Convergence of Rescaled Competing Species Processes to a Class of SPDEs
Convergence of Rescaled Competing Species Processes to a Class of SPDEs
复制标题
将重新调整的竞争物种过程收敛到一类 SPDE
DOI：
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发表时间：
2011
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通讯作者：
Sandra Kliem
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作者：
Sandra Kliem
我们可以构造维$d=1$的选民过程的重标摄动序列，其近似密度是紧的。通过结合扰动中的远程模型和固定核模型，并考虑临界远程情况，对Cox和Perkins（2005）的结果进行了改进。作为一种特殊情况，我们可以考虑具有远程分散和短程竞争的重新标度的Lotka-Volterra模型。仅在远程相互作用的情况下，近似密度收敛到连续的时空密度，从而求解一类SPDEs（随机偏微分方程），即由Fisher-Wright噪声驱动的一类漂移的热方程。若极限SPDE的初始条件可积，则极限的弱唯一性。所得到的结果通过包含扰动扩展了Mueller和Tribe（1995）关于选民模型的结果。特别是，Neuhauser和Pacala（1999）中介绍的Lotka-Volterra模型的空间版本在参数接近1时被涵盖。他们的模型包含了繁殖力参数，并模拟了种内和种间竞争。
One can construct a sequence of rescaled perturbations of voter processes in dimension $d=1$ whose approximate densities are tight. By combining both long-range models and fixed kernel models in the perturbations and considering the critical long-range case, results of Cox and Perkins (2005) are refined. As a special case we are able to consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition. In the case of long-range interactions only, the approximate densities converge to continuous space time densities which solve a class of SPDEs (stochastic partial differential equations), namely the heat equation with a class of drifts, driven by Fisher-Wright noise. If the initial condition of the limiting SPDE is integrable, weak uniqueness of the limits follows. The results obtained extend the results of Mueller and Tribe (1995) for the voter model by including perturbations. In particular, spatial versions of the Lotka-Volterra model as introduced in Neuhauser and Pacala (1999) are covered for parameters approaching one. Their model incorporates a fecundity parameter and models both intra- and interspecific competition.