生物数学是近现代应用数学中有着重大进展和发展潜力的领域，其发展所涉数学工具几乎遍布数学所有分支，其中一个重要工具便是偏微分方程。捕食者-食饵模型是生物数学研究中一类重要模型，然而一般模型都假设物种资源分布在空间中是均匀的。本项目具体研究内容包括：1）鉴于空间因素在种群演化中的重要性，考虑空间非均匀环境，特别是非局部扩散和对流-扩散等生态因素，建立相应的捕食者-食饵扩散模型；2）利用非线性分析、偏微分方程等方法研究捕食者-食饵扩散模型平衡态的存在性、惟一性、稳定性及系统长时间行为等问题，讨论重要参数对模型动力学行为的影响，阐明种群存活、灭绝与共存过程中的动力学规律，揭示空间非均匀环境资源下物种演化的数学机制；3）基于MATLAB平台，建立辅助分析空间种群模型的有限差分算法，从数值仿真角度研究模型动力学行为。该项目研究及其成果对于丰富种群动力学模型研究内涵，推动偏微分方程的应用具有重要意义。

Biological mathematics is a field of great progress and development potential in contemporary and modern applied mathematics. The mathematical tools involved in the development of biological mathematics cover almost all branches of mathematics, and one of the important tools is partial differential equation. The predator-prey model is an important model in the study of biological mathematics, but the general model assumes that the distribution of species resources in space is homogeneous. The contents of this research mainly include three parts: .1) Given the importance of space in population evolution, considering spatially heterogeneous environment, we establish some predator-prey diffusion models, especially the non-local diffusion and convection-diffusion model; .2) By using the nonlinear analysis method and the theories of partial differential equations, we study the results of existence, uniqueness, stability of the solutions and long time behavior of the models, discuss the effects of the main parameters of the models on the dynamic behavior of the models, clarify the kinetics of survival, extinction and coexistence, and expose the mechanism of population evolution in spatially heterogeneous environment; .3) A new finite difference algorithm is established to assist the analysis of spatial and temporal dynamics of the spatial population models. From the perspective of numerical simulation, the dynamical behaviors of the models are studied. .The results are of great significance to enrich the research contents of population dynamics model and promote the application of partial differential equation.