American Institute of Mathematical Sciences

Advanced
December  2019, 24(12): 6481-6494. doi: 10.3934/dcdsb.2019148

Non-oscillation principle for eventually competitive and cooperative systems

Lin Niu and  Yi Wang

School of Mathematical Science, University of Science and Technology of China, Hefei, Anhui 230026, China

Received  November 2018 Revised  January 2019 Published  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.

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

D. Angeli and E. Sontag, Monotone control systems, IEEE Trans. Autom. Control, 48 (2003), 1684-1698.  doi: 10.1109/TAC.2003.817920.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. W. Leontief, Input-output Economics, Oxford University Press on Demand, 1986. doi: 10.1038/scientificamerican1051-15.  Google Scholar

[16]

L. Markus, Structurally stable dynamical systems, Ann. of Math., 73 (1961), 1-19.  doi: 10.2307/1970280.  Google Scholar

[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.  Google Scholar

[18]

S. Newhouse, Lectures on dynamical systems, Progress in Math., 8 (1980), 1-114.   Google Scholar

[19]

L. Niu and Y. Wang, Singularly perturbed competitive systems and applications, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404.  doi: 10.1016/0040-9383(69)90024-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[27]

E. D. Sontag, Monotone and near-monotone biochemical networks, Syst. Synthetic Biol., 1 (2007), 59-87.   Google Scholar

[28]

A. Sootla and A. Mauroy, Operator-Theoretic Characterization of Eventually Monotone Systems, preprint, arXiv: 1510.01149. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

D. Angeli and E. Sontag, Monotone control systems, IEEE Trans. Autom. Control, 48 (2003), 1684-1698.  doi: 10.1109/TAC.2003.817920.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. W. Leontief, Input-output Economics, Oxford University Press on Demand, 1986. doi: 10.1038/scientificamerican1051-15.  Google Scholar

[16]

L. Markus, Structurally stable dynamical systems, Ann. of Math., 73 (1961), 1-19.  doi: 10.2307/1970280.  Google Scholar

[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.  Google Scholar

[18]

S. Newhouse, Lectures on dynamical systems, Progress in Math., 8 (1980), 1-114.   Google Scholar

[19]

L. Niu and Y. Wang, Singularly perturbed competitive systems and applications, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385-404.  doi: 10.1016/0040-9383(69)90024-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[27]

E. D. Sontag, Monotone and near-monotone biochemical networks, Syst. Synthetic Biol., 1 (2007), 59-87.   Google Scholar

[28]

A. Sootla and A. Mauroy, Operator-Theoretic Characterization of Eventually Monotone Systems, preprint, arXiv: 1510.01149. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control & Related Fields, 2020, 10 (1) : 27-45. doi: 10.3934/mcrf.2019028
[2]

D. P. Demuner, M. Federson, C. Gutierrez. The Poincaré-Bendixson Theorem on the Klein bottle for continuous vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 495-509. doi: 10.3934/dcds.2009.25.495
[3]

Jonas Deré. Periodic and eventually periodic points of affine infra-nilmanifold endomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5347-5368. doi: 10.3934/dcds.2016035
[4]

Hunseok Kang. Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 939-959. doi: 10.3934/dcds.2008.20.939
[5]

Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789
[6]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[7]

G. Conner, Christopher P. Grant, Mark H. Meilstrup. A Sharkovsky theorem for non-locally connected spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3485-3499. doi: 10.3934/dcds.2012.32.3485
[8]

Yu-Chung Tsao. Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits. Journal of Industrial & Management Optimization, 2017, 13 (1) : 329-347. doi: 10.3934/jimo.2016020
[9]

Rafael Obaya, Víctor M. Villarragut. Direct exponential ordering for neutral compartmental systems with non-autonomous $\mathbf{D}$-operator. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 185-207. doi: 10.3934/dcdsb.2013.18.185
[10]

G. Donald Allen. A dynamic model for competitive-cooperative species. Conference Publications, 1998, 1998 (Special) : 29-50. doi: 10.3934/proc.1998.1998.29
[11]

Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725
[12]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335
[13]

Jean Lerbet, Noël Challamel, François Nicot, Félix Darve. Kinematical structural stability. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 529-536. doi: 10.3934/dcdss.2016010
[14]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 47-66. doi: 10.3934/dcdss.2020003
[15]

Haiyan Wang, Carlos Castillo-Chavez. Spreading speeds and traveling waves for non-cooperative integro-difference systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2243-2266. doi: 10.3934/dcdsb.2012.17.2243
[16]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
[17]

Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
[18]

Alain Rapaport, Mario Veruete. A new proof of the competitive exclusion principle in the chemostat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3755-3764. doi: 10.3934/dcdsb.2018314
[19]

Roberta Fabbri, Russell Johnson, Carmen Núñez. On the Yakubovich frequency theorem for linear non-autonomous control processes. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 677-704. doi: 10.3934/dcds.2003.9.677
[20]

Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (74)
  • HTML views (175)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences