The optimality Kuhn-Tucker condition and the wolfe duality for the preinvex set-valued optimization are investigated. Firstly, the concepts of alpha-order G-invex set and the alpha-order S-preinvex set-valued function were introduced, from which the properties of the corresponding contingent cone and the alpha-order contingent derivative were studied. Finally, the optimality Kuhn-Tucker condition and the Wolfe duality theorem for the alpha-order S-preinvex set-valued optimization were presented with the help of the alpha-order contingent derivative.

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Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneous steady state is obtained by the linearized theory. A necessary condition for bifurcations in spatially non-uniform solutions in uncoupled and coupled systems is given. Further information about Turing pattern solutions near bifurcation points is obtained by weakly nonlinear theory. Finally, the stability of equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches of the weakly coupled system.