Here are the problems presented at the 2021 Graduate Student Mathematical Modeling Camp, as well as summary reports of the work accomplished.
Ecological modeling of host-parasitoid population dynamics
Brooks Emerick, Kutztown University
Abstract: Host-parasitoid interactions make up an important class of consumer resource dynamics. A parasitoid is an organism that spends most of its life cycle attached to or inside the host. Unlike an actual parasite, the parasitoid eventually kills the host. During a particular season of the year, known as the vulnerable period, the parasitoid injects eggs into the host larvae. Parasitoid larvae, then, emerge from the host, effectively killing it. Discrete-time models are usually employed to simulate host-parasitoid interactions because reproduction occurs at the same fixed date every year. Early models of this phenomenon include the discrete-time Nicholson-Bailey model, which is known to be unstable i.e. coexistence is impossible. More recent models take advantage of a hybrid technique that uses continuous dynamics to model the vulnerable period so that stabilizing characteristics can be mechanistically incorporated. During this camp, we'll review the basic discrete models, explore various dynamic models of the vulnerable period, and study the semi-discrete framework. Along the way, we'll develop the model building process and analyze each model with appropriate linear stability analysis. Ultimately, by the end of the camp, students will be comfortable with the host-parasitoid modeling process and will be able to explore sophisticated modeling scenarios, potentially discovering new models that provide biological insight into the intricate dynamics of the host-parasitoid interaction.
Water flow through plants and its impact on climate
Valeria Barra, California Institute of Technology
To understand the impact of greenhouse gas emissions, it is important to constrain carbon uptake by the land surface, which is currently a big uncertainty in future climate predictions. To do this, we need to understand the terrestrial water cycle, and especially how plants use water. In climate models that simulate the entire Earth system (comprising the atmosphere, ocean, land and ice masses) plants can serve as an important link between the land and the atmosphere. Hence, plant hydraulics, which studies water transport processes through roots, trunks, stems, and leaves, becomes a key link between different modules in modern climate models. Accurately modeling this process with equations based on Darcy’s Law for water flow in a porous medium helps constrain photosynthesis and evapotranspiration (also crucial for temperature extremes and precipitation predictions).In this workshop, we present a simple root-stem-leaf plant model that represents the water flow through plants, allowing water exchange between plants and soil. Our goal for the camp will be to extend the simple model provided, either by increasing the accuracy of the modeled plant response to drought or by taking into account different plant geometries. If time allows it, we will investigate different parameter choices or systemic changes to address several open questions: Up to what height can plants grow with this model? How does the hydraulic conductivity affect the decay of plants?
Mathematical Models and Simulations of Recongurable Flow Networks: Erosion, Deposition, Filtration and Growth
Pejman Sanaei, New York Institute of Technology