Skip to main content
SpringerLink
Log in

Monotone Dynamics

  • Chapter
  • First Online:
Book cover Dynamical Systems in Population Biology

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

  • 2388 Accesses

Abstract

As we illustrated in the Preface, some population models can generate continuous- or discrete-time dynamical systems with monotonicity: Ordered initial states lead to ordered subsequent states. This chapter is aimed at monotone dynamics. We are primarily interested in some global results that may be effectively applied to both discrete-time and periodic biological systems. In Section 2.1 we prove the existence and global attractivity of an order interval defined by two fixed points, and a theorem on fixed points and connecting orbits for continuous and monotone maps on an ordered Banach space E.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. N.P. CÁC, J.A. Gatica, Fixed point theorems for mappings in ordered Banach spaces. J. Math. Anal. Appl. 71, 547–557 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. E.N. Dancer, Some remarks on a boundedness assumption for monotone dynamical systems. Proc. Am. Math. Soc. 126, 801–807 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. E.N. Dancer, P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems. J. Reine Angew. Math. 419, 125–139 (1991)

    MathSciNet  MATH  Google Scholar 

  4. E.N. Dancer, P. Hess, Stable subharmonic solutions in periodic reaction–diffusion equations. J. Differ. Equ. 108, 190–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Daners, Qualitative behavior of an epidemics model. Differ. Integral Equ. 5, 1017–1032 (1992)

    MathSciNet  MATH  Google Scholar 

  6. K. Deimling, Nonlinear Functional Analysis (Springer, New York, 1985)

    Book  MATH  Google Scholar 

  7. W. Ding, X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media. SIAM J. Math. Anal. 47, 855–896 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Fang, X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness. SIAM J. Math. Anal. 46, 3678–3704 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Fang, X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications. J. Eur. Math. Soc. 17, 2243–2288 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. H.I. Freedman, X.-Q. Zhao, Global asymptotics in some quasimonotone reaction–diffusion systems with delay. J. Differ. Equ. 137, 340–362 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. H.I. Freedman, X.-Q. Zhao, Global attractivity in a nonlocal reaction–diffusion model, in Differential Equations with Applications to Biology. Fields Institute Communications, vol. 21 (American Mathematical Society, Providence, RI, 1999), pp. 175–186

    Google Scholar 

  12. K.P. Hadeler, J. Tomiuk, Periodic solutions of difference–differential equations. Arch. Ration. Mech. Anal. 65, 87–95 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  14. X. He, W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity I. Commun. Pure Appl. Math. 69, 981–1014 (2016)

    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, A.C. Lazer, On an abstract competition model and applications. Nonlinear Anal. TMA 16, 917–940 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. M.W. Hirsch, The dynamical systems approach to differential equations. Bull. Am. Math. Soc. 11, 1–64 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. M.W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species. Nonlinearity 1, 51–71 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. M.W. Hirsch, Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383, 1–53 (1988)

    MathSciNet  MATH  Google Scholar 

  21. 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 

  22. S.-B. Hsu, X.-Q. Zhao, A Lotka–Volterra competition model with seasonal succession. J. Math. Biol. 64, 109–130 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. S.-B. Hsu, H.L. Smith, P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces. Trans. Am. Math. Soc. 348, 4083–4094 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Jiang, Sublinear discrete-time order-preserving dynamical systems. Math. Proc. Camb. Philos. Soc. 119, 561–574 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Jiang, S. Yu, Stable cycles for attractors of strongly monotone discrete-time dynamical systems. J. Math. Anal. Appl. 202, 349–362 (1996)

    Article  MathSciNet  Google Scholar 

  26. J. Jiang, X.-Q. Zhao, Convergence in monotone and uniformly stable skew-product semiflows with applications. J. Reine Angew. Math. 589, 21–55 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Jiang, X. Liang, X.-Q. Zhao, Saddle point behavior for monotone semiflows and reaction–diffusion models. J. Differ. Equ. 203, 313–330 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. M.A. Krasnoselskii, Translation Along Trajectories of Differential Equations. Translations of Mathematical Monographs, vol. 19 (American Mathematical Society, Providence, RI, 1968)

    Google Scholar 

  29. U. Krause, R.D. Nussbaum, A limit set trichotomy for self-mappings of normal cones in Banach spaces. Nonlinear Anal. 20, 855–870 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. K.-Y. Lam, D. Munther, A remark on the global dynamics of competitive systems on ordered Banach spaces. Proc. Am. Math. Soc. 144, 1153–1159 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. B. Li, H.F. Weinberger, M.A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. X. Liang, J. Jiang, The dynamical behavior of type-K competitive Kolmogorov systems and its application to 3-dimensional type-K competitive Lotka-Volterra systems. Nonliearity 16,785–801 (2003)

    Article  MATH  Google Scholar 

  33. X. Liang, X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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 

  35. X. Liang, Y. Yi, X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems. J. Differ. Equ. 231, 57–77 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Y. Lou, D. Xiao, P. Zhou, Qualitative analysis for a Lotka–Volterra competition system in advective homogeneous environment. Discrete Contin. Dyn. Syst. 36, 953–969 (2016)

    MathSciNet  MATH  Google Scholar 

  37. R. Lui, Biological growth and spread modeled by systems of recursions I. Mathematical theory. Math. Biosci. 93, 269–295 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. R.H. Martin, H.L. Smith, Abstract functional differential equations and reaction-diffusion systems. Trans. Am. Math. Soc. 321, 1–44 (1990)

