Stochastic modelling and inference for some ecological systems

我们考虑一个确定性Lotka-Volterra微分方程系统的参数的推断问题，该系统描述了猎物和捕食者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，其中捕食者-食饵相互作用的演化被建模为Lotka-Volterra系统的轨迹加上随机误差，这是一个独立同分布的过程。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依赖结构的Ornstein-Uhlenbeck过程。我们提出的数学性质，用于推理，这是推导出这种依赖结构。此外，我们表明，这些更一般和灵活的假设，使我们能够提高预测的人口规模（猎物和捕食者）。在本论文及相关论文中，我们提出了确定性微分方程组参数的一种估计，以及关于这些参数的最强有力的检验。此外，我们研究了所提出的估计和测试的渐近性质。到目前为止，我们只使用了频率论推理。因此，在短期和中期，我考虑改进估计方法，以及预测的人口规模使用贝叶斯方法。也就是说，我将对参数使用无信息先验分布。在我们以前的工作中，我将建立新的估计和测试的渐近性质，通过使用遍历理论。一个长期的目标是发展新的统计方法的基础上扩散过程。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，一致的，无偏的，一致的最小方差。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方法。事实上，我打算处理其他生态模型，如两个或三个物种在社会现象的存在。

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.