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doi: 10.3934/era.2021026
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Instability and bifurcation of a cooperative system with periodic coefficients

Tian Hou 1,, , Yi Wang 1, and  Xizhuang Xie 1,2,

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

* Corresponding author: Tian Hou

Received  November 2020 Revised  January 2021 Early access March 2021

Fund Project: This work is supported by NSF of China No. 11825106, 11871231 and 11771414, CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of 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
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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C. Chicone, Ordinary Differential Equations with Applications, 2nd edition, Springer-Verlag, New York, 2006.  Google Scholar

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B. S. Goh, Stability in models of mutualism, Amer. Natur., 113 (1979), 261-275.  doi: 10.1086/283384.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[8]

K. Josić and R. Rosenbaum, Unstable solutions of nonautonomous linear differential equations, SIAM Review, 50 (2008), 570-584.  doi: 10.1137/060677057.  Google Scholar

[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.  Google Scholar

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R. M. May and A. R. McLean, Theoretical Ecology: Principles and Applications, 3nd edition, Oxford, UK, 2007.  Google Scholar

[11]

J. Mierczyński, Instability in linear cooperative systems of ordinary differential equations, SIAM Review, 59 (2017), 649-670.  doi: 10.1137/141001147.  Google Scholar

[12]

J. D. Murray, Mathematical Biology I: An Introduction, 3nd edition, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[14]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer-Verlag, New York, 2003.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. Chicone, Ordinary Differential Equations with Applications, 2nd edition, Springer-Verlag, New York, 2006.  Google Scholar

[5]

B. S. Goh, Stability in models of mutualism, Amer. Natur., 113 (1979), 261-275.  doi: 10.1086/283384.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[8]

K. Josić and R. Rosenbaum, Unstable solutions of nonautonomous linear differential equations, SIAM Review, 50 (2008), 570-584.  doi: 10.1137/060677057.  Google Scholar

[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.  Google Scholar

[10]

R. M. May and A. R. McLean, Theoretical Ecology: Principles and Applications, 3nd edition, Oxford, UK, 2007.  Google Scholar

[11]

J. Mierczyński, Instability in linear cooperative systems of ordinary differential equations, SIAM Review, 59 (2017), 649-670.  doi: 10.1137/141001147.  Google Scholar

[12]

J. D. Murray, Mathematical Biology I: An Introduction, 3nd edition, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[14]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer-Verlag, New York, 2003.  Google Scholar

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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