Grant: Discovery Grants Program - Individual

The challenge in many mathematical biology models is to understand the consequences of interactions between species on the long-term outcomes of the population. In systems biology cellular networks have species interacting in biochemical reactions. In evolution and ecology, genetic species interact through reproduction and competition for resources. Epidemiology and neuroscience involve interacting individuals, and neurons, respectively. Stochasticity is an important element in all these models as some interactions are essentially stochastic with crucial differences between possible outcomes. Spatial migration also makes these models more complex as local interactions are affected by the time-changing distribution of species. In cellular systems, spatial heterogeneity is due to the localization of certain species, or compartmentalization of certain reactions. In ecological systems, spatial heterogeneity arises from different propagation speeds, spatial sorting and spatial selection. Finally, the dynamics in these models often have a multi-scale nature. For example, species abundances can be of different orders of magnitude (genes vs proteins, mutant vs wild types); interaction rates can be on different time scales (unstable bonds in biochemistry, slow mutations in evolution). The combination of stochasticity, spatial heterogeneity and multiple time scales gives rise to novel mathematical objects and a need to develop new tools to analyze them. My research is on stochastic systems of interacting species, whose dynamics evolve in space and have a multi-scale nature. To analyze them I develop large population limits, derive stationary distributions, and perform model reductions based on their multi-scale nature. With a rigorous mathematical analysis, I derive approximate models with a simpler structure that still retains the essential stochastic features of the original processes. Such approximations provide quantitative predictions of the transient and long-term behaviour of these stochastic processes and enable faster numerical calculations. Each area of biological application is concerned with distinct phenomena impacted by noise and spatial heterogeneity: oscillations and bistability for biochemical species, extinction vs persistence and spatial propagation for ecological ones. The mathematical analysis I propose to study targets a range of models that can provide insight into many biologically relevant questions.