具有聚合和排斥跳跃的空间种群动力学模型动力学方程的数值解算法
Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps
摘要：
本文提出了一种求解动力学方程的数值解的算法，该动力学方程描述了放置在$\mathbb{R}^d (d \geq 1)$中的点物品的无限系统。粒子执行随机跳跃与成对的排斥，在这个过程中，他们也可以合并。动力学方程是一个非线性、非局部的积分微分方程，很难用解析法求解。该算法的推导是基于使用的时空离散，边界条件，复合辛普森和梯形法则，龙格-库塔法，可调系统大小的计划，等等，然后该算法被应用到各种初始条件的一维版本的方程。结果表明，对于特殊的模型参数的选择，解决方案可能会有意想不到的时间行为。对所得结果进行了数值误差分析。
英文摘要：
An algorithm is proposed for finding numerical solutions of a kinetic equation that describes an infinite system of point articles placed in $\mathbb{R}^d (d \geq 1)$. The particles perform random jumps with pair wise repulsion, in the course of which they can also merge. The kinetic equation is an essentially nonlinear and nonlocal integro-differential equation, which can hardly be solved analytically. The derivation of the algorithm is based on the use of space-time discretization, boundary conditions, composite Simpson and trapezoidal rules, Runge-Kutta methods, adjustable system-size schemes, etc. The algorithm is then applied to the one-dimensional version of the equation with various initial conditions. It is shown that for special choices of the model parameters, the solutions may have unexpectable time behaviour. A numerical error analysis of the obtained results is also carried out.
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