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

我们得到了Lotka-Volterra型两种群扩散竞争模型的最小行波波前速度。1989年，A.Okubo等人的论文利用该模型，通过估计田间数据的模型参数，讨论了灰松鼠的入侵速度，并利用启发式分析论证预测了其入侵速度。我们利用奇异摄动和几何方法研究了保证其证明有效的条件，并用数值方法证明了其启发式证明不成立的参数的真实值的存在性。我们还研究了具有传染性的两组分传染病模型的行波解的存在性。模型系统由具有经典Kermack-Mckendric型非线性反应项的反应扩散方程描述。假定感染者和易感者的扩散系数分别为非负常数和d1d2。借助于不变流形理论的打靶论，我们证明了存在一个正常数c^&lt；**&gt；使得对任何大于或等于&gt；c^&lt；**&gt；的c&lt；都存在行波解。证明了最小波速c^&lt；**&gt；与d2=0时的值相同，与d2无关。利用上述方法，我们研究了M.Mimura等人提出的具有与Kermack-McKendric型相同的非线性的细菌菌落模式形成模型。分析了行波方程的局域性质，数值得到了行波四维异宿轨道的性质。这些分析为进一步研究细菌菌落格局的形成奠定了基础。

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.