中文摘要：

生物入侵的机制与控制已经成为生态学关注的重点和研究的热点问题。建立数学模型是解释生物入侵机制的主要手段。对外来生物包括疾病的入侵扩散过程进行模型分析不仅具有重要的理论意义，还有助于风险评估。特别对入侵进行早期预测、控制、科学管理与防治。然而，目前有关生物入侵的数理模型方法还远不能满足实际的需要，本文系统地介绍和分析了生物入侵模型的研究现状。首先分析了对于生物入侵过程建模所要考虑的基本问题：然后讨论了几类典型的入侵数学模型，分析了各自的特点及条件，同时给出了有代表性的实例和最近的研究成果；其次比较列出了入侵模型分析的各类方法：最后探讨了生物入侵建模与求解过程中尚待解决的问题。并对未来研究进行了展望。

英文摘要：

Biological invasion is now one of the six hot research topics in modern ecology. Modeling the invasion processes, the diffusions and the spreading mechanisms of invasive species including diseases is important not only for theoretical purposes, but also for practical purposes, such as in ecological risk assessments, and optimal control and management of those invasive species. In this paper, the widely adopted models of biological invasion are systematically reviewed such as diffusion-reaction equations, integral-difference models, space-discrete models, and stochastic models. The fundamental issues of modeling biological invasions are discussed, and the characteristics and analytic conditions of various models are analyzed with typical examples and recent research results. Various modeling techniques are compared, and some commonly used methods in solving biological invasion models are outlined, including numerical computing and simulating, exact solutions, methods for functional differential equation and qualitative analysis of differential equations. Some key and unsolved problems in building mathematical models of biological invasion, and future research directions are also discussed.