Spatially inhomogeneous population dynamics: beyond the mean field approximation

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.