Abstract
The rhizosphere microbe plays an important role in removing the pollutant generated from industrial and agricultural production. To investigate the dynamics of the microbial degradation, a nonlinear mathematical model of the rhizosphere microbial degradation is proposed based on impulsive state feedback control. The sufficient conditions for existence of the positive order-1 or order-2 periodic solution are obtained by using the geometrical theory of the semicontinuous dynamical system. We show the impulsive control system tends to an order-1 periodic solution or order-2 periodic solution if the control measures are achieved during the process of the microbial degradation. Furthermore, mathematical results are justified by some numerical simulations.
Highlights
The constructed wetland is regarded as the most efficient and cost-effective system that involves microbes consuming pollutant generated from industrial and agricultural production
Degradation ability of the microbe in the constructed wetland is affected by many factors such as temperature, pH, dissolved oxygen, and substrate concentration [1]
Studies have shown that excessive pollutant strongly affects the dynamic behavior of microbe, which can lead to the incidence of the sustained oscillation and decrease degradation ability of the microbe under certain conditions [2]
Summary
The constructed wetland is regarded as the most efficient and cost-effective system that involves microbes consuming pollutant generated from industrial and agricultural production. For the purpose of improving degradation ability of the microbe and decreasing the inhibition effect of the substrate, it is necessary to control the pollutant concentration lower than a certain level. Considering the product inhibition during the microorganism culture, Guo and Chen [16] develop a mathematical model concerning a chemostat with impulsive state feedback control to investigate the periodicity of bioprocess and an order-1 periodic solution is obtained by the existence criteria of periodic solution. The main purpose of this paper is to construct a mathematical model of the microbial degradation with impulsive state feedback control and understand the dynamics of the microbial degradation in the constructed wetland. We give some numerical simulations and a brief discussion
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