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Multi-dimensional Lotka-Volterra systems for carcinogenesis mutations. (English) Zbl 1181.35129
Summary: We consider three classes of models describing carcinogenesis mutations. Every considered model is described by the system of $$(n+1)$$ equations, and in each class three models are studied: the first is expressed as a system of ordinary differential equations (ODEs), the second – as a system of reaction-diffusion equations (RDEs) with the same kinetics as the first one and with the Neumann boundary conditions, while the third is also described by the system of RDEs but with the Dirichlet boundary conditions. The models are formulated on the basis of the Lotka-Volterra systems (food chains and competition systems) and in the case of RDEs the linear diffusion is considered. The differences between studied classes of models are expressed by the kinetic functions, namely by the form of kinetic function for the last variable, which reflects the dynamics of malignant cells (that is the last stage of mutations). In the first class the models are described by the typical food chain with favourable unbounded environment for the last stage, in the second one – the last equation expresses competition between the pre-malignant and malignant cells and the environment is also unbounded, while for the third one – it is expressed by predation term but the environment is unfavourable. The properties of the systems in each class are studied and compared.
It occurs that the behaviour of solutions to the systems of ODEs and RDEs with the Neumann boundary conditions is similar in each class; i.e. it does not depend on diffusion coefficients, but strongly depends on the class of models. On the other hand, in the case of the Dirichlet boundary conditions this behaviour is related to the magnitude of diffusion coefficients. For sufficiently large diffusion coefficients it is similar independently of the class of models, i.e. the trivial solution that is unstable for zero diffusion gains stability.

##### MSC:
 35K51 Initial-boundary value problems for second-order parabolic systems 34C60 Qualitative investigation and simulation of ordinary differential equation models 34D05 Asymptotic properties of solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations 35K57 Reaction-diffusion equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C50 Medical applications (general) 92D25 Population dynamics (general) 35B40 Asymptotic behavior of solutions to PDEs 35K58 Semilinear parabolic equations
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