Executive Summary : | A fundamental question in biology is to understand how interactions among organisms (or even cells) scale to higher levels of organizations like groups, populations and even ecosystems. This question offers interesting mathematical challenges: First, the behaviors of organisms are stochastic. Further, interactions between organisms are local and are driven by both the intrinsic stochastic factors as well as those arising from environmental noise. To address these questions, the tools of nonlinear dynamics, nonequilibrium statistical physics and stochastic processes all come in handy. Therefore, over the past several decades, this theme of research has evoked strong interest among theoretical physicists and applied mathematicians. Despite offering enormous insights to the above broad biological question, the classic mathematical analytical approaches such as mean-field approximations or field-theoretic approaches fail to capture some important biological features. Of these, we primarily highlight two key limitations: (1) First, these approximations often work very well only for large population sizes (since they assume the limit of infinite population sizes). In other words, they ignore the noise arising from finite system sizes, also called intrinsic noise. (2) Secondly, organisms are implicitly assumed to be identical, thus ignoring organism to organism (or cell to cell) variation in their traits. These two characteristics are often not of relevance for physical systems but many recent studies – including those done in our lab – show that ignoring these may lead to missing important biological insights. In this context, the main goal of this proposal is to build and analyse mathematical models in the context of ecosystems that account for intrinsic noise and heterogeneity in organismal trait variations. We will develop stochastic agent-based models in the ecological contexts such as multi-species systems or classic ecosystems like savanna ecosystems which show bistability, hysteresis, and other nonlinear features. We will derive stochastic differential equations that correspond to the above agent-based models. To do so, we will employ the chemical Langevin framework which can explicitly account for stochasticity in the behaviour of organisms, probabilistic nature of interactions between organisms/species and importantly, it can also account for the stochasticity arising from finite sizes of real biological populations. Using a combination of analytical approximations such as chemical Langevin and Fokker Planck equations, and numerical solutions, we will obtain how population dynamics is affected by important biological features of intrinisic noise and heterogeneity. We expect that our work will emphasize analytical techniques from stochastic calculus and nonequilibrium statistical physics to ecological modelling. We expect to obtain novel insights on how intrinsic noise and population trait heterogeneity can influence ecosystem dynamics. |