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Bifurcation phenomena in Biomathematics

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Bifurcation Theory and Applications

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References

  1. Auslander, D., Oster, G. and Huffaker, C. (1974). "Dynamics of interacting populations", J. Franklin Inst. 5: 345–376.

    Article  Google Scholar 

  2. Bailey, N.T.J. (1975). The Mathematical Theory of Infectious Diseases, Second Edition, Hafner, New York.

    MATH  Google Scholar 

  3. Banks, H.T. and Mahaffy, J.M. (1978). "Global asymptotic stability of certain models for protein synthesis and repression", Quart. Appl. Math., 36: 209–221.

    MathSciNet  MATH  Google Scholar 

  4. Bernoulli, D. (1760). "Essai d'une nonvelle analyse de la mortalite causee par la petite verole, e des avantage de l'inoculation pour la prevenir", Memoires de Mathematiques et de Physique, Academic Royale des Science, Paris, 1–45.

    Google Scholar 

  5. Bonner, J.T. (1974). On Development, the Biology of Form, Harvard U. Press, Cambridge, Mass.

    Google Scholar 

  6. Bradley, L. (1971). Smallpox Inoculation, An Eighteenth Century Mathematical Controversy, U. of Nottingham, Nottingham.

    Google Scholar 

  7. Busenberg, S. and Cooke, K. (1982). "Models of vertically transmitted diseases with sequential-continuous dynamics" in Nonlinear Phenomena in Mathematical Sciences, V. Lakshmikantham (Editor), Academic Press, New York: 179–187.

    Chapter  Google Scholar 

  8. Collet, P. and Eckmann, J.P. (1980). Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Boston.

    MATH  Google Scholar 

  9. Fisher, R.A. (1937). "The wave of advance of advantageous genes", Ann. Eugen. London 7: 355–369.

    Article  MATH  Google Scholar 

  10. Guckenheimer, J., Oster, G. and Ipaktchi, A. (1976). "The dynamics of density dependent population models", J. Math. Biol. 4: 101–147.

    Article  MathSciNet  MATH  Google Scholar 

  11. Hastings, S. (1981). "Some mathematical problems arising in neurobiology" in Mathematics of Biology, M. Iannelli (Editor), Liguori, Naples: 179–264.

    Google Scholar 

  12. Kloeden, P.E. (1976). "Chaotic difference equations are dense", Bull. Austral. Math. Soc. 15: 371–379.

    Article  MathSciNet  MATH  Google Scholar 

  13. Levin, S.A. (1981). "Models of population dispersal" in Differential Equations and Applications in Ecology, Epidemics, and Population Problems, S. Busenberg and K. Cooke (Editors), Academic Press, New York: 1–18.

    Chapter  Google Scholar 

  14. Li, T.Y. and Yorke, J.A. (1975). "Period three implies chaos", Amer. Math. Monthly, 82: 985–992.

    Article  MathSciNet  MATH  Google Scholar 

  15. Mackey, M.C., and Glass, L. (1979). "Pathological conditions resulting from instabilities in physiological control systems", Annals N.Y. Acad. of Sci. 316: 214–235.

    Article  MATH  Google Scholar 

  16. May, R.M. (1974). "Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos", Science 186: 645–647.

    Article  Google Scholar 

  17. May, R.M. (1975). Stability and Complexity in Model Ecosystems, 2nd ed., Princeton U. Press, Princeton.

    Google Scholar 

  18. McKendrick, A.G. (1926). "Applications of mathematics to medical problems", Proc. Edin. Math. Soc. 44: 98–130.

    Article  Google Scholar 

  19. Nicholson, A.J. (1957). "The self-adjustment of populations to change", Cold Spring Harbor Symposia on Quantitative Biology 22: 153–173.

    Article  Google Scholar 

  20. Okubo, A. (1980). Diffusion and Ecological Problems: Mathematical Models, Biomathematics Vol. 10, Springer, New York.

    Google Scholar 

  21. Oster, G. (1978). "The dynamics of nonlinear models with age structure" in Studies in Mathematical Biology, Part II: Populations and Communities, S.A. Levin (Editor), Math. Association of America, Providence, RI: 411–438.

    Google Scholar 

  22. Pearson, K. and Blakeman, J. (1906). "Mathematical contributions to the theory of evolution-XV. A mathematical theory of random migration", Draper's Company Research Mem. Biometric Series II., Dept. Appl. Math., Univ. College, University of London.

