  Our motivation comes from the large population approximation of individual
based models in population dynamics and population genetics. We propose a
general method to investigate scaling limits of finite dimensional population
size Markov chains to diffusion with jumps. The statements of tightness,
identification and convergence in law are based on the convergence of suitable
characteristics of the transition of the chain and strongly exploit the
structure of the population processes defined recursively as sums of
independent random variables. These results allow to reduce the convergence of
characteristics of semimartingales to analytically tractable functional spaces.
We develop two main applications. First, we extend the classical Wright-Fisher
diffusion approximation to independent and identically distributed random
environment. Second, we obtain the convergence in law of generalized
Galton-Watson processes with interactions and random environment to the
solution of stochastic differential equations with jumps.
