Study on differential equations arising out of life phenomena in vivo

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(1) We invited Professor M.Chaplain from Dundee University, UK to Japan for a week, September, 2007. On this occasion we organized the international conference" "Workshop on Mathematical Modelling and Analysis of Biological Pattern Formations and the Related Topics" (25-26, September, Nagoya University) (http://www.fujita-hu.ac.jp/~akikubo/).(2) We investigated mathematical model of tumour growth and invasion by M.Chaplain and G.Lolas, 2006. The principal investigator and co-investigators have to spend much time to overcome this difficulty.(3) We succeeded to obtain the mathematically exact relation between Othmer and Stevens model and Anderson and Chaplain model of tumour induced angiogenesis. Our result allow us to connect them from one to another only by formal calculation and substitution.(4) We tried to improve our mathematical way used in a sequence of mathematical models of tumour angiogenesis and tumour invasion by Anderson and Chaplain in order to study the latest model of nonlocal tumour invasion by Gerish and Chaplain (2007).2009(5) The principal investigator and co-investigators of this project studied an initial 0-Neumann boundary problem of nonlinear evolution equations in more general frame work. Finally to complete touch to this project we could give a characterization of a sequence of tumour growth models studied in this project by this framework of existence theorem and properties of the solution of our nonlinear evolution equations(see research activities 1,2,3).(6) The main investigator and his collaborator applied the method used by Levine and Sleeman (1998) in initial Neumann boundary value problems of the evolution equation and succeeded to show the existence of blow up solution under Dirichlet boundary condition(see research activities 4)(7) Based on the mathematical analysis in (2)to (6), we made computer simulations for each problem.