在本文中，一方面是以自催化的化学反应模型为背景，对一类反应扩散系统的Turing-Hopf分支问题进行了研究，另一方面是以捕食者-食饵系统为背景，对具有空间记忆扩散、具有趋向性以及具有非直接趋向性的系统的Hopf分支问题和Turing-Hopf分支问题进行了研究，其中具有趋向性的系统和具有非直接趋向性的系统可以看作是空间记忆扩散系统中的种群向过去某个时间点为零时的其它种群的密度的梯度方向或者相反梯度方向移动的特殊情形，因此可以将对空间记忆扩散系统进行分析的相关方法推广到这两类系统上。本文的主要内容可以概括为以下六个章节。

第一章，介绍了课题的研究背景和意义、国内外研究现状以及一些基本概念和理论。

第二章，研究了具有基因表达时滞的Brusselator反应扩散模型的Turing-Hopf分支问题。首先，利用定性理论和分支理论得到了该模型在正常数稳态解处的Turing分支、Hopf分支以及Turing-Hopf分支出现的条件。然后，将已有的计算带参数的偏泛函微分方程的三阶截断形式的Turing-Hopf分支的规范型的算法应用于该模型，进而通过一些必要的更改，给出了该模型的Turing-Hopf分支的规范型的系数的具体计算公式。最后，通过选取满足该模型的Turing-Hopf分支出现条件的一些参数，利用MATLAB软件进行辅助计算和数值模拟，验证了理论分析的结果。

第三章，研究了一个一般形式的具有空间记忆扩散的系统中由记忆时滞和其它时滞共同诱发的空间齐次Hopf分支问题。首先，在该系统存在正常数稳态解的假设下，利用中心流形定理和规范型理论推导出了计算该系统的Hopf分支的规范型的算法，并且给出了Hopf分支的规范型的系数的具体计算公式。然后，为了验证算法的有效性，对一个具有记忆时滞和妊娠时滞的捕食者-食饵系统进行了研究，利用定性理论和分支理论得到了该系统在正常数稳态解处的Hopf分支出现的条件。最后，通过选取满足该系统的Hopf分支出现条件的一些参数，利用MATLAB软件进行辅助计算和数值模拟，验证了理论分析的结果。

第四章，研究了一个一般形式的具有妊娠时滞、趋向性以及恐惧效应的捕食者-食饵系统的Turing-Hopf分支问题。首先，在该系统存在正常数稳态解的假设下，选取趋向性系数和妊娠时滞作为分支参数，利用中心流形定理和规范型理论推导出了计算该系统的三阶截断形式的Turing-Hopf分支的规范型的算法，并且给出了Turing-Hopf分支的规范型的系数的具体计算公式。然后，为了验证算法的有效性，对一个具有妊娠时滞、趋向性、恐惧效应以及方根功能性反应函数的捕食者-食饵系统进行了研究，利用定性理论和分支理论得到了该系统在正常数稳态解处的Turing-Hopf分支出现的条件。最后，通过选取满足该系统的Turing-Hopf分支出现条件的一些参数，利用MATLAB软件进行辅助计算和数值模拟，验证了理论分析的结果。

第五章，研究了具有非直接捕食者趋向性的Schoener类型的捕食者-食饵系统中由灵敏度系数诱发的空间非齐次Hopf分支问题。首先，利用定性理论和分支理论得到了该系统在正常数稳态解处的Hopf分支出现的条件。然后，利用中心流形定理和规范型理论推导出了计算该系统的Hopf分支的规范型的算法，并且给出了Hopf分支的规范型的系数的具体计算公式。最后，通过选取满足该系统的Hopf分支出现条件的一些参数，利用MATLAB软件进行辅助计算和数值模拟，验证了理论分析的结果。

第六章，进一步概括了本文的主要结论，并且对之后值得研究的一些问题进行了展望。

In this thesis, on the one hand, with the autocatalytic chemical reaction model as the background, the Turing-Hopf bifurcation of a class of reaction-diffusion system is studied, and on the other hand, with the predator-prey system as the background, the Hopf and Turing-Hopf bifurcations of system with spatial memory diffusion, system with taxis and system with indirect taxis are studied. Notably, system with taxis and system with indirect taxis can be seen as special cases that the population in spatial memory diffusion system moves in the gradient direction or the opposite gradient direction of the density of other population when a certain time point in the past was zero, then the relevant methods for analyzing spatial memory diffusion system can be extended to these two types of systems. The main content of this thesis can be summarized in the following six chapters.

