We consider a system of particles that simultaneously move on a two-dimensional periodic lattice at discrete times steps. Particles remember their last direction of movement and may either choose to continue moving in this direction, remain stationary, or move toward one of their neighbors. The form of motion is chosen based on predetermined stationary probabilities. Simulations of this model reveal a connection between these probabilities and the emerging patterns and size of aggregates. In addition, we develop a reaction–diffusion master equation from which we derive a system of ODEs describing the dynamics of the particles on the lattice. Simulations demonstrate that solutions of the ODEs may replicate the aggregation patterns produced by the stochastic particle model. We investigate conditions on the parameters that influence the locations at which particles prefer to aggregate. This work is a two-dimensional generalization of Galante and Levy (2012), in which the corresponding one-dimensional problem was studied.

我们考虑一个以离散时间步长在二维周期晶格上同时移动的粒子系统。粒子会记住它们最后的运动方向，并且可以选择继续沿该方向运动、保持静止或向其邻居之一移动。基于预定的平稳概率选择运动形式。该模型的模拟揭示了这些概率与聚合体的新兴模式和大小之间的联系。此外，我们开发了一个反应扩散主方程，从中我们可以推导出描述晶格上粒子动力学的 ODE 系统。模拟表明 ODE 的解可以复制随机粒子模型产生的聚集模式。我们研究了影响粒子倾向于聚集的位置的参数条件。这项工作是 Galante 和 Levy (2012) 的二维推广，其中研究了相应的一维问题。