ANALYTICAL RESEARCH ON FORMATION OF TIME DEPENDENT INHOMOGENEOUS STRUCTURE IN REACTION-DIFFUSION SYSTEMS

我们研究了Lotka-Volterra型两种扩散竞争模型的最小行波前速度。a . okubo等人在1989年的论文中利用该模型从野外数据中估计模型参数，讨论了灰松鼠的入侵速度，并利用启发式分析方法对其入侵速度进行了预测。我们用奇异摄动和几何方法研究了保证他们的论证有效的条件，并在数值上证明了他们的启发式论证不成立的参数的现实值的存在性。我们还研究了传染-易感二组分流行病模型的行波解的存在性。模型系统用经典Kermack-McKendric型非线性反应项的反应扩散方程来描述。传染物和易感物的扩散系数分别假定为非负常数和d_1 - d_2。借助于不变流形理论，通过射射论证，证明了存在一个正常数c^<**>，使得任意c<大于等于>c^<**>的行波解存在。最小波速c^<**>与d_2无关，其值与d_2=0时相同。利用上述方法，我们研究了M.Mimura等人提出的细菌菌落模式形成模型，该模型与Kermack-McKendric型具有相同的非线性。对行波方程的局部性质进行了分析，得到了行波四维异斜轨道的数值性质。这些分析为进一步研究菌落格局形成奠定了基础。

We are concened with the minimal speed of traveling wave fronts for a two species diffusion-competition model of Lotka-Volterra type. The 1989 paper by A.Okubo et al.employed this model to discuss the speed of invasion of gray squirrel by estimating the model parameters from field data, and predicted its speed by the use of a heuristic analytical argument. We investigate the conditions which assure the validity of their argument by singular perturbations and a geometric method and show numerically the existence of the realistic values of parameters for which their heuristic argument does not hold.We also investigate the existence of traveling wave solutions for the infective-susceptible two component epidemic model. The model system is described by reaction-diffusion equations with the nonlinear reaction term of the classical Kermack-McKendric type. The diffusion coefficients of infectives and susceptibles are assumed to be nonnegative constants and d_1 d_2 respectively. By the shooting argument with the aid of the invariant manifold theory, we prove that there exists a positive constant c^<**> such that traveling wave solutions exist for any c<greater than or equal>c^<**>. The minimal wave speed c^<**> is shown to be independent of d_2 and to have the same value as for the case d_2=0.Using the above method, we study the pattern formation model of bacteria colony proposed by M.Mimura et al.which has the same kind of nonlinearity as the Kermack-McKendric type. We have analyzed the local properties of the traveling wave equations and obtained numerically the properties of the 4-dimensional heteroclinic orbits for traveling waves. These analysis give the foundation of further research of pattern formation of bacteria colony.