Life History of Plants
Professor Glenn Ledder (Department of Mathematics)
An introductory course in ODEs. Familiarity with a scientific programming language is also desirable. All relevant biology will be presented during the early weeks of the project.
Various life history strategies are observed among plants. While most plants can be classified as annual or perennial, there are variations on these themes. Annual plants differ in timing, with some sprouting in the spring and others in the fall. Those that sprout in the fall may set seeds in the fall or overwinter and set seeds the following spring. A small number of annual plants flower in both the fall and the spring. Some perennials reproduce every year, while others save resources for occasional reproduction binges. The distribution of resources between roots and shoots can be adapted to the individual plant's micro-environment. Leaves can be delicate or sturdy, long-lived or short-lived, defended against herbivory by toxic chemicals or not. In all cases, the principle of natural selection suggests that a plant's actual life history is approximately the optimal life history for its environment and ecological niche, subject to limitations in genetic variation. Various mathematical models have been used to determine plant fitness in terms of life history parameters and schedules and to determine the optimal life history for different scenarios, but there is a lot of room for new work.
The students who work on this project will learn some of the models and methods that have already been developed for this area, identify a new feature to include in a model or a new scenario to study, construct an appropriate mathematical model, and analyze the model to see what biological phenomena it predicts. Analytical work may include methods of control theory, optimization and/or dynamic programming, with some scientific computation.
Calculus on Time Scales
Professor Allan Peterson (Department of Mathematics)
An introductory course in ODEs.
The concept of time scales unifies and extends discrete time and continuous time. In this project the students will study calculus and dynamical systems on time scales, which is a natural extension of the more familiar calculus and differential equations in continuous time which the students have already studied. The focus of the project will be on fractional derivatives and fractional differential equations, and their generalization to time scales. Fractional differential equations have applications in numerous diverse fields, including electrical engineering, chemistry, mathematical biology, control theory and the calculus of variations. For instance, the fractional calculus may provide more mathematically accurate epidemic models.
The students will first learn about fractional derivatives and fractional differential equations in continuous time. They will then learn about the fractional calculus on arbitrary time scales. This is a very new topic, and hence has many new research areas to explore.
New Models for Heat Conduction and Elasticity in Structures with Cracks
An introductory course in differential equations; familiarity with a scientific programming language is also desirable.
The classical equations for heat and mass transfer are not well suited for structures where discontinuities (like cracks) appear. An example of such a phenomenon would be the melting of an iceberg. Similarly, the classical theory cannot be applied directly in the formation of cracks in an elastic body subject to an external force.
The very new area of solid mechanics called Peridynamics offers a framework in which one can formulate and investigate mathematical models that take into account the breakage of bonds between different parts of a body.
Students will learn about mathematical modeling of such structures using the general theory of peridynamics, and then investigate analytically and numerically properties of a perdynamic model for a body with evolving material discontinues.