一个用于猎食者-猎物社区的分段确定性模型
A piecewise deterministic model for a prey-predator community
AI总结：
本文提出一个分段确定性模型用于快慢动态的猎食者-猎物社区，证明其具有唯一不变测度并展示数值模拟结果。
摘要：
我们关注的是猎食者种群生长速度远快于猎物种群的猎食者-猎物社区（例如昆虫-树社区）。我们引入了一个分段确定性模型来描述这些猎食者-猎物社区，该模型是当捕食者数量趋于无穷大时微观模型的极限。我们证明了该过程具有唯一的不变概率测度，并且它具有指数性ergodic性。进一步地，我们对捕食者的动态进行重新缩放，以建模较小尺寸的捕食者。这种快慢系统收敛到一个社区过程，其中猎物动态在捕食者平衡点上被平均。该平均过程具有可显式计算的不变概率测度。我们使用数值模拟来研究缩放过程的不变概率测度的收敛性。
英文摘要：
We are interested in prey-predator communities where the predator population evolves much faster than the prey's (e.g. insect-tree communities). We introduce a piecewise deterministic model for these prey-predator communities that arises as a limit of a microscopic model when the number of predators goes to infinity. We prove that the process has a unique invariant probability measure and that it is exponentially ergodic. Further on, we rescale the predator dynamics in order to model predators of smaller size. This slow-fast system converges to a community process in which the prey dynamics is averaged on the predator equilibria. This averaged process admits an invariant probability measure which can be computed explicitly. We use numerical simulations to study the convergence of the invariant probability measures of the rescaled processes.
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