Abstract
The goal of this article is to illustrate several interesting bifurcations that can arise in population biology. These are of interest since it is often through bifurcation phenomena that changes significant enough to be measured occur. For example, a minor change in some environmental parameter can cause a system to change from being at rest to oscillating. We illustrate here the role of several canonical types of bifurcations in population modeling.
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References
F. Albrecht, H. Gatzke, A. Hadad, and N. Wax, The dynamics of two interacting populations, J.Math. Anal. Appl. 46 (1974), 658–670
V.I. Arnol’d, ed., Dynamical Systems V. Bifurcation Theory and Catastrophe Theory. Springer-Verlag, New York, 1994.
G. J. Butler and P. Waltman, Bifurcation from a limit cycle in a two predator-one prey ecosysstem modeled on a chemostat, J. Math. Bio. 12 (1981), 295–310
Cheng, K. -S., Uniqueness of limit cycles for a predator-prey system, SIAM J. Math. Analysis 12 [ 1981 ], 541–548
W. Feller, An Introduction to Probability Theory and its Applications, vols. I, II, Wiley-Interscience, New York.
H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980
J. Glieck, Chaos: Making of a New Science, Viking, 1987.
J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983.
J. Hofbauer and J.W.-H. So, Multiple Limit Cycles for Pedator-Prey Models, Math. Biosciences 99 (1990), 71–75
F. C. Hoppensteadt, A nonlinear renewal equation with periodic and chaotic solutions, Proc. AMS-SIAM Conf. Appl. Math., New York, April, 1976.
F.C. Hoppensteadt, Analysis and Simulation of Chaotic Systems, 2nd ed., Springer Verlag, New York, 2000.
F.C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, Cambridge, 1986.
F.C. Hoppensteadt and J.M. Hyman, Periodic solutions to a discrete logistic equation, SIAM Appl. Math. 32(1977)73–81.
F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks, Springer-Verlag, New York, 1997.
F.C. Hoppensteadt, Z. Jacekewicz, Numerical solution of nonlinear Volterra equations, in preparation.
F.C. Hoppensteadt, C.S. Peskin, Modeling and Simulation in Medicine and the Life Sciences, Springer-Verlag, in press.
F.C. Hoppensteadt, P. Waltman, A problem in the theory of epidemics (1970, 1971)Math. Biosci. 9,71–91, 12, 133–145.
S. B. Hsu, S.P. Hubbell, and P. Waltman, Competing Predators, SIAM J. Appl. Math. 35 (1978), 617–625
S. B. Hsu, S.P. Hubbell, and P. Waltman, A contribution to the theory of competing predators, Ecol. Monographs 48 (1978), 337–349
A. Kolmorgoroff, Sulla teoria di Volterra della latta per l’esistenza, Gi. Inst. Ital. Attuari 7 (1936), 74–80
Yu. Kuznetzov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1995.
J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976
R. M. May, Limit cycles in predator-prey communities, Science 177 (1972), 900–902.
J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1989.
W.E. Ricker, Stock and recruitment, J. Fish. Res. Bd. Canada, 11 (1954) 559–623.
A. Riscigno and I.W. Richardson, On the competitive exclusion principle, Bulletin Math. Biophysics, 27 (1965), 85–89
A. Riscigno and I. W. Richardson,The struggle for life I:two species, Bulletin Math. Biphysics, 29 (1967), 377–388
A.V. Skorokhod, F.C. Hoppensteadt, H.S. Salehi, Random Perturbations of Dynamical Systems, Springer-Verlag, 2002.
H. L. Smith, The interaction of steady state and Hopf bifurcation in a two-predator, one prey competition model, SIAM J. Applied Math. (1982)27–43.
R. Thom, Structural stability and Morphogenesis: An outline of a general theory of models, W.A. Benjamin, Reading, MA, 1975.
P. Waltman, Deterministic Threshold Models in the Theory of Epidemics, Springer-Verlag, New York, 1974.
P. Waltman, Competition Models in Population Biology, Soc. Ind. Appl. Math., Philadelphia, 1983.
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.
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This paper is dedicated to our good friend Professor Willi Jäger on the occasion of his 60th birthday. The three of us have collaborated since a momentous year (for us) at the Courant Institute of Mathematical Sciences in 1969–70
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Hoppensteadt, F., Waltman, P. (2003). Did Something Change? Thresholds in Population Models. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds) Trends in Nonlinear Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05281-5_10
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DOI: https://doi.org/10.1007/978-3-662-05281-5_10
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