We consider a system of two reaction-diffusion equations in a bounded domain of the \( m \)-dimensional space with Neumann boundary conditions on the boundary for which the reaction terms \( f(u,v) \) and \( g(u,v) \) depend on two parameters \( a \) and \( b \). Assume that the system has a spatially homogeneous solution \( (u_{0},v_{0}) \), with \( f_{u}(u_{0},v_{0})>0 \) and \( -g_{v}(u_{0},v_{0})=F(\operatorname{Det}(\operatorname{J})) \), where \( \operatorname{J} \) is the Jacobian of the corresponding linearized system in the diffusionless approximation and \( F \) is a smooth monotonically increasing function. We propose some method for the analytical description of the domain of necessary and sufficient conditions of Turing instability on the plane of system parameters for a fixed diffusion coefficient \( d \). Also, we show that the domain of necessary conditions of Turing instability on the plane \( (\operatorname{Det}(\operatorname{J}),f_{u}) \) is bounded by the zero-trace curve, the discriminant curve, and the locus of points \( \operatorname{Det(\operatorname{J})}=0 \). Explicit expressions are found for the curves of sufficient conditions and we prove that the discriminant curve is the envelope of the family of these curves. It is shown that one of the boundaries of the Turing instability domain consists of the fragments of the curves of sufficient conditions and is expressed in terms of the function \( F \) and the eigenvalues of the Laplace operator in the domain under consideration. We find the points of intersection of the curves of sufficient conditions and show that their abscissas do not depend on the form of \( F \) and are expressed in terms of the diffusion coefficient and the eigenvalues of the Laplace operator. In the special case \( F(\operatorname{Det}(\operatorname{J}))=\operatorname{Det}(\operatorname{J}) \). For this case, the range of wave numbers at which Turing instability occurs is indicated. We obtain some partition of the semiaxis \( d>1 \) into half-intervals each of which corresponds to its own minimum critical wave number. The points of intersection of the curves of sufficient conditions lie on straight lines independent of the diffusion coefficient \( d \). By way of applications of the statements proven, we consider the Schnakenberg system and the Brusselator equations.

我们考虑在\( m \)维空间有界域中的两个反应扩散方程组，其边界上具有诺依曼边界条件，其中反应项 \( f(u,v) \)和 \( g (u,v) \) 取决于两个参数 \( a \)和 \( b \)。假设系统有一个空间齐次解\( (u_{0},v_{0}) \)，其中\( f_{u}(u_{0},v_{0})>0 \)和\( -g_{v}(u_{0},v_{0})=F(\operatorname{Det}(\operatorname{J})) \)，其中\( \operatorname{J} \)是无扩散近似中相应的线性化系统，并且 \( F \) 是一个平滑单调递增函数。我们提出了一些方法来解析描述固定扩散系数\( d \)的系统参数平面上图灵不稳定性的充分必要条件域 。此外，我们还证明了平面上图灵不稳定性的必要条件域\( (\operatorname{Det}(\operatorname{J}),f_{u}) \)由零迹曲线界定，判别式曲线，以及点的轨迹\( \operatorname{Det(\operatorname{J})}=0 \)。找到了充分条件曲线的显式表达式，并证明判别曲线是这些曲线族的包络。结果表明，图灵不稳定域的边界之一由充分条件曲线的片段组成，并用函数 F和所考虑域中拉普拉斯算子的特征值来表示。我们找到了充分条件曲线的交点，并表明它们的横坐标不依赖于 \(F\)的形式，而是用扩散系数和拉普拉斯算子的特征值来表示。在特殊情况下 \( F(\operatorname{Det}(\operatorname{J}))=\operatorname{Det}(\operatorname{J}) \)。对于这种情况，指示了发生图灵不稳定性的波数范围。我们将半轴\( d>1 \)划分为半间隔，每个半间隔对应于其自己的最小临界波数。充分条件曲线的交点位于与扩散系数 \(d\)无关的直线上。通过应用已证明的陈述，我们考虑 Schnakenberg 系统和 Brusselator 方程。