标题： 连续粒子系统的随机演化：扩散与竞争的微观和介观描述 显示英文标题

标题： Stochastic evolution of a continuum particle system with dispersal and competition: micro- and mesoscopic description

摘要： 在$\mathbb{R}^d$维空间中点粒子系统的马尔可夫演化在微观和介观层次上被描述。 粒子在远处自我繁殖（扩散）并死亡，独立进行且受到彼此的影响（竞争）。 微观描述基于相关函数的无限方程链，类似于在连续粒子系统哈密顿动力学中使用的BBGKY层次结构。 介观描述基于通过缩放过程从上述方程链得到的粒子密度的Vlasov型动力学方程。 微观理论的主要结论是，竞争可以防止系统聚集，这使得用密度来描述系统是合理的。 还讨论了长期极限下动力学方程解的可能均质化。 显示英文摘要

摘要： A Markov evolution of a system of point particles in $\mathbb{R}^d$ is described at micro-and mesoscopic levels. The particles reproduce themselves at distant points (dispersal) and die, independently and under the influence of each other (competition). The microscopic description is based on an infinite chain of equations for correlation functions, similar to the BBGKY hierarchy used in the Hamiltonian dynamics of continuum particle systems. The mesoscopic description is based on a Vlasov-type kinetic equation for the particle's density obtained from the mentioned chain via a scaling procedure. The main conclusion of the microscopic theory is that the competition can prevent the system from clustering, which makes its description in terms of densities reasonable. A possible homogenization of the solutions to the kinetic equation in the long-time limit is also discussed.