    MathSciNet  MATH  Google Scholar 

  39. R.H. Martin, H.L. Smith, Reaction–diffusion systems with time delays: monotonicity, invariance, comparison and convergence. J. Reine Angew. Math. 413, 1–35 (1991)

    MathSciNet  MATH  Google Scholar 

  40. R.D. Nussbaum, Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theorem, in Fixed Point Theory. Lecture Notes in Mathematics, vol. 886 (Springer, New York/Berlin, 1981), pp. 309–331

    Google Scholar 

  41. R.D. Nussbaum, Hilbert’s Projective Metric and Iterated Nonlinear Maps. Memoirs of the American Mathematical Society, No. 391 (American Mathematical Society, Providence, RI, 1988)

    Google Scholar 

  42. T. Ogiwara, H. Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry. Discrete Cont. Dyn. Syst. 5, 1–34 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  44. P. Polác̆ik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations. J. Differ. Equ. 79, 89–110 (1989)

    Google Scholar 

  45. P. Polác̆ik, I. Teres̆c̆ák, Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems. Arch. Ration. Mech. Anal. 116, 339–360 (1991)

    Google Scholar 

  46. P. Polác̆ik, I. Teres̆c̆ák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations. J. Dyn. Differ. Equ. 5, 279–303 (1993)

    Google Scholar 

  47. R.J. Sacker, G.R. Sell, Lifting properties in skew-product flows with applications to differential equations, in Memoirs of the American Mathematical Society, No. 190, vol. 11 (American Mathematical Society, Providence, RI, 1977)

    Google Scholar 

  48. 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 

  49. M. Shub, Global Stability of Dynamical Systems (Springer, New-York/Berlin, 1987)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. H.L. Smith, Invariant curves for mappings. SIAM J. Math. Anal. 17, 1053–1067 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  52. H.L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem. SIAM J. Appl. Math. 46, 856–874 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  53. H.L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps. J. Differ. Equ. 64, 165–194 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  54. H.L. Smith, Periodic solutions of periodic competitive and cooperative systems. SIAM J. Math. Anal. 17, 1289–1318 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  55. 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 

  56. H.L. Smith, H.R. Thieme, Monotone semiflows in scalar non-quasi-monotone functional differential equations. J. Math. Anal. Appl. 150, 289–306 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  57. H.L. Smith, H.R. Thieme, Convergence for strongly order-preserving semiflows. SIAM J. Math. Anal. 22, 1081–1101 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  58. H.L. Smith, H.R. Thieme, Strongly order preserving semiflows generated by functional differential equations. J. Differ. Equ. 93, 332–363 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  59. H.L. Smith, H.R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces. J. Differ. Equ. 176, 195–222 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  60. 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 

  61. P. Takác̆, Domains of attraction of generic ω-limit sets for strongly monotone semiflows. Z. Anal. Anwend. 10, 275–317 (1991)

    Google Scholar 

  62. P. Takác̆, Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups. J. Reine Angew. Math. 423, 101–173 (1992)

    Google Scholar 

  63. P. Takác̆, Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems. Proc. Am. Math. Soc. 115, 691–698 (1992)

    Google Scholar 

  64. P. Takác̆, Convergence in the part metric for discrete dynamical systems in ordered topological cones. Nonlinear Anal. TMA 26, 1753–1777 (1996)

    Google Scholar 

  65. P. Takác̆, Discrete monotone dynamics and time-periodic competition between two species. Differ. Integral Equ. 10, 547–576 (1997)

    Google Scholar 

  66. I. Teres̆c̆ák, Dynamics of C 1-smooth strongly monotone discrete-time dynamical systems (preprint)

    Google Scholar 

  67. A.C. Thompson, On certain contraction mappings in a partially ordered vector space. Proc. Am. Math. Soc. 14, 438–443 (1963)

    MathSciNet  MATH  Google Scholar 

  68. Y. Wang, Convergence to periodic solutions in periodic quasimonotone reaction–diffusion systems. J. Math. Anal. Appl. 268, 25–40 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  69. Y. Wang, J. Jiang, The general properties of discrete-time competitive dynamical systems. J. Differ. Equ. 176, 470–493 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  70. Y. Wang, J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems. J. Differ. Equ. 186, 611–632 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  71. Y. Wang, X.-Q. Zhao, Convergence in monotone and subhomogeneous discrete dynamical systems on product Banach spaces. Bull. Lond. Math. Soc. 35, 681–688 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  72. H.F. Weinberger, Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  73. H.F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45, 511–548 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  74. J. Wu, Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences, vol. 119 (Springer, New York, 1996)

    Google Scholar 

  75. J. Wu, X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations. J. Differ. Equ. 186, 470–484 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  76. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  78. 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 

  79. X.-Q. Zhao, P. Zhou, On a Lotka-Volterra competition model: the effects of advection and spatial variation. Calc. Var. Partial Differ. Equ. 55, Art. 73, 25 pp. (2016)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Xiao-Qiang Zhao

Authors
  1. Xiao-Qiang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Zhao, XQ. (2017). Monotone Dynamics. In: Dynamical Systems in Population Biology. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-56433-3_2

Download citation

Publish with us

Policies and ethics

Search

Navigation