This paper considers plant–pollinator–herbivore systems where the plant produces food for the pollinator, the pollinator provides pollination service for the plant in return, while the herbivore consumes both the food and the plant itself without providing pollination service. Based on these resource–consumer interactions, we form a plant–pollinator–herbivore model which includes the intermediary food. Using qualitative method and Kuznetsov theorem, we show global dynamics of the subsystems, uniform persistence of the whole system and periodic oscillation by Hopf bifurcation. Rigorous analysis on the system demonstrates mechanisms by which varying parameters could make the system transition between extinction of herbivore, coexistence of the three species at steady states, coexistence in periodic oscillations and extinction of pollinator. It is shown that (i) in plant–pollinator interactions, the plant would produce food; (ii) in plant–herbivore interactions, the plant would produce toxin; (iii) in the presence of both pollinator and herbivore, the plant would produce both food and toxin, and intermediate productions are analytically given by which the plant can reach its maximal density; and (iv) an appropriate toxin production could drive the herbivore into extinction, an unappropriate one would drive the pollinator into extinction, while too much toxin production will drive the plant itself into extinction. The analysis leads to explanations for experimental observations and provides new insights.

中文翻译：

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植物-传粉媒介-食草动物系统的持久性和振荡

本文考虑植物-授粉媒介-食草动物系统，其中植物为授粉媒介生产食物，授粉媒介反过来为植物提供授粉服务，而食草动物既消耗食物又消耗植物本身，但不提供授粉服务。基于这些资源-消费者的相互作用，我们形成了一个包括中间食物的植物-传粉媒介-草食动物模型。使用定性方法和库兹涅佐夫定理，我们展示了子系统的全局动态、整个系统的均匀持久性和 Hopf 分岔的周期性振荡。对该系统的严格分析证明了不同的参数可以使系统在草食动物灭绝、稳定状态下三种物种共存、周期性振荡共存和传粉媒介灭绝之间过渡的机制。结果表明（i）在植物 - 传粉媒介相互作用中，植物会产生食物；(ii) 在植物与食草动物的相互作用中，植物会产生毒素；(iii) 在传粉媒介和食草动物同时存在的情况下，植物会产生食物和毒素，并且中间产品通过分析给出植物可以达到其最大密度；(iv) 适当的毒素产生会导致食草动物灭绝，不适当的毒素会导致传粉媒介灭绝，而过多的毒素产生会导致植物本身灭绝。该分析导致对实验观察的解释并提供新的见解。(iii) 在传粉媒介和食草动物同时存在的情况下，植物会产生食物和毒素，并且中间产品通过分析给出植物可以达到其最大密度；(iv) 适当的毒素产生会导致食草动物灭绝，不适当的毒素会导致传粉媒介灭绝，而过多的毒素产生会导致植物本身灭绝。该分析导致对实验观察的解释并提供新的见解。(iii) 在传粉媒介和食草动物同时存在的情况下，植物会产生食物和毒素，并且中间产品通过分析给出植物可以达到其最大密度；(iv) 适当的毒素产生会导致食草动物灭绝，不适当的毒素会导致传粉媒介灭绝，而过多的毒素产生会导致植物本身灭绝。该分析导致对实验观察的解释并提供新的见解。而过多的毒素产生会导致植物本身灭绝。该分析导致对实验观察的解释并提供新的见解。而过多的毒素产生会导致植物本身灭绝。该分析导致对实验观察的解释并提供新的见解。