Abstract: In this work, we study the two-species lottery competition model with non-stationary reproduction and mortality rates of  both species is studied.   First a diffusion approximation for the fraction of sites occupied by each adult species is derived as the continuum limit of a classical discrete-time lottery model.   Then a non-autonomous stochastic differential equation on sites occupied by the species, as well as a Fokker-Planck equation on its transitional probability are developed.   Existence, uniqueness, and dynamics of solutions for the resulting stochastic differential equation are investigated, from which sufficient conditions for the existence of a time-dependent limiting process and coexistence of species in the sense of stochastic persistence are established.   A unique classical solution to the Fokker-Planck equation is also proved to exist and shown to be a probability density.  Numerical simulations are presented to illustrate the theoretical results.

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