Nonautonomous Structured Population Models with Application to Amphibians and Associated Diseases

在这个项目中，研究人员开发了一个一般类的结构化模型，其中包括一个系统的一阶偏微分方程耦合到一个系统的常微分方程。对于这类新的模型，他们将研究解的存在唯一性，并开发一个框架来计算解和估计参数，这是将这些工具与观测数据联系起来所必需的。特别是，二阶有限差分方法将被开发，这是有效的计算这些模型的解决方案，具有高精度。这些计划的收敛结果将建立。以前的工作已经表明，属于这一一般类的模型是成功的描述动态的绿色树蛙种群。研究人员将证明，这类模型的应用可以扩展到两栖动物之间的传播Batrachochytrium dendrobaerum（Bd）。最近发现，共生菌青金杆菌（Jl）通过产生抗真菌代谢产物为某些两栖动物物种提供对这种疾病的免疫力。在这个项目中开发的数值方法将用于了解疾病Bd和共生体Jl对两栖动物持久性的影响。此外，该项目的一个重要目标是培养博士。数学生物学多学科领域的学生，以帮助开展拟议的研究。两栖动物种群的减少和灭绝是科学界极为关注的问题。由于两栖动物在陆地和水生生态系统之间旅行，并跨越不同的营养水平，它们通常是气候变化，栖息地丧失，污染和非本地物种造成的大规模环境退化的第一个指标。 例如，引入的疾病减少和削弱了两栖动物的种群，这反过来又改变了生态景观。本项目中对两栖动物和相关疾病的研究将产生结构化模型，以了解种群和疾病动态。计算这些模型的解决方案将需要新的数值方法，这将被开发。在本项目期间进行的模型的广泛的数值模拟将提供洞察力，如何与明智的共生体Jl的引入，以打击Bd的程序可能会进行最有效的。此外，由于在这个项目中建立的模型的一般性和数学技术的抽象性，这些工具无疑将是有用的其他重要的生物学问题。属于已经开发的拟议类别的模型的具体实例包括：（i）红细胞生成中涉及的细胞动力学，（ii）疟疾的宿主内动力学，（iii）海洋分枝杆菌（TB样细菌）在鱼类中的传播动力学。

In this project, the investigators develop a general class of structured models which consists of a system of first-order partial differential equations coupled to a system of ordinary differential equations. For this novel class of models, they will study existence-uniqueness of solutions and develop a framework within which to compute solutions and estimate parameters, which is necessary to connect these tools to observed data. In particular, second order finite difference methods will be developed, which are efficient in computing solutions of these models with high accuracy. Convergence results for these schemes will be established. Previous work has demonstrated that models falling under this general class are successful in describing the dynamics of a green tree frog population. The investigators will demonstrate that the application of this class of models can be extended to the spread of Batrachochytrium dendrobatidis (Bd) among amphibians. It has been recently discovered that the symbiont Janthinobacterium lividum (Jl) provides immunity to this disease for some species of amphibians through the production of anti-fungal metabolites. The numerical methods developed during this project will be used to understand the effects the disease Bd and the symbiont Jl have on the persistence of amphibians. Further, an important aim of this project is to train Ph.D. students in the multidisciplinary field of mathematical biology to help carry out the proposed research.The decline and extinction of amphibian populations are of immense concern to the scientific community. Because amphibians travel between terrestrial and aquatic ecosystems, and across different trophic levels, they are often the first indicators of large-scale environmental degradation caused by climate change, habitat loss, pollution, and non-native species. For example, introduced diseases diminish and weaken amphibian populations, which in turn transform the ecological landscape. The study of amphibians and associated diseases in this project will result in structured models to understand population and disease dynamics. The computation of solutions of these models will require novel numerical methods, which will be developed. The extensive numerical simulations of the models conducted during this project will provide insight as to how a program to combat Bd with the judicious introduction of the symbiont Jl might be carried out most effectively. Furthermore, because of the generalities of the models developed and the abstractness of the mathematical techniques established in this project, these tools will undoubtedly be useful to other important biological problems. Specific examples of models that fall under the proposed class that have already been developed include: (i) cell dynamics involved in erythropoiesis, (ii) intra-host dynamics of malaria, (iii) the transmission dynamics of Mycobacterium marinum (a TB-like bacterium) among fish.