    Google Scholar 

  23. Ross, R. (1911). The prevention of Malaria, 2nd ed., Murray, London.

    Google Scholar 

  24. Scudo, R.M. and Ziegler, J.R. (1978). The Golden Age of Theoretical Biology: 1923–1940, Lecture Notes in Biomath. 22, Springer, Berlin.

    MATH  Google Scholar 

  25. Smoller, J. (1982). Shock Waves and Reaction-Diffusion Equations, Springer, New York.

    MATH  Google Scholar 

  26. Siegberg, H.-W. (1983). "Chaotic difference equations: genetic aspects", Trans. Amer. Math. Soc.: 205–213.

    Google Scholar 

  27. Turing, A.M. (1952). "The chemical basis of morphogenesis", Phil. Roy. Soc. London B237: 37–72.

    Article  MathSciNet  Google Scholar 

  28. Winfree, A.T. (1980). The Geometry of Biological Time, Springer, New York.

    Book  MATH  Google Scholar 

References

  1. Amann, H. (1976). "Fixed point equations and nonlinear eigenvalues in ordered Banach spaces", SIAM Rev. 18: 620–709.

    Article  MathSciNet  MATH  Google Scholar 

  2. Aronsson, G. and Mellander, I. (1980). "A deterministic model in biomathematics. Asymptotic behavior and threshold conditions", Math Biosci. 49: 207–222.

    Article  MathSciNet  MATH  Google Scholar 

  3. Busenberg, S. and Cooke, K.L. (1978). "Periodic solutions of a periodic nonlinear delay differential equation", SIAM J. Appl. Math. 35: 704–721.

    Article  MathSciNet  MATH  Google Scholar 

  4. Busenberg, S. and Cooke, K.L. (1979). "Periodic solutions of delay differential equations arising in some models of epidemics", in Applied Nonlinear Analysis, V. Lakshmikanthan (editor), Academic Press, New York: 67–78.

    Chapter  Google Scholar 

  5. Gatica, J. and Smith, H.L. (1977). "Fixed point techniques in a cone with applications", J. Math. Anal. Appl. 61: 58–71.

    Article  MathSciNet  MATH  Google Scholar 

  6. Greiner, G. (1981). "Zur Perron-Frobenius-Theorie stark stetiger Halbgruppen", Math. Z. 177: 401–423.

    Article  MathSciNet  MATH  Google Scholar 

  7. Hale, J. (1977). Theory of Functional Differential Equations, Springer, New York.

    Book  MATH  Google Scholar 

  8. Kostitzin, V.A. (1939). "Sur les equations integrodifferentielles de la theorie de l'action toxique du milieu", Comples Rendus Ac. des. Sci. 208: 1545–1547.

    MATH  Google Scholar 

  9. Krasnoselskii, M.A. (1964). Positive Solutions of Operator Equations, O. Noordhoff, Groningen.

    Google Scholar 

  10. Krein, M.G. and Rutman, M.A. (1962). "Linear operators leaving invariant a cone in a Banach space", Amer. Math. Soc. Transl. Series 1, 10: 199–325.

    MATH  Google Scholar 

  11. Lajmanovich, A. and Yorke, J.A. (1976). "A deterministic model for gonorrhea in a nonhomogeneous population", Math. Biosci. 28: 221–236.

    Article  MathSciNet  MATH  Google Scholar 

  12. Marcati, P. and Pozio, M.A. (1980). "Global asymptotic stability for a vector disease model with spatial spread", J. Math. Biol. 9: 179–187.

    Article  MathSciNet  MATH  Google Scholar 

  13. Nussbaum, R. (1974). "Periodic solutions of some nonlinear autonomous functional differential equations", Ann. Math. Pura Appl. 10: 263–306.

    Article  MathSciNet  MATH  Google Scholar 

  14. Nussbaum, R. (1978). "A periodicity threshold theorem for some nonlinear integral equations", SIAM J. Math. Anal. 9: 356–376.

    Article  MathSciNet  MATH  Google Scholar 

  15. Pozio, M.A. (1980). "Behavior of solutions of some abstract functional differential equations and applications to predator-prey dynamics", Nonlinear Analysis TMA, 4: 917–938.

    Article  MathSciNet  MATH  Google Scholar 

  16. Scudo, F.M. and Ziegler, J.R. (1978). The Golden Age of Theoretical Ecology, Lecture Notes in Biomathematics 22, Springer, New York.

