Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Competitive processes and comparison differential systems


Author: G. S. Ladde
Journal: Trans. Amer. Math. Soc. 221 (1976), 391-402
MSC: Primary 34D20; Secondary 92A05, 90A99
DOI: https://doi.org/10.1090/S0002-9947-1976-0407401-7
MathSciNet review: 0407401
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions are given for stability and nonnegativity of solutions of a system of differential equations, in particular, of comparison differential equations. Finally, it has been shown that the comparison differential equations represent the mathematical models for competitive processes in biological, physical and social sciences.


References [Enhancements On Off] (What's this?)

  • [1] R. Bellman, Introduction to matrix analysis, McGraw-Hill, New York, 1960. MR 23 #A153. MR 0122820 (23:A153)
  • [2] -, Topics in pharmacokinetics. I: Concentration dependent rates, Math. Biosci. 6 (1970), 13-17.
  • [3] -, Topics in pharmacokinetics. IV: Approximation in process space and fitting by sums of exponentials, Math. Biosci. 14 (1972), 45-47.
  • [4] S. W. Benson, The foundations of chemical kinetics, McGraw-Hill, New York, 1960.
  • [5] F. Brauer, Some refinements of Lyapunov's second method, Canad. J. Math. 17 (1965), 811-819. MR 31 #3670. MR 0179422 (31:3670)
  • [6] H. G. Bray and K. White, Kinetics and thermodynamics in biochemistry, 2nd ed., J. & A. Churchill, London, 1966.
  • [7] R. Grimmer, Stability of a scalar differential equation (to appear). MR 0288369 (44:5567)
  • [8] L. J. T. Grujić and D. D. Šiljak, Asymptotic stability and instability of large-scale systems, IEEE Trans. AC-18 (1973), 636-645. MR 0414186 (54:2290)
  • [9] T. G. Hallam and J. W. Heidel, Structure of the solution set of some first order differential equations of comparison type, Trans. Amer. Math. Soc. 160 (1971), 501-512. MR 43 #7709. MR 0281995 (43:7709)
  • [10] J. R. Hicks, Value and capital, 2nd ed., Oxford Univ. Press, Oxford, 1946.
  • [11] L. A. Metzler, Stability of multiple markets: The Hicks conditions, Econometrica 13 (1945), 277-292. MR 7, 465. MR 0015764 (7:465i)
  • [12] K. K. Krasovskiĭ, Certain problems in the theory of stability of motion, Fizmatgiz, Moscow, 1959; English transl., Stability of motion. Application of Ljapunov's second method to differential systems and equations with delay, Stanford Univ. Press, Stanford, Calif., 1963. MR 21 #5047; 26 #5258.
  • [13] G. S. Ladde, Systems of differential inequalities and stochastic differential equations. II, J. Mathematical Phys. 16 (1975), 894-900. MR 0428442 (55:1463)
  • [14] -, Variational comparison theorem and perturbations of nonlinear systems, Proc. Amer. Math. Soc. 52 (1975), 181-187. MR 0372351 (51:8567)
  • [15] V. Lakshmikantham and S. Leela, Differential and integral inequalities, theory and applications. Vol. I, Academic Press, New York, 1969.
  • [16] P. K. Newman, Some notes on stability conditions, Rev. Econ. Studies 72 (1959), 1-9.
  • [17] D. D. Šiljak, Stability of large-scale systems under structural perturbations, IEEE Trans. Systems, Man and Cybernet. SMC-2 (1972), 657-663. MR 47 #4669. MR 0316121 (47:4669)
  • [18] -, Connective stability of competitive equilibrium, Automatika 11 (1975), 389-400. MR 0419034 (54:7067)
  • [19] K. J. Arrow and F. H. Hahn, General competitive analysis, Holden-Day, San Francisco, Calif., 1971. MR 0439057 (55:11958)
  • [20] G. S. Ladde, Cellular systems. I: Stability of chemical systems, Math. Biosci. (in press). MR 0682070 (58:33092)
  • [21] -, Cellular systems. II: Stability of compartmental systems, Math. Biosci. (in press). MR 0681522 (58:33067)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34D20, 92A05, 90A99

Retrieve articles in all journals with MSC: 34D20, 92A05, 90A99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0407401-7
Keywords: Chemical kinetics, comparison differential systems, compartment, competitive processes, concentration, economic systems, gross substitutability, Hicks matrix, interpretivity, maximal solution, Metzler matrix, nonnegativity, pharmacokinetics, quasimonotone property, reactant, stability
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society