THE STRUCURE OF ATTRACTORS AND TRANSIENT DYNAMICS OF MULTI-COMPONETS REACTION-DIFFUSION SYSTEMS

(1)研究了多种群Lotka-Volterra竞争模型和捕食者-食饵模型。我们主要关注描述外来物种入侵情况的行波及其传播速度。(I)对于两种群Lotka-Volterra竞争模型，我们证明了存在两种类型的行波，即推挽锋，并从解析和数值上讨论了行波速度的参数依赖性。(Ii)对于两种群Lotka-Volterra捕食者-食饵模型，我们从数值上得到了捕食者和被捕食者都能入侵到开放空间的条件。研究了三种群Lotka-Volterra竞争系统和两种群捕食者-食饵系统的解的混沌行为，得到的结果有助于我们对非线性结构的数学理解，从而保证了两种群系统解的相似的复杂行为。(2)研究了两组分自催化反应模型的推拉前锋。(I)对于二阶和三阶混合自催化反应，证明了当反应物不能扩散时，行波的存在。(Ii)对于纯高阶自催化反应，我们考虑了反应物和自催化具有相同扩散系数和反应物不扩散两种情况。对于这两种情况，我们都证明了拉前线的存在，并得到了最小波速的估计，表明当自催化级数趋于无穷大时，最小波速趋于零。(Iii)对于自催化衰变级数小于1级的高阶自催化反应，也证明了类似的结果。

(1) We investigated the multi-species Lotka-Volterra competition model and predator-prey model. Our main concern is travelling waves and its propagation speeds which describe the situation of the invasion of the external species.(i) For the two species Lotka-Volterra competition model, we proved that there exist two types of travelling waves, that is, pushed and pulled fronts, and discussed the parameter dependence of wave speeds analytically and numerically.(ii) For the two species Lotka-Volterra predator-prey model, we obtained the numerically the condition which enable both of predator and prey to invade into an open space. This result gives us an important information on further analytical studies(iii) We confirm numerically the chaotic behavior of solutions of the three species Lotka-Volterra competition system and the two species predator-prey system, the obtains results help our mathematical understanding of the nonlinear structure which assures the similar complex behavior of solutions of both systems.(2) We studies the pushed and pulled fronts of the two components autocatalytic reaction models.(i) For the mixed second and third order autocatalytic reaction, we proved the existence of traveling waves when the reactant cannot diffuse, and gave an estimate of the minimal wave speeds which depends on the ratio of the mixed reactions.(ii) For the pure higher order autocatalytic reaction, we consider the two cases that the reactant and the autocatalysis have the same diffusivity and that the reactant does not diffuse. For both cases, we proved the existence of pulled fronts and obtained the estimate of the minimal wave speeds which showed that the minimal wave speed tends to zero as the order of autocatalysis goes to infinity.(iii) The similar results was proved for the higher order autocatalytic reaction with decay of autocatalysis whose order of decay is less of 1 than the order of autocatalysis.