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

(1)研究了多物种Lotka-Volterra竞争模型和捕食者-猎物模型。我们主要关注的是行波及其传播速度，它描述了外来物种入侵的情况。(i)对于两种Lotka-Volterra竞争模型，证明存在推拉锋面两种行波，并对波速的参数依赖性进行了解析和数值讨论。（ii）对于Lotka-Volterra两种捕食者-猎物模型，我们得到了捕食者和猎物都能入侵开放空间的数值条件。这一结果为进一步的分析研究提供了重要的信息（iii）我们用数值方法证实了三种Lotka-Volterra竞争系统和两种捕食者-猎物系统解的混沌行为，所得结果有助于我们对非线性结构的数学理解，从而保证了两种系统解的复杂行为相似。(2)研究了两组分自催化反应模型的推前和拉前。(1)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性，并给出了与混合反应比例有关的最小波速的估计。（ii）对于纯高阶自催化反应，我们考虑反应物和自催化剂具有相同的扩散率和反应物不扩散的两种情况。对于这两种情况，我们证明了拉锋的存在性，并得到了最小波速的估计，结果表明，当自催化阶数趋于无穷时，最小波速趋于零。（iii）高阶自催化反应的衰变阶数小于1的自催化衰变阶数也得到了类似的结果。

(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.