  We propose a novel method for numerical modeling of spatially inhomogeneous
moment dynamics of populations with nonlocal dispersal and competition in
continuous space. It is based on analytically solvable decompositions of the
time evolution operator for a coupled set of master equations. This has allowed
us -- for the first time in the literature -- to perform moment dynamics
simulations of spatially inhomogeneous systems beyond the mean-field approach
and to calculate the inhomogeneous pair correlation function using the Kirkwood
superposition ansatz. As a result, we revealed a number of new subtle effects,
possible in real populations. Namely, for systems with short-range dispersal
and mid-range competition, strong clustering of entities at small distances
followed by their deep disaggregation at larger separations are observed in the
wavefront of density propagation. For populations in which the competition
range is much shorter than that of dispersal, the pair correlation function
exhibits a long-tail behavior. Remarkably, the latter effect takes place only
due to the spatial inhomogeneity and thus was completely unknown before.
Moreover, both effects get stronger in the direction of propagation. All these
types of behavior are interpreted as a trade-off between the dispersal and
competition in the coexistence of reproductive pair correlations and the
inhomogeneity of the density of the system.
