Taiwanese Journal of Mathematics

A SURVEY OF CONSTRUCTING LYAPUNOV FUNCTIONS FOR MATHEMATICAL MODELS IN POPULATION BIOLOGY

Sze-Bi Hsu

Full-text: Open access

Abstract

In this paper we survey the construction of Lyapunov functions (or functionals) for various ecological models which take the form of ODE systems (or Reaction-Diffusion PDE systems). First we consider the resources-consumers type ecological models which study the competition of $n$ microorgansims for a single limiting resource or two complementary resources in the chemostat. Next we consider the Gause-type predator-prey systems and the Lesile-type predator-prey systems. From the Lyapunov functions of the predator-prey system we construct new Lyapunov functions for three-level food chain models and one prey two predators models. Suppose a Lyapunov function is known for an ecological model which takes the form of an ODE system. Then we construct a Lyapunov functional for the corresponding reaction-diffusion PDE systems. Open problems are indicated when there is gap in the theory.

Article information

Source
Taiwanese J. Math., Volume 9, Number 2 (2005), 151-173.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407791

Digital Object Identifier
doi:10.11650/twjm/1500407791

Mathematical Reviews number (MathSciNet)
MR2142410

Zentralblatt MATH identifier
1087.34031

Subjects
Primary: 34D05: Asymptotic properties 34D20: Stability 92D25: Population dynamics (general)

Keywords
chemostat Lyapunov functions predator-prey systems consumers-resources competition models diffusion-reaction systems

Citation

Hsu, Sze-Bi. A SURVEY OF CONSTRUCTING LYAPUNOV FUNCTIONS FOR MATHEMATICAL MODELS IN POPULATION BIOLOGY. Taiwanese J. Math. 9 (2005), no. 2, 151--173. doi:10.11650/twjm/1500407791. https://projecteuclid.org/euclid.twjm/1500407791


Export citation

References

  • A. Ardito, P. Ricciardi, Lyapunov functions for a generalized Gause-type model, J. Math. Biology, 33 (1995), 816-828.
  • R. A. Armstrong and R. McGehee, Competitive exclusion American Naturalist, 115 (1980), 151.
  • M. Braun, Differential Equations and Their Application, Springer-Verlag, 1975.
  • G. Butler, S. B. Hsu and P. Waltman, Coexistence of competing predator in a chemostat, J. Math. Biology, 17 (1983), 133-151.
  • K. Cheng, S. B. Hsu and S. S. Lin, Some results on global stability of a predator-prey system, J. Math. Biology, 12 (1981), 115-126
  • C. W. Chi, S. B. Hsu and L. I. Wu, On the asymmetric May-Leonard model of three competing species SIAM, J. Appl. Math., 58(1) (1998), 211-226.
  • C. H. Chiu, Lyapunov functions for the global stability of competing predators, J. Math. Anal. Appl., 230(1) (1999), 232-241.
  • C. H. Chiu and S. B. Hsu, Extinction of top predator in a three-level food-chain model, J. Math. Biology, 37 (1998), 372-380.
  • Y. Du and S. B. Hsu, A diffusive predator-prey model: in heterogeneous environment, J. Differential Equations, 203 (2004), no. 2, 331-364.
  • H. I. Freedman, Graphical stability, enrichment, and pestcontrol by a natural enemy, Math. Biosci., 31 (1976), 207-225.
  • B. S. Goh, Global stability in many species system, Amer. Naturalist, 111 (1977), 135-143.
  • J. Hale, Ordinary Differential Equations, Wiley-Interscience, 1969.
  • J. Hale, Asymptotic Behavior of Dissipative Systems, AMS Monographs Number 25, 1988.
  • A. Hasting, Global stability in Lotka-Volterra systems with diffusion, J. Math. Biology, 6 (1978), 163-168.
  • M. Hirsch, Systems of differential equations which are competitive or cooperative I: limitsets, SIAM J. Math. Analysis, 13 (1982), 167-179.
  • S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
  • S. B. Hsu, On global stability of a predator-prey systems, Math. Biosci., 39 (1978), 1-10.
  • S. B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.
  • S. B. Hsu, S. P. Hubbell and P. Waltman, Competing predator, SIAM J. Appl. Math., 135(4) (1978), 617-625.
  • S. B. Hsu, K. S. Cheng and S. P. Hubbell, Exploitative competition of microorganisms for two complementary nutrients in continuous culture, SIAM J. Appl. Math., 41 (1981), 422-444.
  • R. T. Gong and S. B. Hsu, Stability analysis for a class of diffusive coupled system with application to population biology, Can. Appl. Math. Quart., 8 (2000), 79-96.
  • S. B. Hsu and T. W. Hwang, Global stability for a class of predator-prey system, SIAM J. Appl. Math., 55(3) (1995), 763-783.
  • S. B. Hsu and T. W. Hwang, Hopf bifurcation analysis for a predator-prey system of Holling and Lesile type, Taiwanese J. Math., 1 (1999), 35-53.
  • S. B. Hsu and P. Waltman, Competition in chemostat when one competitor produces toxin, Japan J. Industrial and Applied Math., 15(3) (1998), 471-490.
  • S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM J. Appl. Math., 52(2) (1992), 528-540.
  • S. B. Hsu, P. Waltman and G. Wolkowicz, Global analysis of a model of plasmid-bearing, plasmid-free competition in a chemostat, J. Maht. Biol., 32 (1994), 731-742.
  • Z. Lu and G. Wolkowicz, Global dynamics of a mathematical model of competition in the chemostat: general response functions and different death rates, SIAM J. Appl. Math., 52(1) (1992), 222-233.
  • B. Li, Global asymptotic behavior of the chemostat: general response function and different death rates, SIAM J. Appl. Math., 59 (1999), 411-422.
  • B. Li and Hal Smith, How many species can two essential resources support? SIAM J. Appl. Math., 62(1) (2001), 333-366.
  • H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equation, J. Math. Kyoto Univ., 18 (1978), 224-243.
  • J. Mallet-Paret and Hal Smith, The Poincar$\acute{\rm e}$-Bendixson theorem for monotone cyclic feedback systems, J. Dynam. Differential Equations, 2 (1990), 367-421.
  • H. Smith, Monotone Dynamical Systems, An introduction to the theorey of competitive and cooperative systems, AMS Monographs Vol. 41, 1995.
  • H. Smith and P. Waltman, Theory of Chemostat, Cambridge Press, 1995.
  • F. J. Weissing and J. Huisman, Biodiversity of plankton by species oscillations and chaos, Nature, 402 (1999), 407-410.
  • N. Alikakos, An application of the invariance principle to reaction-diffusion equations. J. Diff. Eqns., 33 (1979), 201-225.
  • N. Alikakos, A Liapunov functional for a class of reaction-diffusion systems. Modelling and Differential Equation in Biology, (Conf. Southern Illinois Univ., Carbondale, Ill. 1978), pp. 153-170, Lecture Notes in Pure and Applied Math., 58, Dekker, New York, 1980.