具有体积排斥的非局部生灭竞争动力学
Nonlocal birth-death competitive dynamics with volume exclusion
摘要：
提出了一个具有排斥体积粒子的随机生灭竞争模型。粒子在规则晶格上移动、复制和消亡。当死亡率是常数时，出生率是空间非局部的，并通过依赖于有限距离内的粒子数来实现粒子间的竞争。粒子的有限体积是通过固定一个上限值来计算的，这个上限值是可以占据一个晶格节点的粒子数量，这就影响了出生和运动。我们推导出封闭的宏观方程的粒子密度和空间相关性在两个相邻的网站。在不同的条件下，描述进一步减少到一个单一的方程的粒子密度，其中包含三个条款：扩散，线性死亡，和一个高度非线性和非本地的出生项。稳态的均匀的解决方案，其稳定性，揭示了空间斑图形成，并与时间相关的均匀的解决方案的动力学进行了讨论和比较，在一维的情况下，与粒子系统的数值模拟。
英文摘要：
A stochastic birth-death competition model for particles with excluded volume is proposed. The particles move, reproduce, and die on a regular lattice. While the death rate is constant, the birth rate is spatially nonlocal and implements inter-particle competition by a dependence on the number of particles within a finite distance. The finite volume of particles is accounted for by fixing an upper value to the number of particles that can occupy a lattice node, compromising births and movements. We derive closed macroscopic equations for the density of particles and spatial correlation at two adjacent sites. Under different conditions, the description is further reduced to a single equation for the particle density that contains three terms: diffusion, a linear death, and a highly nonlinear and nonlocal birth term. Steady-state homogeneous solutions, their stability which reveals spatial pattern formation, and the dynamics of time-dependent homogeneous solutions are discussed and compared, in the one-dimensional case, with numerical simulations of the particle system.
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