Impacts of microbial decomposition delays on the stability of paddy ecosystem during fallow season

As we know it takes some time for microorganisms to decompose organic matter into nutrients required by plants. In order to study the impact of the decomposition delay on the stability of paddy ecosystem, we use a three-dimensional differential equation model with two time delays to reflect the complex dynamic relationship of a paddy ecosystem in fallow season composed of weeds, inorganic salts and herbivores, and the two delays respectively reflect the time of microbial decomposition of weeds and herbivores excrement. By constructing the crossing set and the stability crossing curves, sufficient conditions for the ecosystem stability and Hopf bifurcation emerging are obtained. The research shows that the decomposing delay not only affects the stability of the paddy ecosystem, but also leads to Hopf bifurcation. The critical value formulas for Hopf bifurcation in two special cases of degradation and equal delay are also obtained. Numerical calculations show that the fluctuation amplitude of the mean value of inorganic salts decreases with the increase of time delay, and the paddy ecosystem with decomposing delay of herbivores smaller than those of weeds has relatively higher mean value of inorganic salts when ecosystem is in an unstable state. Therefore, it is necessary to adopt composting like techniques to reduce the decomposing delay of weeds.