In this article, a nutrient‐autotroph‐herbivore model with nutrient recycling is constructed. Holling type‐II functional response for the relation between nutrient and autotroph while Beddington‐DeAngelis‐type functional response for autotroph and herbivore relation are considered here. It is plausible that the conversion of nutrient from dead biomass (autotroph and herbivore) by decomposers (i.e., bacteria and fungi) are not instantaneous, which takes times. Hereby, two different discrete time delays for the decomposition process are introduced. The local and global stability behaviours of both nondelayed and delayed models are analysed around the equilibrium points. The stability and direction of Hopf‐bifurcation using normal form theory and centre manifold theorem by taking one delay as a bifurcation parameter while keeping the other one fixed in the stable interval are discussed. It is observed that if the delay increases, the system loses its stability and hence becomes unstable. It is analysed how autotroph‐herbivore ecosystem can be affected by the quantity of input nutrient and the properties of delays. The quantity of nutrient and the length of delays play significant roles in determining the stability of the system since a sufficiently small amount of nutrients or a long enough delay leads to the extinction of a species.

中文翻译：

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具有两个延迟的养分循环的养分自养型食草动物模型的稳定性和分叉性

在本文中，构建了具有养分循环利用的养分自养型食草动物模型。营养和自养关系的Holling II型功能性反应，自养和食草动物关系的Beddington-DeAngelis型功能性反应。可能是分解剂（即细菌和真菌）从死生物量（自养生物和食草动物）转化养分不是瞬时的，这需要时间。因此，引入了两个不同的离散时间延迟用于分解过程。围绕平衡点分析了无延迟和延迟模型的局部和全局稳定性行为。讨论了使用范式理论和中心流形定理，以一个时延为分岔参数，同时将另一个保持在稳定区间内的Hopf分岔的稳定性和方向。观察到，如果延迟增加，则系统失去其稳定性，因此变得不稳定。它分析了输入营养素的数量和延迟特性如何影响自养食草动物生态系统。营养素的数量和延迟的长度在决定系统稳定性方面起着重要作用，因为营养素的量足够少或延迟时间足够长会导致物种的灭绝。它分析了输入营养素的数量和延迟特性如何影响自养食草动物生态系统。营养素的数量和延迟的长度在确定系统稳定性方面起着重要作用，因为营养素的量足够少或延迟时间足够长会导致物种的灭绝。分析了输入营养素的数量和延迟特性如何影响自养食草动物生态系统。营养素的数量和延迟的长度在确定系统稳定性方面起着重要作用，因为营养素的量足够少或延迟时间足够长会导致物种的灭绝。