    MATH  Google Scholar 

  17. Smith, H.L. (1977). "On periodic solutions of a delay integral equation modelling epidemics", J. Math. Biol., 4: 69–80.

    Article  MathSciNet  MATH  Google Scholar 

  18. Smith, H.L. (1981). "An abstract threshold theorem for one parameter families of positive noncompact operators", Funkcial. Ekvac. 24: 141–153.

    MathSciNet  MATH  Google Scholar 

  19. Thieme, H.R. (1983). "Global asymptotic stability in epidemic models", preprint.

    Google Scholar 

  20. Thieme, H.R. (1980). "On the boundedness and the asymptotic behavior of the non-negative solutions of Volterra-Hammerstein integral equations", Manuscr. Math. 31: 379–412.

    Article  MathSciNet  MATH  Google Scholar 

  21. Varga, R.S. (1962). Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs.

    Google Scholar 

  22. Verhulst, P.F. (1845). "Recherches mathematiques sur la loi d'accroissement de la population", Nouveau Mem. Acad. Sci. Bruxelles, 18: 3–41.

    Google Scholar 

  23. Volterra, V. (1939). "The general equations of biological strife in the case of historical actions", Proc. Edinburgh Math. Soc. 6: 4–10.

    Article  MATH  Google Scholar 

  24. Volz, R. (1982). "Global asymptotic stability of a periodic solution to and epidemic model", J. Math. Biol. 15: 319–338.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Allwright, D.J., (1977). "A global stability criterion for simple loops," J. Math. Biol. 4: 363–373.

    Article  MathSciNet  MATH  Google Scholar 

  2. An der Heiden, U. (1979). "Periodic solutions of a nonlinear second order differential equation with delay", J. Math. Anal. Appl. 70: 599–609.

    Article  MathSciNet  MATH  Google Scholar 

  3. Atkin, E. (1983). "Hopf bifurcation in the two locus genetic model", Mem. AMS 284, American Mathematical Society, Providence, RI.

    Google Scholar 

  4. Banks, H.T. and Mahaffy, J.M. (1978). "Global asymptotic stability of certain models for protein synthesis and repression", Quart. Appl. Math. 36: 209–221.

    MathSciNet  MATH  Google Scholar 

  5. Bardi, M. (1982). "Exchange of stability along a branch of periodic solutions of a single specie model" (preprint).

    Google Scholar 

  6. Bellman, R. and Cooke, K.L. (1963). Differential Difference Equations, Academic Press, New York.

    MATH  Google Scholar 

  7. Busenberg, S.N. and Cooke, K.L. (1980). "The effect of integral conditions in certain equations modelling epidemics and population growth", J. Math. Biol. 10: 13–32.

    Article  MathSciNet  MATH  Google Scholar 

  8. Busenberg, S.N. and Travis, C.C. (1982). "On the use of reducible functional differential equations in biological models", J. Math Anal. App. 89: 46–66.

    Article  MathSciNet  MATH  Google Scholar 

  9. Chow, S.-N. and Hale, J.K. (1982). Methods of Bifurcation Theory, Springer Verlag, New York.

    Book  MATH  Google Scholar 

  10. Cooke, K.L. (1967). "Functional-differential equations: some models and perturbation problems" in Differential Equations and Dynamical Systems, J.K. Hale and J.P. LaSalle, editors, Academic Press, New York: 167–183.

    Google Scholar 

  11. Cooke, K.L. and Yorke, J. (1973). "Some equations modelling growth processes and gonorrhea epidemics", Math. Biosciences, 16: 75–101.

    Article  MathSciNet  MATH  Google Scholar 

  12. Fredrikson, A.G. and Stephanopoulos, G. (1981). "Microbial Competition", Science 213: 972–979.

    Article  MathSciNet  MATH  Google Scholar 

  13. Frazer, A. and Tivari, J. (1974). "Genetical feedback repression: II cyclic genetic systems", J. Theor. Biol. 47: 397–412.

    Article  Google Scholar 

  14. Goodwin, B.C. (1963). "Oscillatory behavior of enzymatic control processes", Adv. Enzyme Res. 3: 425–439.

    Article  Google Scholar 

  15. Green, D. (1978). "Self-oscillations for epidemic models", Math. Biosciences 38: 91–111.

    Article  MathSciNet  MATH  Google Scholar 

  16. Grossman, Z. and Gumowski, I. (1976). "Self-sustained oscillations in the Jacob-Monod model of gene regulation", Lecture notes in Computer Science, Vol. 40, Springer, New York: 145–154.

