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Non-oscillation principle for eventually competitive and cooperative systems
School of Mathematical Science, University of Science and Technology of China, Hefei, Anhui 230026, China |
A nonlinear dynamical system is called eventually competitive (or cooperative) provided that it preserves a partial order in backward (or forward) time only after some reasonable initial transient. We present in this paper the Non-oscillation Principle for eventually competitive or cooperative systems, by which the non-ordering of (both $ \omega $- and $ \alpha $-) limit sets is obtained for such systems; and moreover, we established the Poincaré-Bendixson Theorem and structural stability for three-dimensional eventually competitive and cooperative systems.
References:
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C. Altafini, Representing externally positive systems through minimal eventually positive realizations, in Proc IEEE Conf Decision Control, (2015), 6385–6390.
doi: 10.1109/CDC.2015.7403225. |
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C. Altafini and G. Lini,
Predictable dynamics of opinion forming for networks with antagonistic interactions, IEEE Trans. Autom. Control, 60 (2015), 342-357.
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D. Angeli and E. Sontag,
Monotone control systems, IEEE Trans. Autom. Control, 48 (2003), 1684-1698.
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D. Daners,
Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator, Positivity, 18 (2014), 235-256.
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D. Daners, J. Glück and J. B. Kennedy,
Eventually positive semigroups of linear operators, J. Math. Anal. Appl., 433 (2016), 1561-1593.
doi: 10.1016/j.jmaa.2015.08.050. |
[6] |
D. Daners, J. Glück and J. B. Kennedy,
Eventually and asymptotically positive semigroups on Banach lattices, J. Diff. Eqns., 261 (2016), 2607-2649.
doi: 10.1016/j.jde.2016.05.007. |
[7] |
A. Ferrero, F. Gazzola and H.-C. Grunau,
Decay and eventual local positivity for biharmonic parabolic equations, Discrete Contin. Dyn. Syst., 21 (2008), 1129-1157.
doi: 10.3934/dcds.2008.21.1129. |
[8] |
F. Gazzola and H.-C. Grunau,
Eventual local positivity for a biharmonic heat equation in $\mathbb{R}^{n}$, Discrete Contin. Dyn. Syst.-S, 1 (2008), 83-87.
doi: 10.3934/dcdss.2008.1.83. |
[9] |
M. W. Hirsch,
Systems of differential equations which are competitive or cooperative Ⅰ: limit sets, SIAM J. Appl. Math., 13 (1982), 167-179.
doi: 10.1137/0513013. |
[10] |
M. W. Hirsch,
Systems of differential equations which are competitive or cooperative Ⅱ: convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439.
doi: 10.1137/0516030. |
[11] |
M. W. Hirsch,
Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53.
doi: 10.1515/crll.1988.383.1. |
[12] |
M. W. Hirsch,
Systems of differential equations which are competitive or cooperative Ⅳ: Structural stability in three dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234.
doi: 10.1137/0521067. |
[13] |
M. W. Hirsch and H. L. Smith, Monotone Systems, A Mini-review, Proceedings of the First Multidisciplinary Symposium on Positive Systems (POSTA 2003), Luca Benvenuti, Alberto De Santis and Lorenzo Farina (Eds.) Lecture Notes on Control and Information Sciences vol. 294, Springer-Verlag, Heidelberg, 2003.
doi: 10.1007/b79667. |
[14] |
M. W. Hirsch and H. L. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations, Vol. Ⅱ, 239–357, Elsevier B. V., Amsterdam, 2005. |
[15] |
W. W. Leontief, Input-output Economics, Oxford University Press on Demand, 1986.
doi: 10.1038/scientificamerican1051-15. |
[16] |
L. Markus,
Structurally stable dynamical systems, Ann. of Math., 73 (1961), 1-19.
doi: 10.2307/1970280. |
[17] |
S. Newhouse, Nondensity of axiom A(a) on $S^{2}$, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), 191–202 Amer. Math. Soc., Providence, R.I. 1970. |
[18] |
S. Newhouse,
Lectures on dynamical systems, Progress in Math., 8 (1980), 1-114.
