Figure 1.
The intervals $ I_{n} = [t_{0}+nA, b+t_{*}+nA] = [t_n, t_n+E] $ for $ n\in \mathbb{N} $ , where $ E = \vert I_n\vert = \vert I_0\vert $ and $ [a, b] $ is the rising interval
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December
2019, 24(12): 6481-6494.
doi: 10.3934/dcdsb.2019148
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.
Keywords:
Eventually cooperative,
eventually competitive,
non-oscillation principle,
non-ordering,
Poincaré-Bendixson theorem,
structural stability.
Mathematics Subject Classification:
Primary: 34C12, 37C65; Secondary: 58F09, 92A15.
Citation:
Lin Niu, Yi Wang. Non-oscillation principle for eventually competitive and cooperative systems. Discrete and Continuous Dynamical Systems - B,
2019, 24
(12)
: 6481-6494.
doi: 10.3934/dcdsb.2019148
![]()
References:
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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.
doi: 10.1109/TAC.2003.817920. |
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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. |
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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. |
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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. |
Figure 2.
Case (Ⅰ) with $ n_{0}D_{0}\in [A-E, A] $ , where $ c_{n_{0}}\in [t_{n_{0}(k_{0}+1)-1}, t_{n_{0}(k_{0}+1)-1}+E] = I_{n_0(k_0+1)-1} $ with $ n_{0} = 3, k_{0} = 1, l_{*} = 3 $ and $ n_{*} = 5 $
Figure 3.
Case (Ⅱ) with $ n_{0}D_{0}>A $ , where $ c_{n_{0}} = t_{n_0(k_0+1)-1}-D_1\in (t_{n_{0}(k_{0}+1)-2}+E, t_{n_{0}(k_{0}+1)-1}) $ and $ c_{n\cdot n_0} = t_{n\cdot[n_0(k_0+1)-1]}-nD_1 $ , with $ n_{0} = n = 2 $ , $ k_{0} = 1 $ and $ D_1 = n_0D_0-A $
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