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Qualitative Studies of Some Biochemical Models

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Abstract

A computational approach to detecting Andronov–Hopf bifurcations in polynomial systems of ordinary differential equations depending on parameters is proposed. It relies on algorithms of computational commutative algebra based on the Groebner bases theory. The approach is applied to the investigation of two models related to the double phosphorylation of mitogen-activated protein kinases, a biochemical network that occurs in many cellular pathways. For the models, we analyze the roots of the characteristic polynomials of the Jacobians in a steady state and prove that Andronov–Hopf bifurcations are absent for biochemically relevant values of parameters. We also performed a search for algebraic invariant subspaces in the systems (which represent “weak” conservations laws) and find all subfamilies admitting linear invariant subspaces. The search is done using the Darboux method. That, is we look for Darboux polynomials and cofactors as polynomials with undetermined coefficients and then determine the coefficients using the algorithms of the elimination theory.

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Notes

  1. The polynomial was computed using the Eliminate procedure of the Mathematica computer algebra system based on the algorithm resulting from theorem 1.

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Authors and Affiliations

  1. West Virginia University, PO Box 6201, Morgantown, West Virginia, United States

    C. Pantea

  2. University of Maribor, SI-2000, Maribor, Slovenia

    V. G. Romanovski

  3. Center for Applied Mathematics and Theoretical Physics, SI-2000, Maribor, Slovenia

    V. G. Romanovski

Authors
  1. C. Pantea
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  2. V. G. Romanovski
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Correspondence to C. Pantea.

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This work was supported by the Slovenian Research Agency, program P1-0306, projects N1-0063 and BI-US/19-21-058.

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In memory of V.A. Pliss

Translated by K. Gumerov

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Pantea, C., Romanovski, V.G. Qualitative Studies of Some Biochemical Models. Vestnik St.Petersb. Univ.Math. 53, 214–222 (2020). https://doi.org/10.1134/S1063454120020144

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  • DOI: https://doi.org/10.1134/S1063454120020144

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