Figure 2.1.
The red region U (consists of U1 and U2) is the unstable region, the green region S is the asymptotically stable region, and the boundary between them are the Lyapunov stable region
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Tori can't collapse to an interval
doi: 10.3934/era.2021026
Instability and bifurcation of a cooperative system with periodic coefficients
| 1. | School of Mathematical Sciences University of Science and Technology of China, Hefei, Anhui 230026, China |
| 2. | School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China |
In this paper, we focus on a linear cooperative system with periodic coefficients proposed by Mierczyński [SIAM Review 59(2017), 649-670]. By introducing a switching strategy parameter $ \lambda $ in the periodic coefficients, the bifurcation of instability and the optimization of the switching strategy are investigated. The critical value of unstable branches is determined by appealing to the theory of monotone dynamical system. A bifurcation diagram is presented and numerical examples are given to illustrate the effectiveness of our theoretical result.
Keywords:
Instability,
bifurcation,
cooperative system,
periodic coefficients,
principal eigenvalue.
Mathematics Subject Classification:
Primary: 34C12, 34D99, 34A26, 34A40.
Citation:
Tian Hou, Yi Wang, Xizhuang Xie.
Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, doi: 10.3934/era.2021026
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References:
| [1] |
M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt,
On the stability of planar randomly switched systems, Ann. Appl. Probab, 24 (2014), 292-311.
doi: 10.1214/13-AAP924. |
| [2] |
D. H. Boucher, S. James and K. H. Keeler,
The ecology of mutualism, Annual Review of Ecology and Systematics, 13 (1982), 315-347.
doi: 10.1146/annurev.es.13.110182.001531. |
| [3] |
K. C. Chang, K. Pearson and T. Zhang,
Perron-Frobenius Theorem for Nonnegative Tensors, Commun. Math. Sci., 6 (2008), 507-520.
doi: 10.4310/CMS.2008.v6.n2.a12. |
| [4] |
C. Chicone, Ordinary Differential Equations with Applications, 2nd edition, Springer-Verlag, New York, 2006. |
| [5] |
B. S. Goh,
Stability in models of mutualism, Amer. Natur., 113 (1979), 261-275.
doi: 10.1086/283384. |
| [6] |
L. Gurvits, R. Shorten and O. Mason,
On the stability of switched positive linear systems, IEEE Trans. Automat. Control, 52 (2007), 1099-1103.
doi: 10.1109/TAC.2007.899057. |
| [7] |
M. W. Hirsch and H. L. Smith, Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Vol. II, 239–357, Elsevier B. V., Amsterdam, (2005). |
| [8] |
K. Josić and R. Rosenbaum,
Unstable solutions of nonautonomous linear differential equations, SIAM Review, 50 (2008), 570-584.
doi: 10.1137/060677057. |
| [9] |
T. Malik and H. L. Smith,
Does dormancy increase fitness of bacterial populations in time varying environments?, Bull. Math. Biol., 70 (2008), 1140-1162.
doi: 10.1007/s11538-008-9294-5. |
| [10] |
R. M. May and A. R. McLean, Theoretical Ecology: Principles and Applications, 3nd edition, Oxford, UK, 2007. |
| [11] |
J. Mierczyński,
Instability in linear cooperative systems of ordinary differential equations, SIAM Review, 59 (2017), 649-670.
doi: 10.1137/141001147. |
| [12] |
J. D. Murray, Mathematical Biology I: An Introduction, 3nd edition, Springer-Verlag, New York, 2002. |
| [13] |
H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. AMS, Providence, RI, 1995. |
| [14] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer-Verlag, New York, 2003. |
show all references
References:
| [1] |
M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt,
On the stability of planar randomly switched systems, Ann. Appl. Probab, 24 (2014), 292-311.
doi: 10.1214/13-AAP924. |
| [2] |
D. H. Boucher, S. James and K. H. Keeler,
The ecology of mutualism, Annual Review of Ecology and Systematics, 13 (1982), 315-347.
doi: 10.1146/annurev.es.13.110182.001531. |
| [3] |
K. C. Chang, K. Pearson and T. Zhang,
Perron-Frobenius Theorem for Nonnegative Tensors, Commun. Math. Sci., 6 (2008), 507-520.
doi: 10.4310/CMS.2008.v6.n2.a12. |
| [4] |
C. Chicone, Ordinary Differential Equations with Applications, 2nd edition, Springer-Verlag, New York, 2006. |
| [5] |
B. S. Goh,
Stability in models of mutualism, Amer. Natur., 113 (1979), 261-275.
doi: 10.1086/283384. |
| [6] |
L. Gurvits, R. Shorten and O. Mason,
On the stability of switched positive linear systems, IEEE Trans. Automat. Control, 52 (2007), 1099-1103.
doi: 10.1109/TAC.2007.899057. |
| [7] |
M. W. Hirsch and H. L. Smith, Monotone dynamical systems, Handbook of Differential Equations: Ordinary Differential Equations, Vol. II, 239–357, Elsevier B. V., Amsterdam, (2005). |
| [8] |
K. Josić and R. Rosenbaum,
Unstable solutions of nonautonomous linear differential equations, SIAM Review, 50 (2008), 570-584.
doi: 10.1137/060677057. |
| [9] |
T. Malik and H. L. Smith,
Does dormancy increase fitness of bacterial populations in time varying environments?, Bull. Math. Biol., 70 (2008), 1140-1162.
doi: 10.1007/s11538-008-9294-5. |
| [10] |
R. M. May and A. R. McLean, Theoretical Ecology: Principles and Applications, 3nd edition, Oxford, UK, 2007. |
| [11] |
J. Mierczyński,
Instability in linear cooperative systems of ordinary differential equations, SIAM Review, 59 (2017), 649-670.
doi: 10.1137/141001147. |
| [12] |
J. D. Murray, Mathematical Biology I: An Introduction, 3nd edition, Springer-Verlag, New York, 2002. |
| [13] |
H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. AMS, Providence, RI, 1995. |
| [14] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer-Verlag, New York, 2003. |
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