Stronger-than-Lyapunov notions of matrix stability, or how "flowers" help solve problems in mathematical ecology

Linear Algebra and Its Applications

Volumn 398, Issue 1-3, 2005, Pages 75-100

  Logofet, Dmitrii O ^a,b  

^a RUSSIAN ACADEMY OF SCIENCES   (Russian Federation)

^b MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

aD stable; Characterisation; Community matrix; Containment relation; D stable matrices; DaD stable; Diagonal quasi dominant; Diagram; Lotka Volterra equations; Quasi recessive; Recessive; Sign stable; Structurally open set; Tot ally stable; Volterra dissipative

Indexed keywords

ASYMPTOTIC STABILITY; CHARACTERIZATION; DIFFUSION; ECOLOGY; HEURISTIC METHODS; LYAPUNOV METHODS; PROBLEM SOLVING;

AD-STABLE; COMMUNITY MATRIX; D-STABLE MATRICES; DAD-STABLE; DIAGONAL QUASI-DOMINANT; LOTKA-VOLTERRA EQUATIONS; QUASI-RECESSIVE; RECESSIVE; TOTALLY STABLE; VOLTERRA DISSIPATIVE;

MATRIX ALGEBRA;

EID: 13544260500     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2003.04.001     Document Type: Article

References (88)

1. N.S. Abrosov, Analysis of the species structure in a single trophic level (Applied to a continuous polyculture of microorganisms), Doctoral Dissertation, Institute of Biophysics, Krasnoyarsk, 1979 (in Russian)
2. K.J. Arrow, and M. McManus A note on dynamical stability Econometrica 26 1958 448 454
3. E.A. Barbashin Introduction to Stability Theory 1967 Nauka Moscow 223 pp. (in Russian)
4. G.P Barker, A. Berman, and R.J. Plemmons Positive diagonal solutions to the Lyapunov equations Linear and Multilinear Algebra 5 1978 249 256
5. A. Berman, and D. Hershkowitz Characterization of acyclic D-stable matrices Linear Algebra Appl. 58 1984 17 32
6. A. Bhaya, E. Kaszkurewicz, Matrix Diagonal Stability in Systems and Computation, Birkhauser, Boston, 1999, 272 pp
7. B.E. Cain Real, 3 × 3, D-stable matrices J. Res. Nat. Bur. Standards USA 8OB 1976 75 77
8. B.E. Cain Inside the D-stable matrices Linear Algebra Appl. 56 1984 237 243
9. D. Carlson, B.N. Datta, and C.R. Johnson A semi-definite Lyapunov theorem and the characterization of tridiagonal D-stable matrices SIAM J. Algebra Disc. Methods 5 1982 346 350
10. C.E. Clark, and T.G. Hallam The community matrix in three species community models J. Math. Biol. 16 1982 25 31
11. G.W. Cross Three types of matrix stability Linear Algebra Appl. 20 1978 253 263
12. B.N. Datta Stability and D-stability Linear Algebra Appl. 21 1978 135 141
13. B.N. Datta Stability and inertia Linear Algebra Appl. 302-303 1999 563 600
14. D.L. DeAngelis, W. Post, C.C. Travis, Positive Feedbacks in Natural Systems, Biomathematics, vol.15, Springer-Verlag, Berlin, 305 pp., 1986
15. M. Fiedler, and V. Pták On matrices with non-positive off-diagonal elements and positive principal minors Czech. Math. J. 12 1962 382 400
16. F.R. Gantmacher, The Theory of Matrices, vol. 2, Chelsea, New York, 1960 (Chapter 16)
17. B.S. Goh Global stability in many-species systems Amer. Natur. 111 1977 135 143
18. B.S. Goh Robust stability concepts for ecosystem models E. Halfon Theoretical Systems Ecology 1979 Academic Press New York 467 487
19. F. Harary, R.Z. Norman, and D. Cartwright Structural Models: An Introduction to the Theory of Directed Graphs 1965 John Wiley New York (Chapter 7)
20. D.J. Hartfiel Concerning the interior of D-stable matrices Linear Algebra Appl. 30 1980 201 207
21. M.-J. Hernandes Dynamics of transitions between population interactions: a nonlinear interaction α-functions defined Proc. Royal Soc. London B 265 1998 1433 1440
22. D. Hershkowitz Recent directions in matrix stability Linear Algebra Appl. 171 1992 161 186
23. J. Hofbauer, and K. Sigmund The Theory of Evolution and Dynamical Systems 1988 Cambridge University Press Cambridge 341 pp
24. R.A. Horn, and C.R. Johnson Matrix Analysis 1990 Cambridge University Press Cambridge 561 pp
25. R.A. Horn, and C.R. Johnson Topics in Matrix Analysis 1991. Cambridge University Press Cambridge
26. C. Jeffries Qualitative stability and digraphs in model ecosystems Ecology 55 1974 1415 1419
27. C. Jeffries, V. Klee, and P. Van den Driessche When is a matrix sign stable? Can. J. Math. 29 1977 315 326
28. C. Jeffries, V. Klee, and P. Van den Driessche Qualitative stability of linear systems Linear Algebra Appl. 87 1987 1 48
29. C.R. Johnson Sufficient conditions for D-stability J. Econom. Theory 9 1974 53 62
30. C.R. Johnson Second, third and fourth order D-stability J. Res. Nat. Bur. Stand. USA 78B 1974 11 13
31. C.R. Johnson D-stability and real and complex quadratic forms Linear Algebra Appl. 9 1974 89 94
32. C.R. Johnson, D.D. Olesky, and P. Van den Driessche Stability of M-matrix products Linear and Multilinear Algebra 18 1985 67 76
33. G.V. Kanovei, D-stability of matrices and sign-constancy of polynomials, Kandidat-of-Science Dissertation, Mechanical-Mathematical Department of Moscow State University, 2000, 76 pp. (in Russian)
34. G.V. Kanovei, On some necessary conditions and sufficient conditions for D- and aD-stability in matrices, Deposited in VIN ITI (Moscow) 07.02.2000, 1282-BOO, 2000, 73 pp. (in Russian)
35. G.V. Kanovei, and D.O. Logofet D-stability of 4-by-4 matrices Comput. Math. Math. Phys. 38 1998 1369 1374
36. G.V. Kanovei, D.O. Logofet, Relations, properties, and transformations of D- and aD-stable matrices, Vestnik Moskovskogo Universiteta, Ser. 1, Mathematika, Mechanika 6 (2001) 51-54 (in Russian)
37. G.V. Kanovei, V.N. Nefedov, On some necessary conditions and sufficient conditions for a real polynomial in several variables to be positive in the positive orthant, Deposited in VIN ITI (Moscow) 07.02.2000, 1281-B00, 2000 (in Russian)
38. J.F.B.M. Kraaijevanger A characterization of Lyapunov diagonal stability using Hadamard products Linear Algebra Appl. 151 1991 245 254
39. R. Law, and R.D. Morton Permanence and the assembly of ecological communities Ecology 77 1996 762 775
40. W.W. Leontief The Structure of American Economy 1949 Academic Press London 181 pp
41. W.W. Leontief Input-Output Economics 1966 Oxford University Press New York 561 pp
42. D.O. Logofet On the hierarchy of subsets of stable matrices Soviet Math. Dokl. 34 1987 247 250
43. D.O. Logofet Do there exist diagonally stable matrices without dominating diagonal? Soviet Math. Dokl. 38 1989 113 115
44. D.O. Logofet Matrices and Graphs: Stability Problems in Mathematical Ecology 1993 CRC Press Boca Raton, FL 308 pp
