W. M. Schaffer's principal interests involve the application of nonlinear dynamics to biology at a variety of levels. At the biochemical level, he has collaborated with T. V. Bronnikova and others on the the peroxidase-oxidase reaction, which has become a model system for the study of complex dynamical behavior in a biochemical contexts. This work has resulted in the production of the first detailed model which is able to reproduce complicated bifurcation sequences observed in careful laboratory experiments. At the physiological level, he is interested in dynamical scenarios which might account for the natural history of diseases such as epilepsy in which the (overt manifestations) of the pathology are intermittent. This work is still preliminary. At the ecological level, he is concerned with complex population dynamics and with the ups and downs of human epidemics. His work on chaos in childhood diseases, in particular, sparked considerable interest in this topic. Presently, he is studying predator-prey dynamics in the presence of seasonality. This work has interesting mathematical content as well as implications for the control of insect pest species in agroecosystems.
Selected publications of W.M. Schaffer and T.V. Bronnikova
1. Schaffer, W. M. 2009. A surfeit of cycles. Energy and Environment. 20: 985-996.
2. Schaffer, W. M. 1984. Stretching and folding in lynx fur returns: evidence for a strange attractor in nature? Amer. Natur. 124: 798-820.
3. Schaffer, W. M., and D. W. Zeh, S. L. Buchmann, S Kleinhans, M. V. Schaffer and J. Antrim. 1983. Competition for nectar between introduced honey bees and native North American bees and ants. Ecology. 64: 564-577.
4. Allen J. C., Schaffer, W. M. and D. Rosko. 1993. Chaos reduces species extinction by amplifying local population noise. Nature. 364: 229-232.
5. King, A. A. and W. M. Schaffer. 1999. The rainbow bridge: Hamiltonian limits and reso-nance in predator-prey dynamics. J. Math. Biol. 39: 439-469.
6. King, A. A. and W. M. Schaffer. 2001. The geometry of a population cycle: A mechanis-tic model of snowshoe hare demography. Ecology 82: 814-830.
7. Schaffer, W. M., Pederson, B., Moore, B. K., Skarpaas, O., King, A. A. and T. V. Bron-nikova. 2001. Subharmonic resonance and multi-annual oscillations in northern mammal cycles: A nonlinear dynamics perspective. Chaos, Solitons and Fractals. 12: 251-264.
8. Schaffer, W. M. 1985. Can nonlinear dynamics help us infer mechanisms in ecology and epidemiology? IMA J. Math. Appl. Med. Biol. 2: 221-252.
9. Schaffer, W. M. and M. Kot. 1985. Nearly one dimensional dynamics in an epidemic. J. Theor. Biol. 112: 403-427.
10. Olsen, L. F. and W. M. Schaffer. 1990. Chaos vs. noisy periodicity: Alternative hypothe-ses for childhood epidemics. Science. 249:499-504.
11. Tidd, C. W., Olsen, L. F. and W. M. Schaffer. 1993. The case for chaos in childhood dis-eases. II. Predicting historical epidemics with mathematical models. Proc. Biol. Sci. Roy. Soc. London. 254: 257-273.
12. Schaffer, W. M. and T. V. Bronnikova. 2007. Parametric dependence in model epidem-ics. J. Biol. Dynam. 1: 183-195.
13. Schaffer, W. M. and T. V. Bronnikova. 2009. Controlling malaria: Competition, season-ality and "slingshotting" transgenic mosquitoes into natural populations. J. Biol. Dynam. 3: 286-304.
14. Schaffer, W. M. 1968. Character displacement and the evolution of the hominidae. Amer. Natur. 102: 559-571.
15. Schaffer, W. M., and M. L. Rosenzweig. 1978. Homage to the Red Queen. I: Coevolution of predators and their victims. Theor. Pop. Biol. 14: 307-326.
16. Schaffer, W. M. 1978. A note on the theory of reciprocal altruism. Amer. Natur. 112:250-253.
Form and Function.
