AIMS Mathematics, 2020, 5(2): 1532-1549. doi: 10.3934/math.2020105

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Dynamical behaviour of fractional-order atmosphere-soil-land plant carbon cycle system

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

The terrestrial carbon cycle is the most important constitution and plays a prominent role in the global carbon cycle. This paper investigates the dynamical behaviours and mathematical properties of a time fractional-order atmosphere-soil-land plant carbon cycle system. We give a sufficient condition for existence and uniqueness of the solution, and obtain the conditions for local asymptotically stable of the equilibrium points by using fractional Routh-Hurwitz stability conditions. Furthermore, we introduce a discretization process to discretize this fractional-order system, and study the necessary and sufficient conditions of stability of the discretization system. It shows that the stability of the discretization system is impacted by the system’s fractional parameter. Numerical simulations show the richer dynamical behaviours of the fractional-order system and verify the theoretical results.
  Article Metrics


1. M. Reichstein, M. Bahn, P. Ciais, et al. Climate extremes and the carbon cycle, Nature, 500 (2013), 287-295.    

2. S. Honjo, T. I. Eglinton, C. D. Taylor, et al. Understanding the Role of the Biological Pump in the Global Carbon Cycle: An Imperative for Ocean Science, Oceanography, 27 (2014), 10-16.

3. L. G. Anderson, R. W. Macdonald, Observing the Arctic Ocean carbon cycle in a changing environment, Polar Res., 34 (2015), 26891.

4. A. Ito, Modelling of carbon cycle and fire regime in an east Siberian larch forest, Ecol. Model., 187 (2005), 121-139.    

5. R. Causebrook, R. E. Dunsmore, Geological Storage of Carbon Dioxide, Latomus, 123 (2004), 218-221.

6. H. Zhang and K. N. Xu, Impact of Environmental Regulation and Technical Progress on Industrial Carbon Productivity: An Approach Based on Proxy Measure, Sustainability, 8 (2016), 819.

7. B. Pang, C. L. Fang and H. M. Liu, Quantitative Study on the Dynamic Mechanism of Smart LowCarbon City Development in China, Sustainability, 8 (2016), 507.

8. L. L. Golubyatnikov and Y. M. Svirezhev, Life-cycle model of terrestrial carbon exchange, Ecol. Model., 213 (2008), 202-208.    

9. I. G. Enting, Laplace transform analysis of the carbon cycle, Environ. Model. Softw., 22 (2007), 1488-1497.    

10. M. Marchi, F. M. Jorgensen, N. Marchettinia, et al. Modeling the carbon cycle of Siena Province (Tuscany, central Italy), Ecol. Model., 225 (2012), 40-60.    

11. Y. Pan, R. A. Birdsey, J. Fang, et al. A large and persistent carbon sink in the world's forests, Science, 333 (2011), 988-993.    

12. S. Piao, J. Fang, P. Ciais, et al. The carbon balance of terrestrial ecosystems in China, Nature, 458 (2009), 1009-1014.    

13. M. Fu, L. Tian, G. Dong, et al. Modeling on Regional Atmosphere-Soil-Land plant Carbon Cycle Dynamic System, Sustainability, 8 (2016), 303.

14. P. Arena, R. Caponetto, L. Fortuna, et al. Nonlinear noninteger order circuits and systems: An Introduction, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, World Sci. Publishing, River Edge, NJ, 2000.

15. S. Song, B. Zhang, X. Song, et al. Neuro-fuzzy-based adaptive dynamic surface control for fractional-order nonlinear strict-feedback systems with input constraint, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019.

16. X. Wang, Z. Wang, X. Zhu, et al. Stability and Hopf Bifurcation of Fractional-Order ComplexValued Neural Networks With Time-Delay, IEEE Access, 7 (2019), 158798-158807.    

17. S. Song, J. H. Park, B. Zhang, et al. Adaptive hybrid fuzzy output feedback control for fractionalorder nonlinear systems with time-varying delays and input saturation, Appl. Math. Comput., 364 (2020), 124662.

18. R. Hilfer, Applications of fractional calculus in physics, World Sci. Publishing, River Edge, NJ, 2000.

19. E. Ahmed and A. S. Elgazzar, On fractional order differential equations model for nonlocal epidemics, Physica A, 379 (2007), 607-614.    

20. E. S. A. Shahri, A. Alfi and J. A. T. Machado, Stabilization of fractional-order systems subject to saturation element using fractional dynamic output feedback sliding mode control, J. Comput. Nonlinear Dyn., 12 (2017), 31014.

21. A. M. A. El-Sayed, A. E. M. El-Mesiry, H. A. A. El-Saka, On the fractional-order logistic equation, Appl. Math. Lett., 20 (2007), 817-823.    

22. W. M. Ahmad, R. El-Khazali, Fractional-order dynamical models of love, Chaos Solitons Fractals, 33 (2007), 1367-1375.    

23. Z. J. Chen and W. J. Liu, Dynamical behavior of fractional-order energy-saving and emissionreduction system and its discretization, Nat. Resour. Model., 32 (2019), e12203.

24. W. J. Liu and K. W. Chen, Chaotic behavior in a new fractional-order love triangle system with competition, J. Appl. Anal. Comput., 5 (2015), 103-113.

25. Y. D. Liu and W. J. Liu, Chaotic behavior analysis and control of a toxin producing phytoplankton and zooplankton system based on linear feedback, Filomat, 32 (2018), 3779-3789.    

26. D. Q. Chen and W. J. Liu, Chaotic behavior and its control in a fractional-order energy demandsupply system, J. Comput. Nonlin. Dyn., 11 (2016), 061010.

27. P. Song, H. Y. Zhao and X. B. Zhang, Dynamic analysis of a fractional order delayed predator-prey system with harvesting, Theor. Biosci., 135 (2016), 59-72.    

28. M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astron. Soc., 13 (1967), 529-539.    

29. A. E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der Pol-Duffing circuit, Commun. Nonlinear Sci., 16 (2011), 975-986.    

30. D. Matignon, Stability results for fractional differential equations with applications to control Processing, CESA'96 IMACS Multiconference: Computational Engineering in Systems Applications, (1996), 963-968.

31. A. E. Matouk, Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system, Phys. Lett. A, 373 (2009), 2166-2173.    

32. A. E. Matouk, Dynamical behaviors, linear feedback control and synchronization of the fractional order Liu system, J. Nonlinear. Syst. Appl., 1 (2010), 135-140.

33. A. S. Hegazi, E. Ahemd and A. E. Matouk, The effect of fractional order on synchronization of two fractional order chaotic and hyperchaotic systems, J. Fract. Calc. Appl., 1 (2011), 1-15.

34. K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22.    

35. A. E. Matouk, A. A. Elsadany, E. Ahmed, et al. Dynamical behavior of fractional-order HastingsPowell food chain model and its discretization, Commun. Nonlinear Sci., 27 (2015), 153-167.    

36. S. Elaydi, An introduction to difference equations, third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.

37. L. Edelstein-Keshet, Mathematical models in biology, The Random House/Birkhäuser Mathematics Series, Random House, New York, 1988.

38. J. Cao, S. Xue, J. Lin, et al. Nonlinear Dynamic Analysis of a Cracked Rotor-Bearing System With Fractional Order Damping, J. Comput. Nonlin. Dyn., 8 (2011), 031008.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved