Abstract: For a type of reaction diflusion model, the rate of material decomposition is considered as abranching parameter, and the conditions for the instability of spatially uniform solutions are found throughlinear theory of differential equations, that is, the conditions for the existence of unstable modes. Then,the local branch theory was used to prove the existence of local branches in the model. Secondly, by u-sing the asymptotic analysis method, the model is expanded near the constant solution and studied usingthe Fredholm principle to obtain a non-constant approximate solution at a certain moment. Finally, usingsimulation analysis , the approximate solution obtained is compared with the actual solution of the model toverify the correctness of the theory.

Key words:  local braches;asymptotic analysis, non-constant solutions, apprximate solutions