假设个体所面临的种内竞争环境随年龄而异, 提出并分析一类结 合空间扩散和个体年龄等级结构的种群模型, 由非线性二阶偏微分积分方程描述. 在合理的参数条件下, 运用不动点方法 和线性系统的相关结果确立了模型解的存在唯一性及非负有界性, 导出了解的比 较原则, 分析了变量分离型解. 所获成果一方面拓展了一些已有文献的工作, 另一方 面也为研究系统的稳定性、可控性和最优控制问题奠定基础.

Assuming that the internal competition environment is decided by individual's age, we formulate and analyze a class of population model combining the spatial diffusion with the age hierarchy, which is described by the second-order integro-partial differential equation. Based upon some rational assumptions on the model parameters, the existence, uniqueness, non-negativity and boundedness of solutions to the model are established by means of the approach of Banach's fixed-points and the corresponding result in linear systems. In order to go through the process, we first freeze the environment variable in the vital rates and introduce a linear model. Then we make the key technic estimation for the solution mapping, which enables us to define the equivalent norm on the space of the object functions. The fixed point of the solution mapping is exactly the solution for the hierarchical population model. Furthermore, we derive the comparison principle and analyze the separable form of the solutions. The results obtained not only extend some existing works in the literature, but also provide a necessary foundation for the study of stability, controllability and optimal control problems.