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.
