Abstract
The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the economic profit passes through zero, this model exhibits the transcritical bifurcation, the Hopf bifurcation, and the limit cycle. In particular, the system undergoes the singularity induced bifurcation at the positive equilibrium, which can result in impulse. Then, state feedback controllers closer to the actual control strategies are designed to eliminate the unexpected singularity induced bifurcation and stabilize the positive equilibrium under the positive profit. Finally, numerical simulations verify the results and illustrate the effectiveness of the controllers. Also, the model with positive economic profit is shown numerically to have different dynamics.
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This work was supported by National Natural Science Foundation of China (No. 60974004) and Science Foundation of Ministry of Housing and Urban-Rural Development (No. 2011-K5-31).
Ning Li received the B. Sc. and M. Sc. degrees in mathematics from Liaoning Normal University, PRC in 2001 and 2004, respectively, and the Ph.D. degree in control theory and control engineering from Northeastern University, PRC in 2010. Her research interests include bifurcations, chaos and the control in biological systems, and complex networks.
Hai-Yi Sun received the M. Sc. degree in operational research and cybernetics from Northeastern University, PRC in 2006, and the Ph.D. degree in control theory and control engineering from Northeastern University, PRC in 2011. He is an associate professor of Shenyang Jianzhu University. His research interests include synchronization of chaos and complex networks.
Qing-Ling Zhang received the B. Sc. and M. Sc. degrees from the Mathematics Department and the Ph.D. degree from the Automatic Control Department of Northeastern University, PRC in 1982, 1986, and 1995, respectively. He is a professor of Northeastern University, PRC and a member of the University Teaching Advisory Committee of National Ministry of Education. His research interests include chaos control, robust control, networked control system, and singular system.
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Li, N., Sun, HY. & Zhang, QL. The dynamics and bifurcation control of a singular biological economic model. Int. J. Autom. Comput. 9, 1–7 (2012). https://doi.org/10.1007/s11633-012-0609-5
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DOI: https://doi.org/10.1007/s11633-012-0609-5