From Markov jump systems to two species competitive Lotka-Volterra equations with diffusion

In this paper, we start at a random evolution system on biological particles, which is described by a Markov jump system. Under a suitable scaling, we perform a proper approximation procedure. Then the so-called weak convergence of Markov processes and Martingales allow us to establish a (deterministic) two species competitive Lotka-Volterra equation.