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@article{1516797, author = {Hajnová, Veronika and Přibylová, Lenka}, article_location = {Amsterdam}, article_number = {JUN 2019}, doi = {http://dx.doi.org/10.1016/j.mbs.2019.03.008}, keywords = {Rosenzweig–MacArthur model; Bifurcation manifolds; Gröbner basis; Hopf bifurcation; Fold bifurcation; Predator–prey model}, language = {eng}, issn = {0025-5564}, journal = {Mathematical Biosciences}, title = {Bifurcation manifolds in predator–prey models computed by Gröbner basis method}, url = {https://doi.org/10.1016/j.mbs.2019.03.008}, volume = {312}, year = {2019} }
TY - JOUR ID - 1516797 AU - Hajnová, Veronika - Přibylová, Lenka PY - 2019 TI - Bifurcation manifolds in predator–prey models computed by Gröbner basis method JF - Mathematical Biosciences VL - 312 IS - JUN 2019 SP - 1-7 EP - 1-7 PB - Elsevier SN - 00255564 KW - Rosenzweig–MacArthur model KW - Bifurcation manifolds KW - Gröbner basis KW - Hopf bifurcation KW - Fold bifurcation KW - Predator–prey model UR - https://doi.org/10.1016/j.mbs.2019.03.008 L2 - https://doi.org/10.1016/j.mbs.2019.03.008 N2 - Many natural processes studied in population biology, systems biology, biochemistry, chemistry or physics are modeled by dynamical systems with polynomial or rational right-hand sides in state and parameter variables. The problem of finding bifurcation manifolds of such discrete or continuous dynamical systems leads to a problem of finding solutions to a system of non-linear algebraic equations. This approach often fails since it is not possible to express equilibria explicitly. Here we describe an algebraic procedure based on the Gröbner basis computation that finds bifurcation manifolds without computing equilibria. Our method provides formulas for bifurcation manifolds in commonly studied cases in applied research – for the fold, transcritical, cusp, Hopf and Bogdanov–Takens bifurcations. The method returns bifurcation manifolds as implicitly defined functions or parametric functions in full parameter space. The approach can be implemented in any computer algebra system; therefore it can be used in applied research as a supporting autonomous computation even by non-experts in bifurcation theory. This paper demonstrates our new approach on the recently published Rosenzweig–MacArthur predator–prey model generalizations in order to highlight the simplicity of our method compared to the published analysis. ER -
HAJNOVÁ, Veronika and Lenka PŘIBYLOVÁ. Bifurcation manifolds in predator–prey models computed by Gröbner basis method. \textit{Mathematical Biosciences}. Amsterdam: Elsevier, 2019, vol.~312, JUN 2019, p.~1-7. ISSN~0025-5564. Available from: https://dx.doi.org/10.1016/j.mbs.2019.03.008.
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