# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019148
##### References:
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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. 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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

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##### 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
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
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 $n_{*} = 5$
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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