|
[19] |
L. Niu and Y. Wang, Singularly perturbed competitive systems and applications, preprint. |
[20] |
D. Noutsos and M. J. Tsatsomeros,
Reachability and holdability of nonnegative states, SIAM J. Matrix Anal. Appl., 30 (2008), 700-712.
doi: 10.1137/070693850. |
[21] |
D. D. Olesky, M. J. Tsatsomeros and P. van den Driessche,
$M_v$-matrices: A generalization of M-matrices based on eventually nonnegative matrices, Electron. J. Linear Algebra, 18 (2009), 339-351.
doi: 10.13001/1081-3810.1317. |
[22] |
J. Palis,
On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404.
doi: 10.1016/0040-9383(69)90024-X. |
[23] |
S. Smale,
Morse inequalities for a dynamical system, Bull. Amer. Math. Soc., 66 (1960), 43-49.
doi: 10.1090/S0002-9904-1960-10386-2. |
[24] |
H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys and Monographs, 41, Amer. Math. Soc., Providence, Rhode Island 1995. |
[25] |
H. L. Smith,
Monotone dynamical systems: Reflections on new advances and applications, Discrete Contin. Dyn. Syst., 37 (2017), 485-504.
doi: 10.3934/dcds.2017020. |
[26] |
H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() ![]() |
[27] |
E. D. Sontag,
Monotone and near-monotone biochemical networks, Syst. Synthetic Biol., 1 (2007), 59-87.
|
[28] |
A. Sootla and A. Mauroy, Operator-Theoretic Characterization of Eventually Monotone Systems, preprint, arXiv: 1510.01149. |
[29] |
ER. Stern and H. Wolkowicz,
Exponential nonnegativity on the ice cream cone, SIAM J. Matrix Anal. Appl., 12 (1991), 160-165.
doi: 10.1137/0612012. |
[30] |
Y. Wang and J. Jiang,
The general properties of discrete-time competitive dynamical systems, J. Diff. Eqns, 176 (2001), 470-493.
doi: 10.1006/jdeq.2001.3989. |
[31] |
L. Wang and E. D. Sontag,
Singularly perturbed monotone systems and an application to double phosphorylation cycles, J. Nonlinear Sci., 18 (2008), 527-550.
doi: 10.1007/s00332-008-9021-2. |
show all references
References:
[1] |
C. Altafini, Representing externally positive systems through minimal eventually positive realizations, in Proc IEEE Conf Decision Control, (2015), 6385–6390.
doi: 10.1109/CDC.2015.7403225. |
[2] |
C. Altafini and G. Lini,
Predictable dynamics of opinion forming for networks with antagonistic interactions, IEEE Trans. Autom. Control, 60 (2015), 342-357.
doi: 10.1109/TAC.2014.2343371. |
[3] |
D. Angeli and E. Sontag,
Monotone control systems, IEEE Trans. Autom. Control, 48 (2003), 1684-1698.
doi: 10.1109/TAC.2003.817920. |
[4] |
D. Daners,
Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator, Positivity, 18 (2014), 235-256.
doi: 10.1007/s11117-013-0243-7. |
[5] |
D. Daners, J. Glück and J. B. Kennedy,
Eventually positive semigroups of linear operators, J. Math. Anal. Appl., 433 (2016), 1561-1593.
doi: 10.1016/j.jmaa.2015.08.050. |
[6] |
D. Daners, J. Glück and J. B. Kennedy,
Eventually and asymptotically positive semigroups on Banach lattices, J. Diff. Eqns., 261 (2016), 2607-2649.
doi: 10.1016/j.jde.2016.05.007. |
[7] |
A. Ferrero, F. Gazzola and H.-C. Grunau,
Decay and eventual local positivity for biharmonic parabolic equations, Discrete Contin. Dyn. Syst., 21 (2008), 1129-1157.
doi: 10.3934/dcds.2008.21.1129. |
[8] |
F. Gazzola and H.-C. Grunau,
Eventual local positivity for a biharmonic heat equation in $\mathbb{R}^{n}$, Discrete Contin. Dyn. Syst.-S, 1 (2008), 83-87.
