Keller-Segel模型的爆炸解问题的新数值方法

The Keller-Segel system is a chemotaxis model proposed in 1970 to describe aggregation of slime molds in microbiology. It is in the form of a two-dimensional system of nonlinear partial differential equations. That is, a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It is well-known that solutions of the Keller-Segel system may blow up in.finite time or may exhibit a very singular, spiky behavior in addition to travelling wave solutions. In either case, capturing such singular.solutions numerically is a challenging problem although a lot of progress has been made in its qualitative study. Since almost all the theoretical results are obtained by using special structures of its simplest form, the necessity drives us to develop numerical methods for the general form of the Keller-Segel system to meet the requirements of practical engineering. In this project, we will propose and develop a new compact finite difference method with higher order accuracy to the Keller-Segel system. Compact finite difference methods have recently attracted much attention for many nonlinear ordinary and partial differential equations because they can achieve higher order accuracy. The new compact finite difference method that we propose will preserve the positivity property of solutions and total mass conservative law of the system, and will give approximate solutions of third order or higher order accuracy with respect to spatial variables. By taking a semi-discretization to spatial variables, we get a system of nonlinear ordinary differential equations with respect to the temporal variable. Then by using the implicit Runge-Kutta method, we will derive a formula to determine stepwise mesh size so that the approximate solution of the fully discretized system shows the same behavior of the exact solution around the blowup time. Since solutions are singular, the standard Lax-Milgram approach cannot be applied. In this project, we also establish a new numerical analysis approach to get error estimate and super-convergence analysis and so on. Our new method provides a useful tool to take qualitative analysis and has a wide application in the computational fields of many scientific fields such as biology, physics and engineering.

Keller-Segel系统是Keller和Segel在1970年研究微生物学中生物对化学物质的应激性时提出的趋化性数理模型。它是非线性偏微分方程的二元联立方程组，具有聚集解、爆炸解、行波解等丰富的数学结构。相比较于定性理论分析方面的许多研究成果，其数值方法的提案和数值分析的研究还非常少。本项目将开发新型compact finite difference数值方法，既保持Keller-Segel系统的正解性和质量守恒性，同时又实现空间变量的3阶以上的高精度。对时间变量，本项目通过导出非均匀时间剖分的步长公式使得新的数值方法能有效地处理Keller-Segel系统的爆炸解问题。本项目将对开发的新数值方法建立其误差估计、超收敛分析等的数值分析理论。新的数值方法将为Keller-Segel系统的定性分析和定量分析提供强有力的研究工具，在数理生物学、理论物理学和工程应用中获得广泛的应用。

本项目针对Keller-Segel系统数理模型的解具有爆炸解这类数学结构特点，本项目首先研究具有奇异性质的椭圆型偏微分方程的边界值问题，引入多项式和指数伸缩函数，采用网格加密思想分割空间网格，在此网格分割基础上，建立数值方法，并给出相应的严密收敛数值分析。同时在本项目研究期间，课题组对生物学中与Keller-Segel系统相关的，描述捕食现象的Rosenzweig-MacArthur 模型也进行了相关研究。本项目研究分析环境参数对模型的影响、随时间变化的承载能力的影响、狩猎的影响、周期性的狩猎的影响等方面。通过引入衰减函数 ，并探讨了当环境质量的改变时，种群密度是如何变化的。并通过数值仿真得到了Hopf分岔图。