Abstract
Great progress has been made in bifurcation control of systems described by ordinary differential equations. However, the control of Hopf bifurcation and Turing patterns is seldom reported in reaction-diffusion systems, which is formed by partial differential equations. In this paper, a hybrid control synthesis combining state feedback is firstly devised in the reaction-diffusion marine planktonic ecosystem. The Turing instability condition and Hopf bifurcation criterion are derived through carrying out the eigenvalue analysis of the controlled system. The numerical simulations show that the hybrid control strategy can not only suppress the formation of Turing patterns, but also delay or advance the Hopf bifurcation point. Therefore, the desired spatial dynamics behaviors can be generated by manipulate the control gain parameters, so as to achieve the purpose of maintaining the marine ecological balance.
Highlights
I N recent years, some ecosystem models have been developed as important analytical methods in order to better understand marine ecological energy cycles [1], [2]
Kumar et al [31] studied the dynamic behavior of the equilibrium point of an open nonlinear system other than the traditional Turing pattern with cross-diffusion, and found that the Turing instability condition can be changed by a critical control parameter including self-diffusion
The results demonstrate that the hybrid controller can suppress the occurrence of Turing pattern, change the position of Hopf bifurcation critical value, and enhance the stability and controllability of the system, so as to obtain the expected dynamic behaviors
Summary
I N recent years, some ecosystem models have been developed as important analytical methods in order to better understand marine ecological energy cycles [1], [2]. As important dynamic behaviors, Turing instability and Hopf bifurcation frequently appear in actual marine planktonic ecosystems. Kumar et al [31] studied the dynamic behavior of the equilibrium point of an open nonlinear system other than the traditional Turing pattern with cross-diffusion, and found that the Turing instability condition can be changed by a critical control parameter including self-diffusion. With reference to the design methods of state feedback control and parameter adjustment, we propose a new hybrid bifurcation control strategy that simulates human intervention to eliminate or delay the adverse effects caused by the occurrence or formation of water blooms and red tides in marine ecological systems. (3) Combined with the design methods of parameter adjustment and state feedback, the hybrid bifurcation control strategy is applied to the marine planktonic ecosystem with diffusion for the first time.
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