W. M. Schaffer:

Publications.


ARTICLES / BOOK CHAPTERS:

Genetic Fractal. (c) 1998, W. M. Schaffer. [1] Schaffer, W. M. 1965. Charles Lyell and the origin of species. Yale Scientific Magazine 39(8): 10-14.

[2] Schaffer, W. M. 1968a. Character displacement and the evolution of the hominidae. Amer. Natur. 102: 559-571.

[3] Schaffer, W. M. 1968b. Intraspecific combat and the evolution of the Caprini. Evolution. 22: 812-825.

[4] Reed, C. A. and W. M. Schaffer. 1972. How to tell the sheep from the goats. Bull. Field Mus. Nat. Hist. 43: 2-7.

[5] Schaffer, W.M. and C.A. Reed. 1972. The coevolution of social behavior and cranial morphology in sheep and goats (Bovidae, Caprini). Fieldiana (Zoology). 61: 1-88.

[6] Charnov, E. L. and W. M. Schaffer. 1973. Life history consequences of natural selection: Cole's result revisited. Amer. Natur. 107: 791-793.

[7] Schaffer, W. M. and R. H. Tamarin. 1973. Changing reproductive rates and population cycles in lemmings and voles. Evolution. 27: 114-125.

[8] Hirth, H. F. and W. M. Schaffer. 1974. Survival rate of the green turtle, Chelonia mydas, necessary to maintain stable populations. Copeia. 1974: 544-546.

[9] Schaffer, W. M. 1974a. The evolution of optimal reproductive strategies: The effects of age structure. Ecology. 55: 291-303.

[10] Schaffer, W. M. 1974b. Optimal reproductive effort in fluctuating environments. Amer. Natur. 108: 783-790.

[11] Schaffer, W. M. and P. F. Elson. 1975. The adaptive significance of variations in life history among local populations of Atlantic salmon in North America. Ecology. 56: 577-590.

[12] Schaffer, W. M. and M.D. Gadgil. 1975. Selection in for optimal life histories in plants. Pp. 142-157. In, Cody, M., and J. M. Diamond (eds.) Ecology and Evolution of Communities. Harvard University (Belknap) Press. Cambridge, Mass.

[13] Schaffer W. M. and E. G. Leigh. 1976. Prospects for a mathematical theory of plant population biology. Syst. Bot. 1: 209-232.

[14] Schaffer, W. M. 1977a. The evolution of reproductive rate and competitive ability in flowering plants. Theor. Pop. Biol. 11: 90-104.

[15] Schaffer, W. M. 1977b. Ecosystem dynamics and evolution: A comment. Theor. Pop. Biol. 11: 326-329.

[16] Schaffer, W. M., and M. L. Rosenzweig. 1977. Selection for optimal life histories II: Multiple equilibria and the evolution of alternative life histories. Ecology. 58: 60-72.

[17] Schaffer, W. M., and M. V. Schaffer. 1977a. The reproductive biology of Agavaceae I: Pollen and nectar production in four Arizona agaves. S. W. Natur. 22:157-167.

[18] Schaffer, W. M., and M. V. Schaffer. 1977b. The adaptive significance of variation in reproductive habit in Agavaceae. Pp. 261-276. In, Stonehouse, B. and C. M. Perrins (eds.) Evolutionary Ecology. Mac Millan, London.

[19] Rosenzweig, M. L., and W. M. Schaffer. 1978. Homage to the Red Queen. II: Coevolution response to enrichment. Theor. Pop. Biol. 14: 135-157.

[20] Schaffer, W. M. 1978. A note on the theory of reciprocal altruism. Amer. Natur. 112:250-253.

[21] 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.

[22] Gaines, M. S., Schaffer, W. M., and R. K. Rose. 1979. Additional comments on reproductive strategies and population fluctuations in microtines rodents. Ecology. 60: 1284-1286.

[23] Schaffer, W. M. 1979a. The theory of life-history evolution and its application to Atlantic salmon. Proc. Zool. Soc. London. 44: 307-326.

[24] Schaffer, W. M. 1979b. On the equivalence of maximizing reproductive value and fitness. Proc. Nat. Acad. Sci. (USA). 76: 3567-3569.

[25] Schaffer, W. M., Jensen, D. E., Hobbs, D., Gurevitch, J., Todd, J. R. and M. V. Schaffer. 1979. Competition, foraging energetics, and the cost of sociality in three species of bees. Ecology. 60: 976-987.

[26] Schaffer, W. M., and M. V. Schaffer. 1979. The adaptive significance of variation in reproductive habit in Agavaceae. II: Pollinator foraging behavior and selection for increased reproductive effort. Ecology. 60: 1051-1069.

[27] Inouye, R. S. and W. M. Schaffer. 1981. On the meaning of ratio (de Wit) diagrams in plant ecology. Ecology. 62: 1679-1681.

[28] Schaffer, W. M. 1981a. Ecological abstraction: The consequences of reduced dimensionality in ecological models. Ecol. Mongrph. 51: 383-401.

[29] Schaffer, W. M. 1981b. On reproductive value and fitness. Ecology. 62: 1683-85.

[30] Schaffer, W. M., R. S. Inouye, and T. S. Whittam. 1982. Energy allocation in an annual plant when the effects of seasonality on growth and reproduction are decoupled. Amer. Natur. 120: 807-815.

[31] Schaffer, W. M., D. W. Zeh, and M. V. Schaffer. 1982. Competition among social insects. Pp. 14-18. In, Breed, M. D., C. D. Michener, and H. E. Evans (eds.). The Biology of Social Insects. Westview Press. Boulder, Co.

[32] Schaffer, W. N. 1983. On the application of optimal control theory to the general life history problem. Amer. Natur. 121: 418-431.

[33] 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.

[34] Kot, M., and W. M. Schaffer. 1984. The effects of seasonality in discrete models of population growth. Theor. Pop. Biol. 26: 340-360.

[35] Kot, M., W. M. Schaffer and R. D. Holmes. 1984. Seasonality and discrete models of population growth. Proc. 2d Pac. Coast Conf. Math. Mod. Renew. Res. 2: 206-211.

[36] Schaffer, W. M. 1984. Stretching and folding in lynx fur returns: evidence for a strange attractor in nature? Amer. Natur. 124: 798-820.

[37] Cooper, P., Schaffer, W. M. and S. L. Buchmann. 1985. Temperature regulation by honey bees in desert environments. J. Exp. Biol. 114: 1-15.

[38] Schaffer, W. M. 1985a. Order and chaos in ecological systems. Ecology. 66:93-106.

[39] Schaffer, W. M. 1985b. Can nonlinear dynamics help us infer mechanisms in ecology and epidemiology? IMA J. Math. Appl. Med. Biol. 2: 221-252. (1)

[40] Schaffer, W. M. and M. Kot. 1985a. Nearly one dimensional dynamics in an epidemic. J. Theor. Biol. 112: 403-427.

[41] Schaffer, W. M. and M. Kot. 1985b. Do strange attractors govern ecological systems? Bioscience. 36: 342-350.

[42] Kot, M. and W. M. Schaffer. 1986. Discrete-time growth-dispersal models. Math. Biosci. 80: 109-136.

[43] Schaffer, W. M., S. Ellner and M. Kot. 1986. Effects of noise on some dynamical models in ecology and epidemiology. J. Math. Biol. 24: 479-523.

[44] Schaffer, W. M. and M. Kot. 1986a. Differential systems in ecology and epidemiology. Pg. 158-178. In, Holden, A. V. (ed). Chaos: An Introduction. Univ. Manchester Press, UK.

[45] Schaffer, W. M. and M. Kot. 1986b. The coals that Newcastle forgot: Chaos in ecological systems. Trends in Ecology and Evolution. 1: 58-63.

[46] Schaffer, W. M. 1987. Chaos in ecology and epidemiology. Pg. 233-248. In, Holden, A. V., Degn, H. and L. F. Olsen (eds.) Chaos in Biological Systems. NATO Advanced Workshop. Chaos in Biology. Plenum Press N.Y.

[47] Kot, M., Schaffer, W. M., Truty, G. L., Graser, D. J. and L. F. Olsen. 1988. Changing criteria for order. Ecol. Model. 43:75-110.

[48] Olsen, L. F., G. L. Truty and W. M. Schaffer. 1988. A nonlinear dynamics analysis of childhood epidemics in Copenhagen, Denmark. Theor. Pop. Biol. 33: 344-370.

