State feedback is used to stabilize the Turing instability at the unstable equilibrium point of a discrete competitive Lotka-Volterra system. In addition, a regularization method is applied to parameter inversion for the given Turing system and numerical simulation can verify the effectiveness of the algorithm. Furthermore, how less or more sample data and dependence on the initial state affect estimation procedure are tested.

A new difference system is induced from a differential competition system by different discrete methods. We give theoretical analysis for local bifurcation of the fixed points and derive the conditions under which the local bifurcations such as flip occur at the fixed points. Furthermore, one- and two-dimensional diffusion systems are given when diffusion terms are added. We provide the Turing instability conditions by linearization method and inner product technique for the diffusion system with periodic boundary conditions. A series of numerical simulations are performed that not only verify the theoretical analysis, but also display some interesting dynamics.

In this paper, a semidiscrete logistic model with the Dirichlet boundary conditions and feedback controls is proposed. By means of the sub- and supper-solution method and eigenvalue theory, the unique positive equilibrium is proved. By constructing a suitable Lyapunov function, the global asymptomatic stability of the unique positive equilibrium is investigated. Finally, numerical simulations are presented to verify the effectiveness of the main results.