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Department of Mathematics
503 Boston Avenue
Medford, MA 02155
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:
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.
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