Amplitude And Dynamics Analysis Of Reaction-diffusion Sysytems

In this paper, by using computer symbolic computation, we derive theamplitude equation from reaction-diffusion system, and using the amplitudeequation we analysis the pattern formation and selection of a ratio-dependentepidemic model. The main results are as follows:In Chapter 1, we introduce some background knowledge and the state ofstudy of amplitude equation and pattern dynamics. In Chapter 2, we establish anew mechanical algorithm AE_Hopf for calculating the amplitude equation nearHopf bifurcation based on the method of normal form approach in Maple. Andthe results indicate that the algorithm is simple and effective. In Chapter 3, usingmultiple scale analysis we derive the amplitude equation from a ratio-dependentepidemic model near Turing bifurcation. In Chapter 4, based on the previouschapter and using weakly nonlinear analysis, we analysis the pattern formationof the epidemic model. Furthermore, we present novel numerical evidence oftypical Turing patterns, and find that the model dynamics exhibits complexpattern replication: on increasing the control parameter r , the sequence H_Ï€-hexagonsâ†'H_Ï€-hexagon-stripe mixturesâ†'stripesâ†'H₀-hexagon-stripe mixturesâ†'H₀-hexagons is observed. This may enrich the Turing pattern 0 Hformation in the spatial epidemic model. In Chapter 5, the discussions andremarks are given.