Rocky Mountain Journal of Mathematics

Asymptotically Autonomous Differential Equations in the Plane

Horst R. Thieme

Full-text: Open access

Article information

Source
Rocky Mountain J. Math., Volume 24, Number 1 (1993), 351-380.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1181072470

Digital Object Identifier
doi:10.1216/rmjm/1181072470

Mathematical Reviews number (MathSciNet)
MR1270045

Zentralblatt MATH identifier
0811.34036

Subjects
Primary: 34C35
Secondary: 34D05: Asymptotic properties 58F12 92D25: Population dynamics (general) 92D30: Epidemiology

Keywords
Asymptotically autonomous differential equations dynamical systems limit equations equilibria closed (periodic) orbits ω -limit sets domain of attraction global stability cyclical chains undamped Duffing oscillator Poincaré & Bendixson Theorem limit set trichotomy Dulac (divergence) criterion Butler and McGehee Lemma chemostat gradostat epidemics

Citation

Thieme, Horst R. Asymptotically Autonomous Differential Equations in the Plane. Rocky Mountain J. Math. 24 (1993), no. 1, 351--380. doi:10.1216/rmjm/1181072470. https://projecteuclid.org/euclid.rmjm/1181072470


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References

  • A. Artstein, Limiting equations and stability of nonautonomous ordinary differential equations, Appendix to J.P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics 25, SIAM, 1976.
  • J.M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations 27 (1978), 224-265.
  • S.P. Blythe, K.L. Cooke and C. Castillo-Chavez, Autonomous risk-behavior change, and non-linear incidence rate, in models of sexually transmitted diseases, Biometrics Unit Technical Report B -1048-M, Cornell University, Ithaca, N.Y., preprint.
  • S. Busenberg and M. Iannelli, Separable models in age-dependent population dynamics, J. Math. Biol. 22 (1985), 145-173.
  • S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol. 29 (1990), 257-270.
  • G. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations 63 (1986), 255-263.
  • C. Castillo-Chavez and H.R. Thieme, Asymptotically autonomous epidemic models, Proc. $3^\textrd$ Intern. Conf. on Mathematical Population Dynamics (O. Arino, M. Kimmel, eds.).
  • E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math. 35 (1978), 1-16.
  • J.M. Cushing, A competition model for size-structured species, SIAM J. Appl. Math. 49 (1989), 838.
  • C.M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971), 239-252.
  • B. Fiedler and J. Mallet-Paret, A Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rat. Mech. Anal. 107 (1989), 325-345.
  • R.C. Grimmer, Asymptotically almost periodic solutions of differential equations, SIAM J. Appl. Math. 17 (1968), 109-115.
  • W. Hahn, Stability of motion, Springer, New York, 1967.
  • J.L. Hale, Ordinary differential equations, $2^nd$ ed., Krieger Publishing Company, 1980.
  • J.K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), 388-395.
  • M.W. Hirsch, Systems of differential equations that are competitive or cooperative. IV: Structural stability in $3$-dimensional systems, SIAM J. Math. Anal. 21 (1990), 1225-1234.
  • M.W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, Academic Press, 1974.
  • L. Hsiao, Y. Su, and Z.P. Xin, On the asymptotic behavior of solutions of a reacting-diffusing system: a two predator-one prey model, SIAM J. Appl. Math. 18 (1987), 647-669.
  • S.-B. Hsu, Predator-mediated coexistence and extinction, Math. Biosci. 54 (1981), 231-248.
  • S.-B. Hsu, K.-S. Cheng and S.P. Hubbell, Exploitative competition of microorganisms for two complementary nutrients in continuous cultures, SIAM J. Appl. Math. 41 (1981), 422-444.
  • W. Jäger, J.W.-H. So, B. Tang and P. Waltman, Competition in the gradostat, J. Math. Biol. 25 (1987), 23-42.
  • J. Mallet-Paret and H.L. Smith, The Poincaré-Bendixson theorem for monotone cyclic feedback systems, J. Dynamics Differential Equations 2 (1990), 367-421.
  • L. Markus, Asymptotically autonomous differential systems, in Contributions to the Theory of Nonlinear Oscillations III (S. Lefschetz, ed.), 17-29. Annals of Mathematics Studies 36, Princeton Univ. Press, 1956.
  • R.K. Miller, Almost periodic differential equations as dynamical systems with applications to the existence of a.p. solutions, J. Differential Equations 1 (1965), 337-345.
  • R.K. Miller and G.R. Sell, Volterra integral equations and topological dynamics, AMS Memoirs 102 (1970).
  • G.R. Sell, Nonautonomous differential equations and topological dynamics I. The basic theory, II. Limiting equations, Trans. Amer. Math. Soc. 127 (1967), 241-262, 263-283.
  • --------, Topological dynamical techniques for differential and integral equations. Ordinary Differential Equations, L. Weiss, ed., 287-304, Proceedings of the NRL-NCR conference 1971, Academic Press, 1972.
  • H.L. Smith, Convergent and oscillatory activation dynamics for cascades of neural nets with nearest neighbor competitive or cooperative interactions, Neural Networks 4 (1991), 41-46.
  • H.L. Smith and B. Tang, Competition in the gradostat: the role of the communication rate, J. Math. Biol. 27 (1989), 139-165.
  • H.L. Smith, B. Tang and P. Waltman, Competition in an n-vessel gradostat, SIAM J. Appl. Math. 51 (1991), 1451-1471.
  • J. Smoller, Shock waves and reaction-diffusion equations, Springer, New York, 1983.
  • H.R. Thieme, Persistence under relaxed point-dissipativity (with applications to an endemic model), SIAM J. Math. Anal. 24 (1993), 407-435.
  • --------, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol. 30 (1992), 755-763.
  • W.R. Utz and P. Waltman, Asymptotic almost periodicity of solutions of a system of differential equations, Proc. Amer. Math. Soc. 18 (1967), 597-601.
  • P. Waltman, Coexistence in chemostat-like models, Rocky Mountain J. Math. 20 (1990), 777-807.
  • S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer, New York, 1990.