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Nonautonomous Semiflows

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Dynamical Systems in Population Biology

Part of the book series: CMS Books in Mathematics ((CMSBM))

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Abstract

There are many nonautonomous models that describe the population dynamics in a fluctuating environment. Solutions of these systems can generate nonautonomous semiflows on phase spaces. The purpose of this chapter is to develop the theory of nonautonomous semiflows. It is well known that the existence and stability of periodic solutions of a periodic differential system are equivalent to those of fixed points of its associated Poincaré map (see, e.g., [152]). In Section 3.1 we introduce the concept of periodic semiflows and prove that uniform persistence of a periodic semiflow also reduces to that of its associated Poincaré map under a general abstract setting. To illustrate the applications of the theory of monotone discrete dynamical systems to periodic problems, we then discuss periodic cooperative ordinary differential systems and scalar parabolic equations. In particular, we establish threshold dynamics in terms of principal multipliers and eigenvalues, and show how to obtain corresponding results for autonomous cases of these systems. Two practical examples are also provided.

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References

  1. W.G. Aiello, H.I. Freedman, A time-delay model of single-species growth with stage structure. Math. Biosci. 101, 139–153 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Aronsson, R.B. Kellogg, On a differential equation arising from compartmental analysis. Math. Biosci. 38, 113–122 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Benaïm, M.W. Hirsch, Asymptotic pseudotrajectories and chain recurrent flows, with applications. J. Dyn. Differ. Equ. 8, 141–176 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. G.J. Butler, P. Waltman, Persistence in dynamical systems. J. Differ. Equ. 63, 255–263 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. R.S. Cantrell, C. Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments. Proc. R. Soc. Edinb. Sect. A 112, 293–318 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. R.S. Cantrell, C. Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments II. SIAM J. Math. Anal. 22, 1043–1064 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Capasso, Mathematical Structures of Epidemic Systems (Springer, Berlin, 1993)

    Book  MATH  Google Scholar 

  8. C. Dafermos, Semiflows generated by compact and uniform processes. Math. Syst. Theory 8, 142–149 (1975)

    Article  MATH  Google Scholar 

  9. E.N. Dancer, P. Hess, On stable solutions of quasilinear periodic–parabolic problems Annali Sc. Norm. Sup. Pisa 14, 123–141 (1987)

    MathSciNet  MATH  Google Scholar 

  10. S.R. Dunbar, K.P. Rybakowski, K. Schmitt, Persistence in models of predator–prey populations with diffusion. J. Differ. Equ. 65, 117–138 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. H.I. Freedman, Q.-L. Peng, Uniform persistence and global asymptotic stability in periodic single-species models of dispersal in a patchy environment. Nonlinear Anal. 36, 981–996 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. J.K. Hale, Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25 (American Mathematical Society, Providence, RI, 1988)

    Google Scholar 

  13. J.K. Hale, O. Lopes, Fixed point theorems and dissipative processes. J. Differ. Equ. 13, 391–402 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  14. J.K. Hale, P. Waltman, Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 388–395 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840 (Springer, Berlin, 1981)

    Google Scholar 

  16. P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics Series, vol. 247 (Longman Scientific and Technical, Harlow, 1991)

    Google Scholar 

  17. P. Hess, On the asymptotically periodic Fisher equation, in Progress in Partial Differential Equations: Elliptic and Parabolic Problems. Pitman Research Notes in Mathematics Series, vol. 266 (Longman Scientific and Technical, Harlow, 1992), pp. 24–33

    Google Scholar 

  18. M.W. Hirsch, Systems of differential equations that are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16, 423–439 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. M.W. Hirsch, Positive equilibria and convergence in subhomogeneous monotone dynamics, in Comparison Methods and Stability Theory. Lecture Notes in Pure and Applied Mathematics, No. 162 (Marcel Dekker, New York, 1994), pp. 169–187

    Google Scholar 

  20. S.-B. Hsu, P. Waltman, On a system of reaction–diffusion equations arising from competition in an unstirred chemostat. SIAM J. Appl. Math. 53, 1026–1044 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. V. Hutson, K. Mischaikow, P. Polác̆ik, The evolution of dispersal rates in a heterogeneous time-periodic environment. J. Math. Biol. 43, 501–533 (2001)

