In this paper, three fuzzy nonlinear mathematical models, corresponding to crisp models described in Devi and Gupta (2018), are formulated to study the comparative behavior of carbon dioxide concerning vegetation biomass in crisp and fuzzy environments. Logistic growth and regrowth of vegetation biomass are considered in the first two models, and delay is incorporated in the regrowth of vegetation biomass in the third one. Conditions for boundedness and persistence of the solutions, the existence of equilibrium points, and the stability of equilibrium points are carried out for each model. We also derived the formula for the critical value of time delay. Further, we also analyzed the existence, stability, and direction of the Hopf bifurcation. Numerical simulations were performed to complement analytical findings and to see the comparison between crisp and fuzzy environments. In fuzzy environment, it is observed that increment in the minimum value of regrowth rate shifts the destabilizing value of delay forward while if the maximum value of regrowth rate decreases, variables will become unstable before the estimated time. The overall analysis demonstrates that the behavior of the models in crisp and fuzzy environments is quite different.

中文翻译：

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脆弱和模糊环境下不同植被生物量的生长对CO2影响的比较研究

本文建立了三个模糊非线性数学模型，分别对应于Devi和Gupta（2018）中描述的脆模型，以研究二氧化碳在脆和模糊环境中与植被生物量的比较行为。前两个模型考虑了植被生物量的对数增长和再生，而第三个模型考虑了植被生物量的延迟增长。对每个模型进行解的有界和持久性的条件，平衡点的存在以及平衡点的稳定性。我们还导出了时延临界值的公式。此外，我们还分析了Hopf分叉的存在，稳定性和方向。进行了数值模拟，以补充分析结果并查看清晰和模糊环境之间的比较。在模糊环境中，可以观察到再生速率最小值的增加会使延迟的不稳定值向前移动，而如果再生速率最大值减小，变量将在估计时间之前变得不稳定。整体分析表明，在清晰和模糊的环境中，模型的行为是完全不同的。