17. Schaffer, W. M. 1968. Intraspecific combat and the evolution of the Caprini. Evolution. 22: 812-825.
18. Cooper, P., Schaffer, W. M. and S. L. Buchmann. 1985. Temperature regulation by honey bees in desert environments. J. Exp. Biol. 114: 1-15.
19. Charnov, E. L. and W. M. Schaffer. 1973. Life history consequences of natural selection: Cole's result revisited. Amer. Natur. 107: 791-793.
20. Schaffer, W. M. 1974b. Optimal reproductive effort in fluctuating environments. Amer. Natur. 108: 783-790.
21. Schaffer, W. M. 1977. The evolution of reproductive rate and competitive ability in flow-ering plants. Theor. Pop. Biol. 11: 90-104.
22. Schaffer, W. M. 1979. On the equivalence of maximizing reproductive value and fitness. Proc. Nat. Acad. Sci. (USA). 76: 3567-3569.
23. Schaffer, W. M. 2004. Life histories, evolution and salmonids. Pp. 20-51. In, Stearns, S. and A. Hendry (eds.) Evolution Illuminated: Salmon and their Relatives. Oxford Univer-sity Press.
24. Mukaetova-Ladinska, E. B., Schaffer, W. M., Bronnikova, T. V., Westwood, J. and E. K. Perry. 2011. Cholinergic therapy for autistic spectrum disorders: Review and case report. Pp. 33-66. In, White. C. J. and J. E. Tait. (eds). Cholinesterase: Production, Uses and Health Effects. Nova Publ. NY.
25. Fed'kina, V. R., Ataullakhanov, F. I., Bronnikova, T. V. and N. K. Balabaev. 1978. Ex-perimental investigation and computer simulation of transitional behaviors in peroxidase-oxidase reaction. I. The model. Studia Biophysica. 72: 195-202.
26. Fed'kina, V. R., Bronnikova, T. V., and F. I. Ataullakhanov. 1984. Computer simulation of sustained oscillations in peroxidase-oxidase reaction. Biophysical Chemistry. 19: 259-264.
27. Fed'kina, V. R., Ataullakhanov, F. I. and T. V. Bronnikova. 1988. Stimulated regimens in the peroxidase-oxidase reaction. Theoretical and Experimental Chemistry. 24: 165-172.
28. Fed'kina, V. R., Bronnikova, T. V. and F. I. Ataullakhanov. 1992. Phase space structure for peroxidase-oxidase reaction. Biophysics. 37: 781-789.
29. Fed'kina, V. R. and T. V. Bronnikova. 1994. Bistability and birhythmicity in peroxidase-oxidase reaction. Biophysics. 39: 605-613.
30. Olsen, L. F., Valeur, K., Geest, T., Tidd, C. W., Schaffer, W. M. and D. Lloyd. 1994. Nonlinear forecasting of non-uniform chaotic attractors in an enzyme reaction. Phil. Trans. Roy. Soc. Lond. 348: 421-430.
31. Bronnikova, T. V., Fed'kina, V. R., Schaffer, W. M. and L. F. Olsen. 1995. Period-doubling bifurcations in a detailed model of the peroxidase-oxidase reaction. J. Phys. Chem. 99: 9309-9312.
32. Fed'kina, V. R. and T. V. Bronnikova. 1995. Complex oscillatory regimes in peroxidase-oxidase reaction. Biophysics. 40: 36-47.
33. Bronnikova, T. V., Schaffer, W. M. and L. F. Olsen. 1996. Quasiperiodicity in a detailed model of the peroxidase-oxidase reaction. J. Chem. Phys. 105: 10849-10859.
34. Bronnikova, T. V., Schaffer, W. M., and L. F. Olsen. 2001. Nonlinear dynamics of the peroxidase-oxidase reaction: I. Bistability and bursting at low enzyme concentrations. J. Phys. Chem. B. 105: 310-321.
35. Schaffer, W. M. and T. V. Bronnikova. 2012. Peroxidase-ROS interactions. Nonlinear Dynamcs. 68: 413-430.