Organisms often need to adapt more efficiently and devise new strategies for surviving difficult ecological circumstances. Mammals indeed spend the winter in hibernation to conserve energy, food,... Show moreOrganisms often need to adapt more efficiently and devise new strategies for surviving difficult ecological circumstances. Mammals indeed spend the winter in hibernation to conserve energy, food, etc., for future purposes. Microbial populations also possess similar characteristics, where organisms enter into a state of low metabolic activity in response to adverse environmental conditions. In plant populations, the analogous strategy is the suspension of seed germination for an extended period of time. Several studies suggest that this bet-hedging strategy has important evolutionary consequences and plays a crucial role in maintaining genetic diversities in a population. In this thesis, we draw motivations from biological populations featuring this trait and investigate its effect in a probabilistic framework. In particular, we introduce a mathematical notion of dormancy in several well-known stochastic interacting systems and study how it changes the qualitative and quantitative properties of the systems by characterizing their behaviors in the long run. The construction of our model is built upon a well-known stochastic process in mathematical population genetics called the Moran model. The Moran model describes the genetic evolution of a single, reproductively active, finite population without seed-bank. We modify the model to include dormancy and extend it to the context of spatially structured populations with varying sizes. Show less
We study stochastic models of non-equilibrium which are exactly solvable with the technique of duality and self-duality. The models include a new class of particle systems which are bosonic, i.e.,... Show moreWe study stochastic models of non-equilibrium which are exactly solvable with the technique of duality and self-duality. The models include a new class of particle systems which are bosonic, i.e., models where there is an attractive interaction between the particles and as a consequence condensation phenomena can occur. Our models belong to the class of interacting particle systems, or systems of interacting diffusions. Via the technique of duality, we connect models of interacting diffusions (e.g. Brownian Momentum Process) to simpler interacting particle systems (e.g. Symmetric Inclusion Process), both in equilibrium and non-equilibrium settings. Part of the thesis is devoted to develop a general formalism for duality. Show less