[49] Schaffer, W. M. 1988. Perceiving order in the chaos of nature. Pp. 313-350. In, Boyce, M. (ed.) Life Histories: Theory and Patterns from Mammals. Yale University Press. New Haven.

[50] Schaffer, W. M., Olsen, L. F. Truty, G. L., Fulmer, S. L. and D. J. Graser. 1988. Periodic and chaotic dynamics in childhood infections. Pp. 331-347. In, Markus, M., Müller, S. and Nicolis, G. (eds.) From Chemical to Biological Organization. Springer-Verlag. Berlin.

[51] Schaffer, W. M. and G. L. Truty. 1989. Chaos vs. noise-driven dynamics. Pp. 77-96. In, Hastings, A. (ed.) Some Mathematical Questions in Biology. AMS. Providence, R.I.

[52] Olsen, L. F. and W. M. Schaffer. 1990. Chaos vs. noisy periodicity: Alternative hypotheses for childhood epidemics. Science. 249:499-504.

[53] Schaffer, W. M. and L. F. Olsen. 1990. Chaos in childhood epidemics. Pp. 187-190. In, Abraham, N B., Albano, A. M., Passamante, A. and P. E. Rapp. Measures of Complexity and Chaos. Plenum Press.

[54] Schaffer, W. M., Olsen, L. F., Truty, G. L. and S. L. Fulmer. 1990. The case for chaos in childhood epidemics. Pp. 139-167. In, Krasner, S. (ed.). The Ubiquity of Chaos. AAAS. Washington, D.C.

[55] Schaffer, W. M. 1990. Reply to Professor Bartlett. Proc. Roy. Stat. Soc.

[56] Krukonis, G., and W. M. Schaffer. 1991. Periodicity and body size in mammals and birds: Is there an allometric relation? J. Theor. Biol. 148: 469-493.

[57] Olsen, L. F., Frisch, L.- L. H., and W. M. Schaffer. 1991. The peroxidase-oxidase reaction: a case for chaos in the biochemistry of the cell. Pp. 299-316. In, Baier, G. and M. Klein. (eds.) A Chaotic Hierarchy. World Scientific. Singapore.

[58] Olsen, L. F., Steinmetz, C. G., Tidd, C. W. and W. M. Schaffer. 1991. Childhood infections - Examples of chaos in the wild. Pp. 359-375, In, Moselkilde, E. and L. Moselkilde (eds.) Complexity, Chaos and Biological Evolution. Plenum Press. N.Y.

[59] Allen J. C., Schaffer, W. M. and D. Rosko. 1993. Chaos reduces species extinction by amplifying local population noise. Nature. 364: 229-232.

[60] Geest, T., Olsen, L. F., and W. M. Schaffer. 1993. Lyapunov exponents and the fractal dimension of chaos in the peroxidase-oxidase reaction. J. Phys. Chemistry. 97: 8431-8441.

[61] Kendall, B. E., Schaffer, W. M. and C. W. Tidd. 1993. Transient periodicity in chaos. Physics Letters A. 177: 13-20.

[62] Schaffer, W. M., Kendall, B. E., Tidd, C. W. and L. F. Olsen. 1993. Transient periodicity and episodic predictability in biological dynamics. IMA J. Math. Appl. Med. Biol. 10: 227-247.

[63] Tidd, C. W., Olsen, L. F. and W. M. Schaffer. 1993. The case for chaos in childhood diseases. II. Predicting historical epidemics with mathematical models. Proc. Biol. Sci. Roy. Soc. London. 254: 257-273.

[64] Kendall, B., Schaffer, W. M., Olsen, L. F., Tidd, C. W. and B. L. Jorgensen. 1994. Using chaos to understand biological dynamics. Pp. 184-203. In, Grassman, J. and G. van Straten (eds.) Predictability and Nonlinear Modelling in Natural Sciences and Economics. Kluwer Academic Publ. Dordrecht.

[65] Olsen, L. F., Valeur, K., Geest, T., Tidd, C. W., Schaffer, W. M. and D. Lloyd. 1994. Nonlinear forecasting of non-uniform chatotic attractors in an enzyme reaction. Phil. Trans. Roy. Soc. Lond. 348: 421-430.

