AUTHORS: Akisato Kubo, Yuto Miyata

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ABSTRACT: In the present paper we investigate local and non-local type of mathematical models of tumour invasion with proliferation in order to understand the mechanism of the non-local tumour invasion. We consider the nonlocal tumour invasion model proposed by Gerisch and Chaplain and an approximation model by expanding the non-local term into Taylor series, which is closely related to Chaplain and Lolas model describing the local tumour invasion. We prove the global existence in time and asymptotic profile of the solution to the initial boundary value problem for the approximation model in one spacial dimension, by applying known mathematical results of the local tumour invasion model. Finally we show by computer simulations of the approximation model, which are verified by our mathematical analysis, the time dependent change of the non-local tumour invasion process and observe the relationship between the value of Taylor coefficients and the tumour cell density or so.

KEYWORDS: Non-local model, mathematical analysis, tumour invasion, Taylor expansion, computational simulation.

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