Multi-dimensional Lotka-Volterra systems for carcinogenesis mutations.

*(English)*Zbl 1181.35129Summary: 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.

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 |

##### Keywords:

food chain; benign cells; malignant cells; local and global stability; dissipativity; Lyapunov functional; invariant set
PDF
BibTeX
XML
Cite

\textit{U. Foryś}, Math. Methods Appl. Sci. 32, No. 17, 2287--2308 (2009; Zbl 1181.35129)

Full Text:
DOI

##### References:

[1] | Cancer Modeling and Simulation (2003) |

[2] | Horn, Special Issue on Mathematical Models in Cancer, Discrete and Continuous Dynamical Systems, Series B 4 (1) (2004) |

[3] | Bhat, Three step food chains in Gompertz and Lotka-Volterra models, Journal of Theoretical Biology 91 (3) pp 429– (1981) |

[4] | Gard, Persistence in food webs. I. Lotka-Volterra food chains, Bulletin of Mathematical Biology 41 (6) pp 877– (1979) · Zbl 0422.92017 |

[5] | So, A note on the global stability and bifurcation phenomenon of a Lotka-Volterra food chain, Journal of Theoretical Biology 80 (2) pp 185– (1979) |

[6] | Ahangar, Multistage evolutionary model for carcinogenesis mutations, Electronic Journal of Differential Equation Conference 10 pp 33– (2003) · Zbl 1014.92018 |

[7] | A Survey of Models for Tumor-immune System Dynamics (1997) · Zbl 0874.92020 |

[8] | Bodnar, Three types of simple DDE’s describing tumour growth, Journal of Biological Systems 15 (4) pp 453– (2007) · Zbl 1151.92015 |

[9] | Drasdo, Individual based approaches to birth and death in avascular tumours, Mathematical Computer Modelling 37 (11) pp 1163– (2003) · Zbl 1047.92023 |

[10] | Foryś, Logistic equation in tumour growth modelling, Journal of Applied Mathematics and Computing 13 (3) pp 317– (2003) |

[11] | Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations, Journal of Applied Analysis 11 (2) pp 200– (2005) · Zbl 1101.34033 |

[12] | Foryś, Proceedings of the X National Conference on Mathematics Applied in Biology and Medicine (2004) |

[13] | Marciniak-Czochra, Modelling of early lung cancer progression: influence of growth factor production and cooperation between partially transformed cells, Mathematical Models and Methods in Applied Sciences 17 (Suppl.) pp 1693– (2007) · Zbl 1135.92019 |

[14] | Hale, Ordinary Differential Equations (1969) · Zbl 0186.40901 |

[15] | Rothe, Global Solutions of Reaction-Diffusion Systems (1984) |

[16] | Henry, Geometric Theory of Semilinear Parabolic Equations (1981) · Zbl 0456.35001 |

[17] | Smoller, Shock Waves and Reaction Diffusion Equations (1994) · Zbl 0807.35002 |

[18] | Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana University Mathematics Journal 6 pp 373– (1977) · Zbl 0368.35040 |

[19] | Hale, Theory of Functional Differential Equations (1997) · Zbl 1092.34500 |

[20] | Evans, Partial Differential Equations (1998) · Zbl 0902.35002 |

[21] | Britton, Reaction-diffusion Equations and their Application to Biology (1986) · Zbl 0602.92001 |

[22] | Hofbauer, The Theory of Evolution and Dynamical Systems (1998) |

[23] | Murray, Mathematical Biology. 1, An Introduction (2002) |

[24] | Chueh, Positively invariant regions for systems of non-linear diffusion equations, Indiana University Mathematics Journal 6 pp 353– (1977) · Zbl 0368.35040 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.