Qualitative analysis on positive steady-state solutions for an autocatalysis model with high order

This paper is concerned with an autocatalysis model with high order under Neumann boundary conditions. Firstly, the stability of the equilibrium is discussed and the effect of diffusion coefficients on Turing instability is described. Next by maximum principle and Poincaré inequality, a priori estimates and some characters of positive steady-state solutions are given. Moreover, the bifurcations at both simple and double eigenvalues are intensively investigated. Using the bifurcation theory, we establish the global structure of the bifurcation from simple eigenvalues and obtain some conditions to determine the bifurcation direction. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues. Finally, some numerical simulations are shown to supply and supplement the analytical results.