Stochastic modelling and inference for some ecological systems

研究了描述捕食者与被捕食者生态相互作用的确定性Lotka-Volterra微分方程组的参数推断问题。到目前为止，我们推广了Froda和Colavita(2005)的方法，其中捕食者-食饵相互作用的演化被建模为Lotka-Volterra系统加随机误差的轨迹，这是一个独立的同分布过程。在Froda和Nkurunziza(2005)以及我的论文中，我们假设分量误差服从具有特殊相依结构的Ornstein-Uhlenbeck过程。我们给出了用于推理的数学性质，这些性质是从这种依赖结构中推导出来的。此外，我们表明，这些更一般和更灵活的假设使我们能够改进对种群大小(猎物和捕食者)的预测。在本文及相关文献中，我们给出了确定性微分方程组参数的一个估计量，并给出了关于这些参数的最强检验。此外，我们还研究了所提出的估计量和检验的渐近性质。到目前为止，我们只使用了频率论者的推论。因此，在短期和中期内，我考虑改进估计方法以及使用贝叶斯方法预测人口规模。也就是说，我将使用非信息性的先验分布作为参数。和我们以前的工作一样，我将利用遍历理论建立新估计和检验的渐近性质。一个长期目标是开发仅基于扩散过程的新统计方法。在扩散过程方法中，我将考虑与确定性方程相对应的随机微分方程。这种方法应该允许我们驱动最大似然估计器，在某些条件下，这些估计器是一致的、无偏的、具有一致最小方差的。此外，对于比两种群捕食者-食饵系统更一般的其他生态模型，这种方法可能是最合适的。事实上，我计划讨论其他的生态模型，比如在存在社会现象的情况下的两种或三种物种。

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.