Abstract-A delay differential equation model of paddy ecosystem was put forward to reveal the interaction among rice, weeds and inorganic fertilizer on the system. The results show that, the system exists a rice and weed extinction equilibrium, and it also exists a rice extinction or weed extinction equilibrium. Their stable and unstable conditions are obtained. Moreover, Hopf bifurcations occur at the rice extinction or weed extinction equilibrium as the delay crosses some critical values. According to the conditions, some measures to increase rice yield were recommended.

By introducing a delayed fractional-order differential equation model, we deal with the dynamics of the stability and Hopf bifurcation of a paddy ecosystem with three main components: rice, weeds, and inorganic fertilizer. In the system, there exists an equilibrium for rice and weeds extinction and an equilibrium for rice extinction or weeds extinction. We obtain sufficient conditions for the stability and Hopf bifurcation by analyzing their characteristic equation. Some numerical simulations validate our theoretical results.

Genetically modified (GM) crops have brought huge economic benefits to mankind, however, at the same time, their safety issues are drawing growing attention. This investigation was conducted to assess whether the long-term cultivation of GM glyphosate resistant maize CC-2 effects bacterial communities in the rhizosphere soil. A 2-year follow-up trial was conducted, and soils were sampled at various plant developmental stages. The bacterial community structure of the rhizosphere soil was analysed by the high-throughput sequencing and compared with the near-isogenic non-GM maize Zheng 58. We showed here that long-term cultivation of CC-2 has no significant effect on the structure and diversity of bacterial communities, while different growth stages had significant effect. These results provided a reliable theoretical basis for the future cultivation and increased commercialisation of CC-2.

Since bifurcation makes it difficult to manage a paddy ecosystem, controlling bifurcation is an important management tool. In this paper, the stability and bifurcation control for a fractional order paddy ecosystem in the fallow season with time delay are investigated. Firstly, a paddy ecosystem model formulated by two-dimensional delayed fractional order differential equations with linear delayed feedback controller is proposed to reveal the interaction between weeds and inorganic fertilizers in paddy systems. Using the time delay as the bifurcation parameter, the sufficient conditions for stability of the system and the existence of Hopf bifurcation are obtained by analyzing the relevant characteristic equations. The results show that the time delay can heavily affect the dynamics of the system, and the feedback gain and the fractional order have significant impact on the control effect. Finally, the verification of the accuracy and validity of these conclusions is made by two examples, the control effect of the feedback gain and the fractional order on Hopf bifurcation are illustrated intuitively by a contour map.

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