45. D.O. Logofet Structurally open subsets of stable matrices Doklady Mathematics 54 1996 528 530
46. D.O. Logofet Svicobians of the compartment models and DaD-stability of the Svicobians: aggregating "0-dimensional" models of global biogeochemical cycles Ecol. Modell. 104 1997 29 47
47. D.O. Logofet Svicobians of the compartment models and DaD-stability of the Svicobians Doklady Mathematics 57 1998 371 373
48. D.O. Logofet Paths and cycles as tools for characterizing some classes of matrices Dokl. Math. 60 1999 46 49
49. D.O. Logofet, and N.B. Ul'janov Necessary and sufficient conditions for sign stability of matrices Soviet Math. Dokl. 25 1982 676 680
50. D.O. Logofet, and N.B. Ulianov Sign stability in model ecosystems: a complete class of sign-stable patterns Ecol. Modell. 16 1982 173 189
51. A.J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, MD, 1925 (reissued as Elements of Mathematical Biology, Dover, New York, 1956, 465 pp.)
52. A.J. Lotka, Elements of Physical Biology, Williams & Wilkins, Baltimore, MD, 1925 (reissued as Elements of Mathematical Biology, Dover, New York, 1956, 465 pp.)
53. M. Marcus, and A. Minc A Survey of Matrix Theory and Matrix Inequalities 1964. Allyn & Bacon Boston
54. R.M. May Stability in model ecosystems Quant. Ecol. 6 1971 18 56
55. R.M. May Qualitative stability in model ecosystems Ecology 54 1973 638 641
56. R.M. May, Stability and Complexity in Model Ecosystems (Monographs in Population Biology, vol. 6), 2nd ed., Princeton University Press, Princeton, NJ, 1974, 265 pp
57. J.S. Maybee, D.D. Olesky, P. Van den Driessche, and G. Wiener Matrices, digraphs, and determinants SIAM J. Matrix Anal. Appl. 10 1989 500 519
58. L. Metzler Stability of multiple markets: the Hicks conditions Econometrica 13 1945 277 292
59. P.J. Moylan Matrices with positive principal minors Linear Algebra Appl. 17 1977 53 58
60. J.D. Murray Mathematical Biology 1990. Springer Berlin
61. V.N. Nefedov Polynomial optimization problems Comput. Math. Math. Phys. 27 13 1987 13 21
62. V.N. Nefedov Estimation of error in convex polynomial optimization problems Comput. Math. Math. Phys. 30 11 1990 147 158
63. A.-M. Neutel, J.A.P. Heesterbeek, and P.C. DeRuiter Stability in real food webs: weak links in long loops Science 296 2002 1120 1123
64. H. Nikaido Convex Structures and Economic Theory 1968 Academic Press New York (Chapter 7)
65. E.P. Odum Basic Ecology 1983 Saunders College Publishing Philadelphia (Chapter 7)
66. A. Ostrowski Uber die Determinanten mit uberwiegender Hauptdiagonale Comment. Math. Helv. 10 1937 1937 1938
67. S.L. Pimm Food Webs 1982 Chapman & Hall London 219 pp
68. W.M. Post, and S.L. Pimm Community assembly and food web stability Math. Biosci. 64 1983 169 192
69. Yu.A. Pykh Equilibrium and Stability in Models of Population Dynamics 1983 Nauka Moscow 182 pp. (in Russian)
70. J.P. Quirk, and R. Ruppert Qualitative economics and the stability of equilibrium Rev. Econom. Studies 32 1965 311 326
71. J. Quirk, and R. Saposnick Introduction to General Equilibrium and Welfare Economics 1968 McGrow-Hill New York 221 pp
72. R. Redheffer Volterra multipliers II SIAM J. Algebra Disc. Methods 6 1985 612 623
73. P.A. Samuelson Foundations of Economic Analysis 1947 Harvard University Press Athenaeum, NY 1971.
74. F.M. Scudo, J.R. Ziegler, The Golden Age of Theoretical Ecology: 1923-1940. A Collection of Works by V. Volterra, V.A. Kostitzin, A. A. Lotka, A.N. Kolmogoroff, Lecture Notes in Biomathematics, vol. 22, Springer, Berlin, 1978, 490 pp
75. A. Seidenberg A new decision method for elementary algebra Ann Math. Ser. 2 60 1954 356 374
76. D.D. Šiljak Connective stability of complex ecosystems Nature (London) 249 1974 280.
77. D.D. Šiljak When is a complex system stable Math. Biosci. 25 1975 25 50
78. Yu.M. Svirezhev On some general properties of trophic networks Ecol. Modell. 99 1997 7 17
79. Yu.M. Svirezhev, and D.O. Logofet Ustoychivost bilogicheskih soobschestv 1978 Nauka Moscow 352 pp (in Russian; revised English translation: Stability of Biological Communities, Mir Publishers, Moscow, 1983, 319 pp.)
80. Y. Takeuchi Global Dynamic Properties of Lotka-Volterra Systems 1996. World Scientific Singapore
81. Y. Takeuchi, N. Adachi, and H. Tokumara Global stability of ecosystems of the generalized Volterra type Math. Biosci. 42 1978 119 136
82. A. Tarski A Decision Method for Elementary Algebra and Geometry 2nd ed. 1951. University of California Press Berkeley, Los Angeles
83. O. Taussky Programmation en mathematiques numeriques Colloq. Int. du Centre Natl. de la Rech. Sci., Paris, Sept. 1966 1966 75 88
84. Y. Togawa A geometrical study of the D-stability problem Linear Algebra Appl. 33 1980 133 151
85. A. Turing The chemical basis of morphogenesis Philosophical Transactions of the Royal Society B 237 1952 37 72
86. V. Volterra, Leçons sur la Théorie Mathématique de la Lutte pour la Vie. Gauthier-Villars, Paris, 1931, 214 pp
87. Yodzis, P., 1978. Competition for Space and the Structure of Ecological Communities, Lecture Notes in Biomathematics, vol. 25, Berlin, Springer, 191 pp
88. N.N. Zavalishin, and D.O. Logofet Modelling ecological systems according to a given 'stores-flows' diagram Matematicheskoe Modelirovanie 9 19 1997 3 17 (in Russian)