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

Received  November 2018 Revised  January 2019 Published  December 2019 Early access  July 2019

Fund Project: This work is partially supported by NSF of China No.11825106, 11771414 and Wu Wen-Tsun Key Laboratory.

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.

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:
[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. DanersJ. 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. DanersJ. 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. FerreroF. 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. OleskyM. 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. DanersJ. 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. DanersJ. 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. FerreroF. 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. OleskyM. 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 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
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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