[66] Rosko, D. J., Schaffer, W. M. and J. C. Allen. 1994. Chaos and metapopulation persistence. Pp. 115-144. In, Yamaguti, M. (ed.) Towards the Harnessing of Chaos. Elsevier.

[67] 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.

[68] Olsen, L. F., Valeur, K. R., Geest, T., Tidd, C. W. and W. M. Schaffer. 1995. Predicting non-uniform chaotic attractors in an enzyme reaction. Pp. 161-174. In, Tong, H. (ed.) Chaos and Forecasting. World Scientific. Singapore.

[69] Bronnikova, T. V., W. M. Schaffer and L. F. Olsen. 1996. Quasiperiodicity in a detailed model of the peroxidase-oxidase reaction. J. Chem. Phys. 105: 10849-10859.

[70] King, A., W. M. Schaffer, C. Gordon, J. Treat and M. Kot. 1996. Weakly dissipative predator-prey systems. Bull. Math. Biol. 58:835-860.

[71] Kendall, B. E., Schaffer, W. M. and C. W. Tidd. 1997. The impact of chaos on biology. Pp. 190-218. In, Grebogi, C. and J. A. Yorke (eds.) The Impact of Chaos on Science and Society. United Nations Press. Tokyo.

[72] Hauser, M. J. B., Olsen, L. F., Bronnikova, T. V. and Schaffer, W. M. 1997. Routes to chaos in the peroxidase-oxidase reaction: I. Period-doubling and period-adding. J. Phys. Chem. 101:5075-5083.

[73] Bronnikova, T. V., Schaffer, W. M., Hauser, M. J. B. and L. F. Olsen. 1998. Routes to chaos in the peroxidase-oxidase reaction: II. The fat torus scenario. J. Phys. Chem. B. 102:632-640.

[74] King, A. A. and W. M. Schaffer. 1999. The rainbow bridge: Hamiltonian limits and resonance in predator-prey dynamics. J. Math. Biol. 39: 439-469.

[75] Schaffer, W. M. 2000. Foreword. Pp. vii-x. In, Perry, J. N., Smith, R. H., Woiwod, I. P. and D. R. Morse. (eds.) Chaos in Real Data. The Analysis of Nonlinear Dynamics from Short Ecological Time series. Kluver Academic Publ. Dordrecht. The Netherlands.

[76] 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.

[77] King, A. A. and W. M. Schaffer. 2001. The geometry of a population cycle: A mechanistic model of snowshoe hare demography. Ecology 82: 814-830.

[78] King, A. A. and W. M. Schaffer. 2001. Nondimensionalization of the minimal model (Appendix A). Ecol. Arch. E-082-007.

[79] Rai, V. and W. M. Schaffer. 2001. Preface to the Special Edition on Chaos in Ecology. Chaos, Solitons and Fractals. 12: 197-203.

[80] Schaffer, W. M., Pederson, B., Moore, B. K., Skarpaas, O., King, A. A. and T. V. Bronnikova. 2001. Subharmonic resonance and multi-annual oscillations in northern mammal cycles: A nonlinear dynamics perspective. Chaos, Solitons and Fractals. 12: 251-264.

[81] Schaffer, W. M., Bronnikova, T. V. and L. F. Olsen. 2001. Nonlinear dynamics of the peroxidase-oxidase reaction: II. Compatibility of an extended mechanistic model with previously reported model-data correspondences. J. Phys. Chem. B. 105: 5331-5340.

[82] Olsen, L. F., Bronnikova, T. V. and W. M. Schaffer. 2002. Secondary quasiperiodicity in the peroxidase-oxidase reaction. Phys. Chem. Chem. Phys. (Nonlinear Kinetics Special Issue) 4: 1292-1298.

[83] Schaffer. W. M. 2002. Foreword: An inordinate fondness for beetles. Pp. ix-xi, In, Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais, R. A. and S. Henson. Chaos in Ecology: Experimental Nonlinear Dynamics. Academic Press, San Diego.

[84] 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 University Press.

[85] Schaffer, W. M. 2007. Predicting Climate. The American Thinker. Published Online 10/26/2007 (http://www.americanthinker.com/blog/2007/10/predicting_climate_1.html)

[86] Schaffer, W. M. and T. V. Bronnikova. 2007a. Parametric dependence in model epidemics. J. Biol. Dynam. 1: 183-195.

