We propose and study a model describing an infinite population of point entities arriving in and departing from \(X=\mathbb {R}^d\), \(d\ge 1\). The already existing entities force each other to leave the population (repulsion) and attract the newcomers. The evolution of the population states is obtained by solving the corresponding Fokker-Planck equation. Without interactions, the evolution preserves states in which the probability \(p(n,\Lambda )\) of finding n points in a compact vessel \(\Lambda \subset X\) obeys the Poisson law. As we show, for pure attraction the decay of \(p(n,\Lambda )\) with \(n\rightarrow +\infty \) may be essentially slower. The main result is the statement that in the presence of repulsion—even of an arbitrary short range—the evolution preserves states in which the decay of \(p(n,\Lambda )\) is at most Poissonian. We also derive the corresponding kinetic equation, the numerical solutions of which can provide more detailed information on the interplay between attraction and repulsion. Further possibilities in studying the proposed model are also discussed.

我们提出并研究了一个模型，该模型描述了到达和离开\(X=\mathbb {R}^d\) , \(d\ge 1\)的无限数量的点实体。已经存在的实体相互迫使对方离开人口（排斥）并吸引新来者。种群状态的演化是通过求解相应的福克-普朗克方程获得的。在没有相互作用的情况下，进化保留了在紧凑容器\(\Lambda \subset X\)中找到n个点的概率\(p(n,\Lambda ) \)遵守泊松定律的状态。正如我们所展示的，对于纯粹的吸引力，\(p(n,\Lambda )\)与\(n\rightarrow +\infty \)的衰减可能本质上更慢。主要结果是这样的陈述，即在存在排斥的情况下——即使是任意短距离——演化保持了\(p(n,\Lambda )\)的衰减至多是泊松分布的状态。我们还推导出相应的动力学方程，其数值解可以提供有关吸引力和排斥力之间相互作用的更详细信息。还讨论了研究所提出模型的进一步可能性。