We consider a system of two reaction-diffusion equations ina bounded domain of the $ m $ -dimensional spacewith Neumann boundary conditionson 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 Jacobianof 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 domainof necessary and sufficient conditions ofTuring instability on the plane of system parametersfor a fixed diffusion coefficient $ d $ .Also, we show that the domainof necessary conditions of Turing instability onthe 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 ofsufficient conditions and we prove that the discriminant curve isthe envelope of the family of these curves.It is shown that one of the boundaries of the Turing instability domainconsists of the fragments of the curves of sufficient conditionsand is expressed in terms of the function $ F $ and the eigenvaluesof the Laplace operator in the domain under consideration.We find the points of intersection of the curves of sufficient conditionsand show that their abscissas do not depend onthe form of $ F $ and are expressed in terms ofthe 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-intervalseach of which corresponds to its own minimum critical wave number.The points of intersection of the curves of sufficient conditions lieon 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.
