Research on the existence and property of the solution for some partial differential equations appeared in the field of the life science and medicine

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2004 Improving Levine and Sleeman's way used for Othmer and Stevens model(1997), we could reduce the mathematical model of tumour angiogenesis by Anderson-Chaplain(1998) to a single equation. The result implies some possibility to discuss the equivalence between these models by the relationship and similarity of the reduced equations of these models, though these models are proposed independently.2005 In fact, we can prove the existence of the solution of Anderson-Chaplain model (A-C) and obtain the asymptotic profile of the solution by the similar way used in. Kubo and Suzuki(2004) for Othmer-Stevens model(O-S). By the above result we may investigate the mathematical structure common to these models. Introducing the uptake case in O-S, it is seen that the reduced equations of the uptake case in O-S and A-C belong to the same class and O-S and A-C can be identified in such sense. Hence we find a mathematical structure common to them and there is an essential mathematical structure of tumour angiogenesis in this point.2006 We study the above argument more precisely. In fact, we could modify A-C to the form of the uptake in O-S directly. Therefore the mathematical structure common to these models and some essential point of tumour angiogenesis can be found in such formal mathematical expression, which assures us that we can study some contact point between them mathematically. That is, A-C can be studied from the standpoint of statistical mechanics and it also gives the medical understanding of O-S while.A-C and O-S arise from the field of medicine and the theory of reinforeced random walk respectively.On the other hand, it assures us that we can proceed the numerical computation of Anderson-Chaplain model based on the theory of reinforced random walk.