    Google Scholar 

  17. Hadeler, K.P. (1976). "On the stability of the stationary state of a population growth equation with time lag", J. Math. Biology, 3: 197–201.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hale, J.K. (1977). Theory of Functional Differential Equations, Springer, New York.

    Book  MATH  Google Scholar 

  19. Hale, J.K. (1983). "Introduction to dynamic bifurcation", this volume, CIME.

    Google Scholar 

  20. Hale, J.K. (1974). "Behavior near constant solutions of functional differential equations", J. Diff. Eq. 15: 278–294.

    Article  MathSciNet  MATH  Google Scholar 

  21. Hastings, A. (1981). "Multiple limit cycles in predator-prey models", J. Math. Biology 11: 51–63.

    Article  MathSciNet  MATH  Google Scholar 

  22. Hastings, S.P., Tyson, J.J. and Webster, D. (1977). "Existence of periodic solutions for negative feedback cellular control systems", J. Differential Eq. 25: 39–64.

    Article  MathSciNet  MATH  Google Scholar 

  23. Hoppensteadt, F. and Waltman, P. (1970). "A problem in the theory of epidemics", Math. Biosciences, 9: 71–91.

    Article  MathSciNet  MATH  Google Scholar 

  24. Hsu, S.B., Hubbell, S.P., and Waltman, P. (1978). "Competing predators", SIAM J. Appl. Math. 35: 617–625.

    Article  MathSciNet  MATH  Google Scholar 

  25. Keener, J.P. (1982). "Oscillatory coexistence in the chemostat; a codimension two unfolding", Prepring 153, Sonderforschungsbereich, University of Heidelberg.

    Google Scholar 

  26. Mahaffy, J.M. (1980). "Periodic solutions of certain protein synthesis models", J. Math. Anal. Appl. 74: 72–105.

    Article  MathSciNet  MATH  Google Scholar 

  27. Marsden, J.E. and MacCracken, M. (1976). The Hopf Bifurcation and its Applications, Springer, New York.

    Book  Google Scholar 

  28. MacDonald, N. (1978). Time Lag in Biological Models, Lecture Notes in Biomathematics, Vol. 27, Springer, New York.

    Book  MATH  Google Scholar 

  29. Monod, J. (1942). Researches sur la Croissence des Cultures Bacteriennes, Hermann, Paris.

    Google Scholar 

  30. Negrini, P. and Salvadori, L. (1979). "Attractivity and Hopf Bifurcation", Nonlinear Analysis 3: 87–99.

    Article  MathSciNet  MATH  Google Scholar 

  31. Powell, E.O. (1958). "Criteria for the growth of contaminants and mutants in continuous culture", J. Gen. Microbiology, 18: 259–268.

    Article  Google Scholar 

  32. Rapp, P.E., (1983). "Biochemical and physiological switching behavior", in Encyclopedia of Systems and Control, M.P. Singh, editor, Pergammon Press, Oxford (to appear).

    Google Scholar 

  33. Smith, H.L. (1982). "The interaction of steady state and Hopf bifurcations in a two-predator-one-prey competition model", SIAM J. Appl. Math. 42: 27–43.

    Article  MathSciNet  MATH  Google Scholar 

  34. Volterra, V. (1939). "The general equations of biological strife in the case of historical actions", Proc. Edinburgh Math. Soc. 6: 4–10.

    Article  MATH  Google Scholar 

  35. Walther, H.O. (1976). "On a transcendental equation in the stability analysis of a population growth model", J. Math. Biology, 3: 187–195.

    Article  MathSciNet  MATH  Google Scholar 

  36. Winfree, A.T. (1980. The Geometry of Biological Time, Springer, New York.

    Book  MATH  Google Scholar 

References

  1. Aronson, D.G., Crandall, M.G., and Peletier, L.A. (1982). "Stabilization of solutions of a degenerate nonlinear diffusion equation", Nonlin. Anal. TMA 10: 1001–1022.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsch, K.A. and Hilhorst, D. (1983). A density dependent diffusion equation in population dynamics: stabilization to equilibrium, preprint.

    Google Scholar 

  3. Bertsch, M.A., Gurtin, M.E., Hilhorst, D., Peletier, L.A., (1983). "On interacting populations that disperse to avoid crowding: the effects of a sedentary colony", preprint.

