Research Interests

My research interests are at the intersection of differential equations and population ecology. These include

predator-prey dynamics, food chain dynamics, invasive species spread, invasive species and pest control, biological control, modeling and analysis of the above via dynamical systems methods, semi linear and quasi linear parabolic PDE.

I am also interested in social dynamics. These include language dynamics, language extinction and code mixing.

My current and past research is funded by the National Science Foundation via NSF DMS 1715377, NSF DMS 1839993.

A more detailed description of some of my research follows.

1) Food chain dynamics: A food chain is a linear sequence of links in a food web. I am

interested in how effects at one trophic level could influence other levels, and thus the

entire chain. For example in past work we show how prey refuge in middle predator/prey

influences top predator dynamics, in a three species food chain. This is currently being

investigated via the ISMaRT program at ISU.

2) Invasive species control: The spread and control of alien species is an important problem in ecology.

I am interested in biological control methods to this end. Herein, a secondary population is introduced into an ecosystem

with a goal to control a target population. This could be via depredation, such as in pest control by natural enemies. This could

also be via manipulation of the mating system, such as the via the Trojan Y Chromosome strategy. Via this method a YY super male or YY feminized super male is introduced into an invasive population with an XX-XY (female-male) sex determinism. Mating with the YY individual only leads to further males or super males, leading to a decline of the female population, and local extinction.

3) Spatial ecology and directed movements: Classical spatially explicit problems in biology and ecology are modeled

via semi linear PDE, where the species are assumed to randomly diffuse, resulting in classical reaction diffusion equations.

I am interested in setting where this movement can be "directed" such as toward a prey/food source, or away from a predator. These result in quasi linear PDE, such as the SKT model or chemotaxis type models. The analysis of such models proves to be

mathematically challenging, with finite time blow up or population "explosion" as a possibility.

4) Language dynamics: Languages compete, just like species, for speakers in a population. I am interested in how such competition is affected by language mixing or code-mixing. I am also interested in language extinction or death, and

how mathematical modeling can inform language preservation efforts.