In Chapter 1, the research background and significance, the current state of research both domestically and internationally, as well as basic concepts and theories relevant to the study are introduced.

In Chapter 2, the Turing-Hopf bifurcation of a Brusselator reaction-diffusion model with gene expression delay is studied. Firstly, this model is analyzed using qualitative and bifurcation theories, and conditions for the existence of Turing, Hopf and Turing-Hopf bifurcations at the positive constant steady state are obtained. Then, the existing algorithm for calculating the third-order truncated normal form of Turing-Hopf bifurcation of the partial functional differential equation with parameters is applied to this model, and specific calculation formulas for the coefficients of the normal form of Turing-Hopf bifurcation of this model are given through some necessary changes. Finally, by selecting some parameters that satisfy the conditions for the existence of Turing-Hopf bifurcation of this model, the MATLAB software is used for auxiliary calculations and numerical simulations, which verifies the results of the theoretical analysis.

In Chapter 3, the spatial homogeneous Hopf bifurcation induced by memory and other delays in a general form of system with spatial memory diffusion is studied. Firstly, under the assumption that this system has positive constant steady state, the algorithm for calculating the normal form of Hopf bifurcation of this system is derived by using the center manifold theorem and normal form theory, and the specific calculation formulas for the coefficients of the normal form of Hopf bifurcation are given. Then, in order to verify the effectiveness of the derived algorithm, a predator-prey system with memory delay and gestation delay is studied. This system is analyzed by using qualitative and bifurcation theories, and the conditions for the existence of Hopf bifurcation at the positive constant steady state are obtained. Finally, by selecting some parameters that satisfy the conditions for the existence of Hopf bifurcation of this system, the MATLAB software is used for auxiliary calculations and numerical simulations, which verifies the results of the theoretical analysis.

In Chapter 4, the Turing-Hopf bifurcation of a general form of predator-prey system with gestation delay, taxis and fear effect is studied. Firstly, under the assumption that this system has positive constant steady state, by selecting the taxis coefficient and gestation delay as bifurcation parameters, the algorithm for calculating the third-order truncated normal form of Turing-Hopf bifurcation of this system is derived by using the center manifold theorem and the normal form theory, and the specific calculation formulas for the coefficients of the normal form of Turing-Hopf bifurcation are given. Then, in order to verify the effectiveness of the derived algorithm, a predator-prey system with gestation delay, taxis, fear effect and square root functional response function is studied. This system is analyzed by using qualitative and bifurcation theories, and the conditions for the existence of Turing-Hopf bifurcation at the positive constant steady state are obtained. Finally, by selecting some parameters that satisfy the conditions for the existence of Turing-Hopf bifurcation of this system, the MATLAB software is used for auxiliary calculations and numerical simulations, which verifies the results of the theoretical analysis.

In Chapter 5, the spatial inhomogeneous Hopf bifurcation induced by the sensitivity coefficient in a Schoener's type predator-prey system with indirect predator-taxis is studied. Firstly, this system is analyzed by using qualitative and bifurcation theories, and the conditions for the existence of Hopf bifurcation at the positive constant steady state are obtained. Then, the algorithm for calculating the normal form of Hopf bifurcation of this system is derived by using the center manifold theorem and the normal form theory, and the specific calculation formulas for the coefficients of the normal form of Hopf bifurcation are given. Finally, by selecting some parameters that satisfy the conditions for the existence of Hopf bifurcation of this system, the MATLAB software is used for auxiliary calculations and numerical simulations, which verifies the results of the theoretical analysis.

In Chapter 6, the main conclusions of this thesis are further summarized, and some problems worthy of further study are prospected.