具有Holling-III的捕食者-食饵系统的简明严格解析解

 捕食关系是自然界中普遍存在的物种间相互作用的基本关系之一，也是生态学界和生物数学界研究的一个重要课题。本文主要对一类比率依赖的、具有Holling第三类功能性反应且具有捕捞的两种群竞争模型进行了巧妙的简化，得到了该模型系统可能存在的代数显式简明严格解析解，它们对提高理论，发展各种数值计算方法具有相应的价值。不同学科的各种基本方程的解析解，历史上对学科的发展曾起过关键的作用。例如各种不可压位流与常系数导热的解析解，就曾是早年流体力学与传热学的基础，后来因为满足一定的初始和边界条件（尤其对复杂的基本方程）的解析解很难找到，再加上电子计算机与数值计算方法的飞速发展，在解决具体问题时就基本转向计算流体力学和计算传热学。但解析解还是有其不可代替的理论价值的，而且它还可以作为标准解推动各种计算方法的发展，以及用来检验各种计算方法与程序的准确性、收敛性与稳定性，也可以启发工作者改进各种差分格式，网格生成技术等。

Exact Analytical Solutions of a Ratio-dependent Capture Predator-Prey System with Holling Type III

The predator-prey relationship is not only a fundamental relation between species in nature, but also very important in ecology and bio-mathematics. In most research fields, which deal with definite objects, the traditional approach is to derive some basic rules based on experiments and observations, then to obtain mathematical equations and their analytical solutions. Numerical solutions are commonly required due to the difficulties to obtain analytical solutions. But analytical solutions have important merits in theory, especially as the benchmark solutions to check the accuracy, the convergence and the stability of various numerical computational methods and to improve their difference schemes, grid generations and so on. Some exact analytical solutions of a ratio-dependent capture predator-prey system with Holling type III functional response are derived in this paper. They can be used as the benchmark solutions to verify the numerical solutions and even to develop various numerical methods.