    Google Scholar 

  22. V. Hutson, W. Shen, G.T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators. Proc. Am. Math. Soc. 129, 1669–1679 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. J.P. LaSalle, The Stability of Dynamical Systems (Society for Industrial and Applied Mathematics, Philadelphia, 1976)

    Book  MATH  Google Scholar 

  24. X. Liang, X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259, 857–903 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Z. Lu, Y. Takeuchi, Global asymptotic behavior in single-species discrete diffusion systems. J. Math. Biol. 32, 67–77 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. K. Mischaikow, H.L. Smith, H.R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Liapunov functions. Trans. Am. Math. Soc. 347, 1669–1685 (1995)

    Article  MATH  Google Scholar 

  27. R.D. Nussbaum, Some nonlinear weak ergodic theorems. SIAM J. Math. Anal. 21, 436–460 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)

    Book  MATH  Google Scholar 

  29. G. Sell, Topological Dynamics and Ordinary Differential Equations (Van Nostrand Reinhold, London, 1971)

    MATH  Google Scholar 

  30. W. Shen, Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows. Memoirs of the American Mathematical Society, No. 647, vol. 136 (American Mathematical Society, Providence, RI, 1998)

    Google Scholar 

  31. W. Shen, Y. Yi, Convergence in almost periodic Fisher and Kolmogorov models. J. Math. Biol. 37, 84–102 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. H.L. Smith, Cooperative systems of differential equations with concave nonlinearities. Nonlinear Anal. TMA 10, 1037–1052 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  33. H.L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41 (American Mathematical Society, Providence, RI, 1995)

    Google Scholar 

  34. J. So, P. Waltman, A nonlinear boundary value problem arising from competition in the chemostat. Appl. Math. Comput. 32, 169–183 (1989)

    MathSciNet  MATH  Google Scholar 

  35. P. Takác̆, Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications in biology. Nonlinear Anal. TMA 14, 35–42 (1990)

    Google Scholar 

  36. H.R. Thieme, Asymptotic proportionality (weak ergodicity) and conditional asymptotic equality of solutions to time-heterogeneous sublinear difference and differential equations. J. Differ. Equ. 73, 237–268 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  37. H.R. Thieme, Convergence results and Poincaré–Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30, 755–763 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. H.R. Thieme, Persistence under relaxed point-dissipativity (with application to an epidemic model). SIAM J. Math. Anal. 24, 407–435 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  39. H.R. Thieme, Aymptotically autonomous differential equations in the plane. Rocky Mt. J. Math. 24, 351–380 (1994)

    Article  MATH  Google Scholar 

  40. H.R. Thieme, Aymptotically autonomous differential equations in the plane (II): strict Poncaré–Bendixson type results. Differ. Integral Equ. 7, 1625–1640 (1994)

    MATH  Google Scholar 

  41. H.R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows. Proc. Am. Math. Soc. 127, 2395–2403 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. H.R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166, 173–201 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  43. X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications. Can. Appl. Math. Q. 3, 473–495 (1995)

    MathSciNet  MATH  Google Scholar 

  44. X.-Q. Zhao, Global attractivity and stability in some monotone discrete dynamical systems. Bull. Aust. Math. Soc. 53, 305–324 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  45. X.-Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications. Commun. Appl. Nonlinear Anal. 3, 43–66 (1996)

    MathSciNet  MATH  Google Scholar 

  46. X.-Q. Zhao, Global asymptotic behavior in a periodic competitor–competitor–mutualist parabolic system. Nonlinear Anal. TMA 29, 551–568 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  47. X.-Q. Zhao, Uniform persistence in processes with application to nonautonomous competitive models. J. Math. Anal. Appl. 258, 87–101 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems. J. Differ. Equ. 187, 494–509 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  49. X.-Q. Zhao, Z.-J. Jing, Global asymptotic behavior of some cooperative systems of functional differential equations. Can. Appl. Math. Q. 4, 421–444 (1996)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada

    Xiao-Qiang Zhao

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Zhao, XQ. (2017). Nonautonomous Semiflows. In: Dynamical Systems in Population Biology. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-56433-3_3

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