[87] Schaffer, W. M. and T. V. Bronnikova. 2007b. Parametric dependence in model epidemics. II: Non-contact related parameters. J. Biol. Dynam. 1: 231-248.

[88] Schaffer, W. M. 2008. Human population and carbon dioxide. Energy Policy. 36: 2761-2764.

[89] Schaffer, W. M. and T. V. Bronnikova. 2009. Controlling malaria: Competition, seasonality and "slingshotting" transgenic mosquitoes into natural populations. J. Biol. Dynam. 3: 286-304.

[90] Schaffer, W. M. 2009. A surfeit of cycles. Energy and Environment. 20: 985-996.

[91] 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.

[92] Schaffer, W. M. and T. V. Bronnikova. 2012. Peroxidase-ROS interactions. Nonlinear Dynamcs. 68: 413-430.


IN PREPARATION (Books):

Schaffer, W. M. and T. V. Bronnikova. From the Pendulum to the Lynx and the Hare: An Introduction to Nonlinear Biological Dynamics.

Schaffer, W. M. Discovering Evolution: Implications for Contemporary Science.


BOOK REVIEWS:

[1] Schaffer, W. M. 1977. Population Biology (Emmel, T. C.). Am. Sci. 65: 779.

[2] Schaffer, W. M. 1978. Mountain Monarchs. Wild Sheep and Goats of the Himalaya. (Schaller, G. B.) Am. Sci. 66: 622.

[3] Schaffer, W. M. 1983. Cave Biology (Culver, D. C.). Biosci. 33: 214.

[4] Schaffer, W M. 1984. Population Biology of Infectious Diseases. (Anderson, R. M. and R. M. May). Math. Biosci.

[5] Schaffer, W. M. and G. L. Truty. 1988. Chaos in the Classroom: Nonlinear Dynamics and Dynamical Software. Academic Computing. March/April:34.

[6] Schaffer, W. M. 1989. Deterministic Chaos (G. Schuster). Quart. Rev. Biol. 64:187-188.

[7] Schaffer, W. M. 1989. From Clocks to Chaos. (L. Glass and M. C. Mackey). Science. 243: 675-676.

[8] Schaffer, W. M. and G. A. Fox. 1989. Plant Strategies and the Dynamics and Structure of Plant Communities (D. Tillman). J. Math. Biol. 53:409-411.

[9] Schaffer, W. M. and G. L. Truty. 1989. Nonlinear dynamics on your desktop computer. Nonlinear News. No. 6. Pp. 1-3. Manchester University Press.

[10] Schaffer, W. M. 1991. Mathematical and Statistical Approaches to AIDS Epidemiology. (C. Castillo-Chavez). Bull. Math. Biol. 53:160-161.

[11] Schaffer, W. M. 1996. Dynamical Disease. Mathematical Analysis of Human Illness. (J. Bélair et al.) Bull. Math. Biol. 58:606-609.

[12] Schaffer. W. M. 1997. Global Dynamical Properties of Lotka-Volterra Systems. (Y. Takeuchi.) Bull. Math. Biol.


COMPUTER SOFTWARE:(2)

[1] Schaffer, W. M. 1986. Dynamical Software: I. User's Manual and Introduction to Chaotic Systems. Pp. iv+193. 14 Disks.

[2] Schaffer, W. M., G. L. Truty and S. L. Fulmer. 1987. Dynamical Software: II. User's Manual and Introduction to Chaotic Systems. Pp. iv+240. 11 Disks.

[3] Schaffer, W. M. 1989. Chaos in the Classroom: I. Maps and Bifurcations. Pp. v+82. 2 Disks.

[4] Tidd, C. and W. M. Schaffer. 1989. Scanner: Automated Equation Entry For Dynamical Software: I.4/II.2. Pp. iv+64. 6 Disks.

[5] Schaffer, W. M. and C. W. Tidd. 1990a. Chaos in the Classroom: II. Fractals and Julia Sets. Pp. iii+65. 1 Disk.

[6] Schaffer, W. M. and C. W. Tidd. 1990b. NLF: Nonlinear Forecasting for Dynamical Systems. Pp. iv+154. 5 Disks.


1. Reprinted in Hao, B.-L. 1990. Chaos II. World Scientific. Singapore.

2. Published by Dynamical Systems, Inc., Tucson, Arizona.