doi: 10.3934/dcdss.2008.1.83. |
[9] |
M. W. Hirsch,
Systems of differential equations which are competitive or cooperative Ⅰ: limit sets, SIAM J. Appl. Math., 13 (1982), 167-179.
doi: 10.1137/0513013. |
[10] |
M. W. Hirsch,
Systems of differential equations which are competitive or cooperative Ⅱ: convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439.
doi: 10.1137/0516030. |
[11] |
M. W. Hirsch,
Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-53.
doi: 10.1515/crll.1988.383.1. |
[12] |
M. W. Hirsch,
Systems of differential equations which are competitive or cooperative Ⅳ: Structural stability in three dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234.
doi: 10.1137/0521067. |
[13] |
M. W. Hirsch and H. L. Smith, Monotone Systems, A Mini-review, Proceedings of the First Multidisciplinary Symposium on Positive Systems (POSTA 2003), Luca Benvenuti, Alberto De Santis and Lorenzo Farina (Eds.) Lecture Notes on Control and Information Sciences vol. 294, Springer-Verlag, Heidelberg, 2003.
doi: 10.1007/b79667. |
[14] |
M. W. Hirsch and H. L. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations, Vol. Ⅱ, 239–357, Elsevier B. V., Amsterdam, 2005. |
[15] |
W. W. Leontief, Input-output Economics, Oxford University Press on Demand, 1986.
doi: 10.1038/scientificamerican1051-15. |
[16] |
L. Markus,
Structurally stable dynamical systems, Ann. of Math., 73 (1961), 1-19.
doi: 10.2307/1970280. |
[17] |
S. Newhouse, Nondensity of axiom A(a) on $S^{2}$, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), 191–202 Amer. Math. Soc., Providence, R.I. 1970. |
[18] |
S. Newhouse,
Lectures on dynamical systems, Progress in Math., 8 (1980), 1-114.
|
[19] |
L. Niu and Y. Wang, Singularly perturbed competitive systems and applications, preprint. |
[20] |
D. Noutsos and M. J. Tsatsomeros,
Reachability and holdability of nonnegative states, SIAM J. Matrix Anal. Appl., 30 (2008), 700-712.
doi: 10.1137/070693850. |
[21] |
D. D. Olesky, M. J. Tsatsomeros and P. van den Driessche,
$M_v$-matrices: A generalization of M-matrices based on eventually nonnegative matrices, Electron. J. Linear Algebra, 18 (2009), 339-351.
doi: 10.13001/1081-3810.1317. |
[22] |
J. Palis,
On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404.
doi: 10.1016/0040-9383(69)90024-X. |
[23] |
S. Smale,
Morse inequalities for a dynamical system, Bull. Amer. Math. Soc., 66 (1960), 43-49.
doi: 10.1090/S0002-9904-1960-10386-2. |
[24] |
H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys and Monographs, 41, Amer. Math. Soc., Providence, Rhode Island 1995. |
[25] |
H. L. Smith,
Monotone dynamical systems: Reflections on new advances and applications, Discrete Contin. Dyn. Syst., 37 (2017), 485-504.
doi: 10.3934/dcds.2017020. |
[26] |
H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() ![]() |
[27] |
E. D. Sontag,
Monotone and near-monotone biochemical networks, Syst. Synthetic Biol., 1 (2007), 59-87.
|
[28] |
A. Sootla and A. Mauroy, Operator-Theoretic Characterization of Eventually Monotone Systems, preprint, arXiv: 1510.01149. |
[29] |
ER. Stern and H. Wolkowicz,
Exponential nonnegativity on the ice cream cone, SIAM J. Matrix Anal. Appl., 12 (1991), 160-165.
doi: 10.1137/0612012. |
[30] |
Y. Wang and J. Jiang,
The general properties of discrete-time competitive dynamical systems, J. Diff. Eqns, 176 (2001), 470-493.
doi: 10.1006/jdeq.2001.3989. |
[31] |
L. Wang and E. D. Sontag,
Singularly perturbed monotone systems and an application to double phosphorylation cycles, J. Nonlinear Sci., 18 (2008), 527-550.
doi: 10.1007/s00332-008-9021-2. |



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