We prove a sufficient condition for the stability of a stationary solution to a system of nonlinear partial differential equations of the diffusion model describing the growth of malignant tumors. We also numerically simulate stable and unstable scenarios involving the interaction between tumor and immune cells.
We note from a general point of view that adding diffusion terms to ordinary differential equations, for example, to logistic ones, can in some cases improve sufficient conditions for the stability of a stationary solution. We give examples of models in which the addition of diffusion terms to ordinary differential equations changes the stability conditions of a stationary solution.
We investigate a class of nonlinear time‐partial differential equations describing the growth of glioma cells. The main results show sufficient conditions for the stability of stationary solutions for these kind of equations. More precisely, we study different spatial variables involving radial or axial symmetries. In addition, we also numerically simulate the system based on three distinct scenarios by considering symmetry across all spatial variables. The numerical results confirm the presence of possible stable states.
In this paper, we describe hypothetical possibilities for applying nonlinear dynamics methods to linguistic research. We suggest to refer to a large text corpus (meta-book of the writer, national language, language of the branch of knowledge) as a fractal object and to measure its fractal dimension (by Hausdorff). We consider the question of practical calculation of fractal dimension, as well as the question of motivation for applying the fractal concept to the language. In addition, we propose to apply some methods of the qualitative theory of differential equations to modelling the growth of the natural language vocabulary.
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