Tilman模型中的若干悖论：如何避免或接受它们
Some paradoxes in the Tilman model, how to avoid or accept them
AI总结：
本文分析Tilman资源竞争模型中的三类反直觉现象，并提出结合不稳定稳态与瞬态时间尺度的扩展稳定性分析，以解释、避免或接受这些生态模型悖论。
摘要：
在为生物系统建立模型时，人们期望能够把握生物学机制，并能够运用直觉预测结果。本文讨论的是预期结果与模型分析所得结果之间的差异，即悖论。消费者竞争资源的Tilman模型具有一些可能看似违反直觉的方面。在单消费者、单资源的标准Tilman模型中，当一个非常高效的消费者迅速吃掉所有可获得的食物时，资源密度变为零；但如果没有食物，消费者如何生存？该悖论可以通过以下认识来解决：在短期内，快速消费确实可能导致饥饿和消费者种群下降，但在长期内，仍会保留较小的资源密度和消费者密度。一个以两种必需营养物为生的单一消费者，会使非限制性营养物的密度保持在其临界水平之上，那么多余的食物去了哪里？我们解释说，多余的食物并不会仅仅因为可获得就被消费，事实上这些额外食物根本没有被消费，因为消费者对其没有用途。对于两个消费者物种竞争单一营养物的模型，其中一个消费者最终将会消失，即使存活下来的那个消费者的食物经济性更差，而且仍有大量食物被浪费。这两者都是模型的固有方面；当消费者物种之间的差异较小时，可以避免该悖论，在这种情况下达到平衡需要非常长的时间。我们认为，一般而言，扩展的稳定性分析不仅考虑渐近稳定状态，还考虑不稳定稳态以及瞬态动力学中涉及的所有时间尺度，能够更深入地理解相关动力学，从而有助于避免或接受生态模型中明显的矛盾。
英文摘要：
In making models for biological systems one expects to grasp the biology and be able to use ones intuition to predict the outcome. This paper is about the discrepancy between what is expected and what is the outcome of the analysis of the model, a paradox. The Tilman model of consumers competing for resources has some aspects that may appear counterintuitive. Within the standard Tilman model for a single consumer and a single resource, when a very efficient consumer rapidly eats all available food, the resource density becomes zero, but if there is no food, how can the consumer survive? The paradox can be resolved by realising that in the short term indeed rapid consumption may lead to starvation and a decline in the consumer population, but in the long term a small resource and consumer density remain. A single consumer living on two essential nutrients leaves the density of the non-limiting nutrient above its critical level, so what is done with the extra food? We explain that the extra food is not consumed, just because it is available, but that in fact the additional food is not consumed at all, because the consumer has no use for it. For a model with two consumer species competing for a single nutrient one of the consumers will eventually disappear, even if the food economics of the surviving one is worse and there is still much food wasted. Both are inherent aspects of the model, and the paradox can be avoided if the difference between the consumer species is small, in which case it will take very long to reach equilibrium. We argue that in general an extended stability analysis, in which not only the asymptotically stable state is considered, but also the unstable steady states and all time scales involved in the transient dynamics, gives more insight in the dynamics involved and thus can help in avoiding apparent contradictions in ecological models, or accepting them.
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