Abstract
We study for any $NınN$N∈N a population of about $N$N individuals in $R^d$Rd, which is divided into $K$K subpopulations. The individuals may move around in $R^d$Rd, may die, or give birth to new individuals, and may change their subpopulation. We rescale the interaction between the individuals in a suitable (moderate) way, as the population size tends to infinity. Essentially, this means that for any fixed particle the drift coefficients, the birth, death and transition rates depend on the configuration of the remaining particles in a neighbourhood, which is macroscopically small, i.e., its volume tends to 0 as $N\toınfty$N→∞, and microscopically large, i.e., it contains an arbitrarily large number of individuals as $N\toınfty$N→∞. It is shown that for large $N$N the empirical processes of the different subpopulations converge to the solution of a system of reaction diffusion equations. For that we consider regularized versions of these empirical processes and study their asymptotic properties as $L_2(R^d)$L2(Rd)-valued stochastic processes.
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