    Google Scholar 

  4. Busenberg, S. and Iannelli, M. (1983). "A class of nonlinear diffusion problems in age-dependent population dynamics", Nonlin. Anal. TMA 7: 501–529.

    Article  MathSciNet  MATH  Google Scholar 

  5. Busenberg, S. and Iannelli, M. (1983). "A degenerate nonlinear diffusion problem in age-structured population dynamics", Nonlin. Anal. TMA, to appear.

    Google Scholar 

  6. Busenberg, S. and Iannelli, M. (1983). "Nonlinear diffusion problems in age-structured population dynamics", preprint.

    Google Scholar 

  7. Busenberg, S. and Travis, C. (1983). "Epidemic models with spatial spread due to population migration", J. Math. Biol. 16: 181–198.

    Article  MathSciNet  MATH  Google Scholar 

  8. Feller, W. (1941). "On the integral equation of renewal theory", Ann. Math. Statistics 12: 243–267.

    Article  MathSciNet  MATH  Google Scholar 

  9. Fife, P.C. (1979). Mathematical Aspects of Reacting and Diffusion Systems, Lectures Notes in Biomathematics 28, Springer, New York.

    Book  MATH  Google Scholar 

  10. Fisher, R.A. (1937). "The advance of advantageous genes", Ann. of Eugenics 7: 355–369.

    Article  MATH  Google Scholar 

  11. Gurtin, M.E. and Pipkin, A.C. (1983). "On the interacting populations that disperse to avoid crowding", preprint.

    Google Scholar 

  12. Hadeler, K.P. (1981). "Diffusion equations in biology", in Mathematics of Biology, M. Iannelli, editor, Liguori, Naples: 149–177.

    Google Scholar 

  13. Hodgkin, A.L. and Huxpy, A.F. (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerves", J. Physiology 117: 500–544.

    Article  Google Scholar 

  14. Hoppensteadt, F. (1975). Mathematical Theory of Population Demographics, Genetics and Epidemics, CBMS-NSF Regional Conference Series in Applied Mathematics 20, Society for Industrial and Applied Mathematics, Philadelphia.

    Book  MATH  Google Scholar 

  15. Kolmogorov, A., Petrovskii, I., Piscunov, N. "Etude de l'equation de la diffusion avec croissance de la quantite de matiere et son application a une probleme biologique", Bull. Univ. Moscow, Ser. Int. Sect. Al 6: 1–25.

    Google Scholar 

  16. Lotka, A.J. (1939). "A contribution to the theory of self-renewing aggregates with special reference to industrial replacement", Ann. Math. Statistics 10: 1–25.

    Article  MATH  Google Scholar 

  17. MacCamy, R.C. (1981). A population model with nonlinear diffusion, J. Diff. Eq. 39: 52–72.

    Article  MathSciNet  MATH  Google Scholar 

  18. Meinhart, H. (1982). Models of Biological Pattern Formation, Academic Press, New York.

    Google Scholar 

  19. Mimura, M. (1981). Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J. 11: 621–635.

    MathSciNet  MATH  Google Scholar 

  20. Okubo, A. (1980). Diffusion and Ecological Problems: Mathematical Models, Springer, New York.

    MATH  Google Scholar 

  21. Rashevsky, N. (1938). Mathematical Biophysics, revised edition, U. of Chicago Press, Chicago.

    MATH  Google Scholar 

  22. Shigesada, N. (1980). Spatial distribution of dispersing animals, J. Math. Biol. 9: 85–96.

    Article  MathSciNet  MATH  Google Scholar 

  23. Shigesada, N., Kawasaki, K., and Teramoto, E., (1979). Spatial segregation of interacting species, J. Theor. Biol. 79: 83–99.

    Article  MathSciNet  Google Scholar 

  24. Turing, A.M. (1952). "The chemical basis of morphogenesis", Phil. Trans. Roy. Soc. B 237: 37–72.

    Article  MathSciNet  Google Scholar 

  25. Weinberger, H.F. (1982). "A simple system with a continuum of stable inhomogeneous steady states", Institute of Mathematics and its Applications, Univ. of Minnesota, preprint #16.

    Google Scholar 

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  1. Department of Mathematics, Harvey Mudd College, 91711, Claremont, California

    Stavros N. Busenberg

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Busenberg, S.N. (1984). Bifurcation phenomena in Biomathematics. In: Salvadori, L. (eds) Bifurcation Theory and Applications. Lecture Notes in Mathematics, vol 1057. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098593

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