具有自由边界和Logistic源的趋化系统的消失和扩散动力学的数值研究
Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary
摘要：
本文研究一维空间中具有自由边界的Logistic型趋化性模型的数值逼近问题。这种具有自由边界的模型描述了新物种或入侵物种在具有代表传播前沿的自由边界的环境中受某些化学物质影响的传播（参见Bao和Shen [1]，[2]）。数值研究中的主要挑战在于追踪运动的自由边界和化学非线性项。为了克服这些问题，引入了前固定框架与有限差分法耦合。讨论了该方法的精度、解的正性和格式的稳定性，数值模拟结果与理论结果如消失扩散二分法、局部持久性和稳定性等一致.这些模拟也验证了我们未来理论研究中的一些结果，如消失-扩散二分法对初始解u 0、初始生境h 0、移动速度ν和趋化敏感性系数\c{hi} 1、\c{hi} 2的依赖性.
英文摘要：
The current paper is to investigate the numerical approximation of logistic type chemotaxis models in one space dimension with a free boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front (see Bao and Shen [1], [2]). The main challenges in the numerical studies lie in tracking the moving free boundary and the nonlinear terms from chemical. To overcome them, a front fixing framework coupled with finite difference method is introduced. The accuracy of the proposed method, the positivity of the solution, and the stability of the scheme are discussed.The numerical simulations agree well with theoretical results such as the vanishing spreading dichotomy, local persistence, and stability. These simulations also validate some conjectures in our future theoretical studies such as the dependence of the vanishing-spreading dichotomy on the initial solution u0, initial habitat h0, the moving speed ν and the chemotactic sensitivity coefficients \c{hi}1,\c{hi}2.
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