This paper considers pollination‐mutualism systems, including intermediary resources, which are represented by nectar for convenience. In the plant–nectar–pollinator system, we focus on the case in which the plant cannot persist without pollination and the pollinator can live independently. When nectar is assumed to rapidly approach a steady state, the plant–nectar–pollinator system can be described by its approximate plant–pollinator equations. Using dynamical system theory, we exhibit conditions under which the three‐dimensional system persists and provide rigorous proof for global dynamics of the approximate equations. The analysis demonstrates mechanisms by which the decay rate of nectar, the nectar‐consumption rate by a pollinator, the nectar‐production cost, and the rate of the plant can result in the persistence of pollination‐mutualisms. As an example, (a) when the decay rate is low, the plant survives in the system and pollination‐mutualisms persist, in which the pollinator approaches an enhanced density. (b) When the decay rate is intermediary, the plant can survive in the system only if the initial density of nectar is not below a threshold. (c) When the decay rate is large, the plant goes to extinction even though the pollinator persists. A novel finding in comparing the plant–nectar–pollinator system and its approximate plant–pollinator equations is that varying the initial value of nectar can lead to the persistence/extinction of pollination‐mutualisms. Our results are consistent with real observations, and numerical simulations confirm and extend the results.

中文翻译：

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中间资源影响下的授粉相互关系的持久性

本文考虑了授粉-互惠系统，包括中间资源，为方便起见以花蜜为代表。在植物-花蜜-传粉媒介系统中，我们关注的情况是植物没有授粉就无法生存，传粉媒介可以独立生活。当假定花蜜迅速接近稳态时，植物-花蜜-授粉系统可以用其近似的植物-授粉方程来描述。使用动力学系统理论，我们展示了三维系统持续存在的条件，并为近似方程的全局动力学提供了严格的证明。分析表明，花蜜的腐烂率，传粉者的花蜜消耗率，花蜜生产成本和植物生长率可导致授粉相互关系的持续存在的机制。例如，（a）当腐烂率低时，植物在系统中幸存下来，并且授粉-互生关系持续存在，其中传粉者接近增强的密度。（b）当腐烂率处于中间水平时，仅当花蜜的初始密度不低于阈值时，植物才能在系统中存活。（c）当腐烂率很大时，即使传粉媒介持续存在，植物也会灭绝。比较植物-花蜜-授粉系统及其近似植物-授粉方程的一个新发现是，改变花蜜的初始值会导致授粉-互生关系的持久/消灭。我们的结果与实际观察结果一致，数值模拟证实并扩展了结果。传粉者接近增强的密度。（b）当腐烂率处于中间水平时，仅当花蜜的初始密度不低于阈值时，植物才能在系统中存活。（c）当腐烂率很大时，即使传粉媒介持续存在，植物也会灭绝。比较植物-花蜜-授粉系统及其近似植物-授粉方程的一个新发现是，改变花蜜的初始值会导致授粉-互生关系的持久/消灭。我们的结果与实际观察结果一致，数值模拟证实并扩展了结果。传粉者接近增强的密度。（b）当腐烂率处于中间水平时，仅当花蜜的初始密度不低于阈值时，植物才能在系统中存活。（c）当腐烂率很大时，即使传粉媒介持续存在，植物也会灭绝。比较植物-花蜜-授粉系统及其近似植物-授粉方程的一个新发现是，改变花蜜的初始值会导致授粉-互生关系的持久/消灭。我们的结果与实际观察结果一致，数值模拟证实并扩展了结果。即使传粉媒介持续存在，植物也会灭绝。比较植物-花蜜-授粉系统及其近似植物-授粉方程的一个新发现是，改变花蜜的初始值会导致授粉-互生关系的持久/消灭。我们的结果与实际观察结果一致，数值模拟证实并扩展了结果。即使传粉媒介持续存在，植物也会灭绝。比较植物-花蜜-授粉系统及其近似植物-授粉方程的一个新发现是，改变花蜜的初始值会导致授粉-互生关系的持久/消灭。我们的结果与实际观察结果一致，数值模拟证实并扩展了结果。