当今，生物群落之间有着许多复杂的关系，因此，人们对许多生物关系建立了数学模型，运用现代数学理论对其不断研究，取得了很多丰富的研究成果. 种群捕食关系的存在以及重要性，更是受到了国内外学者的关注. 在研究大多数生物数学模型时，通常运用反应扩散方程理解种群的动力学行为.

       本文应用偏微分方程理论中的特征值理论、分歧理论等理论方法，研究了空间非均匀环境中的一类捕食被捕食模型的动力学行为. 首先，研究扩散速率与空间非均匀环境的联合作用如何影响该系统半平凡解的稳定性. 在该模型中，我们假设被捕食者的内部增长率在空间上是均匀的，而栖息地的承载能力在空间上是非均匀的. 我们发现，与空间均匀环境中捕食被捕食模型相比，此类模型的半平凡解的稳定性具有很大的差别. 研究表明，该半平凡解的稳定性取决于一个具有Nuemann边界条件下的特征值问题的主特征值的符号. 当被捕食者的死亡率很小时，对于任意的捕食者与被捕食者的扩散速率，该半平凡解都是不稳定的；当捕食者的死亡率处于某个中间区域时，随着被捕食者的扩散速率由小变大，半平凡解的稳定性至少可以改变一次，而半平凡解的稳定性与捕食者的扩散速率无关；当捕食者的死亡率处于其它中间区域时，存在一个特定的捕食者扩散速率，当捕食者扩散速率大于这个特定值时，该半平凡解是稳定的，当捕食者扩散速率大于这个特定值时，随着被捕食者的扩散速率由小变大，半平凡解的稳定性至少可以改变一次；当被捕食者的死亡率很大时，对于任意的捕食者与被捕食者的扩散速率，该半平凡解都是稳定的.

       然后，基于上述研究的空间非均匀环境中的一类捕食被捕食模型的半平凡解的稳定性以及合理的假设，首先我们以被捕食者的扩散系数作为一个分歧参数，运用局部分歧理论得到该模型多个正稳态解以及局部稳定性；其次，我们以捕食者的扩散系数作为一个分歧参数，应用局部和全局分歧理论，获得该模型多个正稳态解以及正稳态解的局部稳定性，同时借助先验估计等，我们可以将局部正稳态解延拓到全局正稳态解. 研究发现在非均匀环境中，对于一定范围的捕食者死亡率和扩散系数，模型的半平凡解可以分歧出多个正稳态解，而在均匀环境中，该模型的半平凡解只分歧一个正稳态解.

       最后，运用有限差分法对一类空间非均匀环境中捕食被捕食模型的正稳态解进行数值模拟并生成可视化图形，对该模型动力学行为进行验证并分析. 另一方面，对广义上的一类空间非均匀环境中捕食被捕食模型的正稳态解，进行数值模拟并对该模型的一些动力学行为进行预测.

       Nowadays, there are many complex relationships between biological communities, so people have established mathematical models of many biological relationships, used modern mathematical theories to continuously study them, and achieved many rich research results. The existence and importance of population predation relationships have attracted the attention of scholars at home and abroad. When studying most biomathematical models, reaction-diffusion equations are often employed to understand the dynamic behavior of populations.

       In this paper, the eigenvalue theory and bifurcation theory in partial differential equation theory are applied to study the local dynamic behavior of a class of predator-prey models in a spatial heterogeneous environment. Firstly, the combined effect of diffusion rate and spatial heterogeneous environment is studied how the stability of the semi-trivial solution of the system is studied. In this model, we assume that the internal growth rate of predators is spatially homogeneous, while the carrying capacity of habitats is spatially non-uniform. We find that the stability of semi-trivial solutions of such models is very different from that of predator-prey models in a spatially homogeneous environment. It is shown that the stability of this semi-trivial solution depends on the sign of a principal eigenvalue problem with the Nuemann boundary condition. When the mortality rate of the predator is small, the semi-trivial solution is unstable for arbitrary predators and the rate of dispersion of predators; When the mortality rate of the predator is in a certain intermediate region, the stability of the semi-trivial solution can be changed at least once as the predator's diffusion rate changes from small to large, while the stability of the semi-trivial solution has nothing to do with the predator's diffusion rate. When the mortality rate of the predator is in other intermediate regions, there is a specific predator diffusion rate, when the predator diffusion rate is greater than this specific value, the semi-trivial solution is stable, when the predator diffusion rate is greater than this specific value, as the predator diffusion rate changes from small to large, the stability of the semi-trivial solution can be changed at least once; When the mortality rate of predators is large, this semi-trivial solution is stable for both arbitrary predators and the rate of spread of predators.

       Then, based on the stability and reasonable assumptions of the semi-trivial solution of a type of predator-prey model in the spatial heterogeneous environment of the above research, firstly, we take the predator diffusion coefficient as a bifurcation parameter, and use the local bifurcation theory to obtain multiple positive steady state solutions and local stability of the model. Secondly, we take the predator diffusion coefficient as a bifurcation parameter, apply local and global bifurcation theory, and obtain multiple positive steady state solutions and the local stability of positive steady state solutions of the model, and with the help of prior estimation, we can extend the local positive steady state solution to the global positive steady state solution. It is found that in a heterogeneous environment, the semi-trivial solution of the model can diverge into multiple positive steady state solutions for a certain range of predator mortality and diffusion coefficient, while in the heterogeneous environment, only one positive steady state solution is diverged by the semi-trivial solution of the model.

       Finally, the finite difference method is used to numerically simulate the positive solution of the predator-prey model in a class of spatial heterogeneous environment and generate a visual graph, and the dynamic behavior of the model is verified and analyzed. On the other hand, the positive solution of the predation model in a generalized spatial heterogeneous environment is numerically simulated and some dynamic behaviors of the model are predicted.

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