  Motivated by the traditional Lotka-Volterra competitive models, this paper
proposes and analyzes a class of stochastic reaction-diffusion partial
differential equations. In contrast to the models in the literature, the new
formulation enables spatial dependence of the species. In addition, the noise
process is allowed to be space-time white noise. In this work, wellposedness,
regularity of solutions, existence of density, and existence of an invariant
measure for stochastic reaction-diffusion systems with non-Lipschitz and
non-linear growth coefficients and multiplicative noise are considered. By
combining the random field approach and infinite integration theory approach in
SPDEs for mild solutions, analysis is carried out. Then this paper develops a
Lotka-Volterra competitive system under general setting; longtime properties
are studied with the help of newly developed tools in stochastic calculus.
