In the paper, we introduce a differential equations model of paddy ecosystems in the fallow season to study the effect of weeds removal from the paddy fields. We found that there is an unstable equilibrium of the extinction of weeds and herbivores in the system. When the intensity of weeds removal meets certain conditions and the intrinsic growth rate of herbivores is higher than their excretion rate, there is a coexistence equilibrium state in the system. By linearizing the system and using the Routh–Hurwitz criterion, we obtained the local asymptotically stable conditions of the coexistence equilibrium state. The critical value formula of the Hopf bifurcation is presented too. The model demonstrates that weeds removal from paddy fields could largely reduce the weeds biomass in the equilibrium state, but it also decreases the herbivore biomass, which probably reduces the content of inorganic fertilizer in the soil. We found a particular intensity of weeds removal that could result in the minimum content of inorganic fertilizer, suggesting weeds removal should be kept away from this intensity.

中文翻译：

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淡季稻田生态系统除草效果的定性分析

在本文中，我们介绍了休耕季节稻田生态系统的微分方程模型，以研究除草对稻田的影响。我们发现系统中杂草和草食动物的灭绝存在不稳定的平衡。当杂草的去除强度满足一定条件并且草食动物的内在生长速度高于其排泄速度时，系统中就会存在一个共存的平衡状态。通过线性化系统并使用Routh-Hurwitz准则，我们获得了共存平衡状态的局部渐近稳定条件。给出了Hopf分支的临界值公式。该模型表明，从稻田中除草可以在平衡状态下大大减少杂草的生物量，但同时也减少了草食动物的生物量，这可能会减少土壤中无机肥料的含量。我们发现杂草的去除强度特别高，可能导致无机肥料的含量降至最低，这表明杂草去除应远离此强度。