The thesis consists of two projects. In the first part, we develop well-posedness theories for infinite dimensional Markov semigroups associated with a non-degenerate interacting diffusion process. Particularly, we present
a construction of infinite volume semigroups in a domain of unbounded mannifold with respect to the L 2 -norm without specifying a reference measure. Moreover, we obtain smoothing estimates on a countable product of smooth, compact manifolds. In the second part, we study nonlinear reaction-diffusion systems that model population dynamics undergoing Lotka-Volterra type
competitions in a heterogeneous environment. We propose a regularisation strategy of the PDEs to avoid the possibilities of singularities, and prove the global existence of a very weak solution.