Bifurcation manifolds in predator–prey models computed by
Gröbner basis method

J 2019

Bifurcation manifolds in predator–prey models computed by Gröbner basis method

HAJNOVÁ, Veronika and Lenka PŘIBYLOVÁ

Basic information

Original name

Bifurcation manifolds in predator–prey models computed by Gröbner basis method

Name in Czech

Bifurkační variety v modelech predátor-kořist vypočítané pomocí Gröbnerových bazí

Authors

HAJNOVÁ, Veronika (203 Czech Republic, guarantor, belonging to the institution) and Lenka PŘIBYLOVÁ (203 Czech Republic, belonging to the institution)

Edition

Mathematical Biosciences, Amsterdam, Elsevier, 2019, 0025-5564

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10102 Applied mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

Full Text

Impact factor

Impact factor: 1.649

RIV identification code

RIV/00216224:14310/19:00109406

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1016/j.mbs.2019.03.008

UT WoS

000469895200001

Keywords (in Czech)

Rosenzweigův–MacArthurův model; Bifurkační variety; Gröbnerovy báze; Hopfova bifurkace; Fold bifurkace; Model predátor-kořist

Keywords in English

Rosenzweig–MacArthur model; Bifurcation manifolds; Gröbner basis; Hopf bifurcation; Fold bifurcation; Predator–prey model

Abstract

V originále

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.

Links

MUNI/A/1204/2017, interní kód MU

Name: Matematické statistické modelování 2 (Acronym: MaStaMo2)

Investor: Masaryk University, Category A

MUNI/A/1503/2018, interní kód MU

Name: Matematické statistické modelování 3 (Acronym: MaStaMo3)

Investor: Masaryk University, Category A

Citovat

HAJNOVÁ, Veronika and Lenka PŘIBYLOVÁ. Bifurcation manifolds in predator–prey models computed by Gröbner basis method. 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.

@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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