Department of Mathematics, Arizona State University, Tempe, AZ, USA.
J Biol Dyn. 2008 Jul;2(3):286-96. doi: 10.1080/17513750701769881.
One of the simplest predator-prey models that tracks the quantity and the quality of prey is the one proposed by [I. Loladze, Y. Kuang, and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol. 62 (2000) pp. 1137-1162.] (LKE model). In it, the ratio of two essential chemical elements, carbon to phosphorus, C:P, represents prey quality. However, that model does not explicitly track P neither in the prey nor in the media that supports the prey. Here, we extend the LKE model by mechanistically deriving and accounting for P in both the prey and the media. Bifurcation diagrams and simulations show that our model behaves similarly to the LKE model. However, in the intermediate range of the carrying capacity, especially near the homoclinic bifurcation point for the carrying capacity, quantitative behaviour of our model is different. We analyze positive invariant region and stability of boundary steady states. We show that as the uptake rate of P by producer becomes infinite, LKE models become the limiting case of our model. Furthermore, our model can be readily extended to multiple producers and consumers.
最简单的跟踪猎物数量和质量的捕食者-猎物模型之一是由 [I. Loladze、Y. Kuang 和 J.J. Elser 提出的](I. Loladze、Y. Kuang 和 J.J. Elser，“生产者-食草动物系统中的化学计量学：将能量流与元素循环联系起来”，《数学生物学通报》62 (2000)，第 1137-1162 页) (LKE 模型)。在这个模型中，两种必需化学元素，碳和磷的比例，C:P，代表猎物的质量。然而，该模型并没有明确跟踪猎物和支持猎物的介质中的磷。在这里，我们通过从机制上推导出并考虑到猎物和介质中的磷来扩展 LKE 模型。分岔图和模拟表明，我们的模型与 LKE 模型的行为相似。然而，在承载能力的中间范围内，特别是在承载能力的同宿分岔点附近，我们模型的定量行为是不同的。我们分析了正不变区域和边界稳定状态的稳定性。我们表明，当生产者对磷的吸收率变得无限大时，LKE 模型成为我们模型的极限情况。此外，我们的模型可以很容易地扩展到多个生产者和消费者。
