Stochastic modelling and inference for some ecological systems

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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.