Stochastic modelling and inference for some ecological systems

本文研究了一类确定性Lotka-Volterra微分方程组参数的推理问题，该方程组描述了捕食者与猎物之间的生态相互作用。到目前为止，我们推广了Froda和Colavita（2005）的方法，该方法将捕食者-猎物相互作用的进化建模为Lotka-Volterra系统的轨迹加上随机误差，这是一个独立的同分布过程。在Froda和Nkurunziza（2005）以及我的论文中，我们假设分量误差遵循具有特殊依赖结构的Ornstein-Uhlenbeck过程。我们给出了用于推理的数学性质，这些性质是由这种依赖结构推导出来的。此外，我们表明，这些更一般和灵活的假设使我们能够改进对种群规模（猎物和捕食者）的预测。在本文和相关论文中，我们提出了确定性微分方程系统参数的估计量，以及关于这些参数的最有力的检验。进一步，我们研究了所提估计量的渐近性质并进行了检验。到目前为止，我们只使用了频率推理。因此，在短期和中期，我考虑改进估计方法以及使用贝叶斯方法预测总体大小。也就是说，我将使用参数的非信息先验分布。在我们之前的工作中，我将利用遍历理论建立新的估计量和检验的渐近性质。一个长期目标是发展仅基于扩散过程的新的统计方法。在扩散过程方法中，我将考虑与确定性方程相对应的微分随机方程。这种方法应该允许我们驱动最大似然估计，在某些条件下，它是一致的，无偏的，具有一致的最小方差。此外，这种方法对于其他比两种捕食者-猎物系统更普遍的生态模型也是最合适的。事实上，我打算在存在社会现象的情况下处理其他生态模型，如两种或三种。

We consider inference problems concerning the parameters of a deterministic Lotka-Volterra system of differential equations, which describes the ecological interaction between prey and predator. So far, we generalized the Froda and Colavita (2005) method where the evolution of predator-prey interactions is modeled as the trajectory of a Lotka-Volterra system plus a random error, which is an independent and identically distributed process. In Froda and Nkurunziza (2005), as well as in my thesis, we assume that the components error follow Ornstein-Uhlenbeck processes with special dependence structure. We present mathematical properties, used in inference, which are deduced from this dependence structure. Moreover, we show that these more general and flexible assumptions allow us to improve the prediction of the population sizes (prey and predators). In the thesis and the related papers, we propose an estimator of the parameters of the deterministic system of differential equations, as well as the most powerful tests concerning these parameters. Furthermore, we study the asymptotic properties of the proposed estimator and tests. So far, we have used only frequentist inference. Thus, in the short and medium term, I consider improving the estimation method as well as the prediction of the population sizes by using Bayes methods. Namely, I will use a noninformative prior distributions for the parameters. As in our previous work, I will establish the asymptotic properties of the new estimators and tests by using ergodic theory. A long-term objective is to develop new statistical methods based only on diffusion processes. In the diffusion processes approach, I will consider differential stochastic equations corresponding to deterministic equations. This approach should allow us to drive maximum likelihood estimators that are, under some conditions, consistent, unbiased, with uniformly minimum variance. Moreover, this approach could be the most appropriate one for other ecological models which are more general than the two-species predator-prey system. In fact, I plan to deal with other ecological models such as two- or three-species in the presence of social phenomena.