- Department of Mathematics
Stochastic processes, probability, and mathematical ecology.
I study ecological systems with the help of mathematical models. I am especially interested in the growth and dynamics of populations, competitive and predator-prey behavior in food webs, and optimal harvesting.
One of the major tasks of mathematical ecology is to describe the dynamics of populations. In most ecosystems hundreds of different species interact in complex ways. Even a system with two species can exhibit complicated dynamics due to dispersion, seasonal differences, and other factors. The dynamics of species is inherently stochastic due to the random fluctuations of environmental factors. The combined effects of biotic interactions (competition, predation, mutualism) and environmental fluctuations (precipitation, temperature, sunlight) are key when trying to determine species richness. Sometimes biotic effects can result in species going extinct. However, if one adds the effects of a random environment extinction might be reversed into coexistence. In other instances deterministic systems that coexist become extinct once one takes into account environmental fluctuations. A successful way of studying this interplay is modelling the populations as discrete or continuous time Markov processes and looking at the long-term behavior of these processes.
I have used tools from SDE (stochastic differential equations), RDS (random dynamical systems), ODE (ordinary differential equations), PDE (partial differential equations) and optimal control in order to explore the dynamics of populations that are affected by random environmental noise. Some recent projects I have worked on are:
- Lotka-Volterra food chains. These are trophic chains consisting of predators and their prey. What conditions are sufficient and necessary for species to persist or to go extinct?
- Stochastic harvesting. What are the optimal harvesting strategies one should use in order to maximize the harvesting yield of a population living in a stochastic environment?
- Evolutionarily stable strategies (ESS). An ESS is a strategy such that if a population plays this strategy then it cannot get invaded by another population playing a different strategy. Once a population plays an evolutionarily stable strategy, mutant strategies fail to invade. Can we characterize the ESS for populations in stochastic environments?
I am also interested in the long-term behavior of Markov processes (stationary distributions, quasistationary distributions, ergodic properties), diffusion limits of kinetic gases and some aspects of mathematical finance.