Cyclic data and planar ordinary differential equations: statistical modeling and inference

目前的程序是关于解决一个广泛的问题，那就是估计常微分方程解的参数，这些常微分方程组在生物统计学应用中存在循环解。我们从生态学开始，重点放在捕食者和猎物之间的相互作用；在实践中，许多动物种群表现出振荡行为。因此，在他们的模型中，生态学家正在考虑允许极限环或螺旋达到平衡的系统。在我们的建议中，我们介绍了一种简单的方法，用于估计由ODE系统的定性分析提出的元素，如等倾线，例如，只要存在周期性，该方法就可以工作。一旦执行了估计，人们还可以测试各种模型。本方案的原创性是在相平面内工作，方法是Froda&Colavita(2005)和Froda&Nkurunziza(2006)提出的方法的延伸。从应用的角度来看，这一建议引起了生态学研究的各个领域的关注。首先，我们考虑在一个地理位置选择一个合适的捕食者-食饵模型。后来，我们应该能够扩展我们的方法，并在集合种群中适应捕食者-猎物模型，即当栖息地支离破碎时(例如，在栖息地被破坏后)。了解栖息地破坏对动物行为的影响在保护生物学中极其重要。值得注意的是，我们的方法允许区分不同的模型，这是生态学家之间争论的问题，也是该领域的核心。最后，从长远来看，可以考虑其他应用领域，如流行病模型的研究。

The present program is on solving a broad question, which is to estimate parameters of solutions to ODEs (ordinary differential equations) that occur in biostatistical applications and have cyclic solutions. We start with ecology, and the emphasis is on predator-prey interactions; in practice, many animal populations exhibit oscillatory behaviour. Therefore, in their models, ecologists are considering systems that admit limit cycles or spiral to an equilibrium. In our proposal we introduce a simple method for estimating elements that are put forward by the qualitative analysis of the ODE system, like isoclines, e.g. The method works whenever periodicity is present. Once the estimation is performed, one can test various models as well. The originality of the present program is to work in the phase plane, and the methodology is an extension of the one developed in Froda & Colavita (2005) and Froda & Nkurunziza (2006).From the applied point of view, various  areas of ecological research are concerned by this proposal. At first, we consider the choice of an appropriate predator-prey model at one geographical site. Later,  we should be able to extend our methodology and fit predator-prey models in metapopulations, i.e. when the habitat is fragmented (e.g. after habitat destruction). Understanding the impact of habitat destruction on animal behaviour is extremely important in conservation biology.It is to be noted that our method allows to discriminate among alternative models, a problem which is debated among ecologists, and is central to the field.Finally, in the long run, other fields of application  can be considered, like the study of epidemic models.