 1970, vol 1. Philosophy of Science Association, Boston, pp 241–256

  • Wimsatt W (1987) False models as means to Truer theories. In: Nitecki M, Hoffman A (eds) Neutral models in biology. Oxford University Press, New York, pp 23–55

    Google Scholar 

  • Wright S (1921) Correlation and causation. J Agric Res 20:557–585

    Google Scholar 

Download references

Acknowledgements

A Josephine De Kármán fellowship supported this research. Thanks to Alexander Moffett, Michael Weisberg, and an anonymous reviewer for helpful comments.

Author information

Authors and Affiliations

  1. Philosophy Department, University of Texas, University Station, Austin, TX, 78713, USA

    James Justus

Authors
  1. James Justus
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James Justus.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10539-006-9050-x

Keywords

Search

Navigation