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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F19%3A00109406" target="_blank" >RIV/00216224:14310/19:00109406 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1016/j.mbs.2019.03.008" target="_blank" >https://doi.org/10.1016/j.mbs.2019.03.008</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.mbs.2019.03.008" target="_blank" >10.1016/j.mbs.2019.03.008</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Mathematical Biosciences

ISSN

0025-5564

e-ISSN

1879-3134

Svazek periodika

312

Číslo periodika v rámci svazku

JUN 2019

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

7

Strana od-do

1-7

Kód UT WoS článku

000469895200001

EID výsledku v databázi Scopus

2-s2.0-85063868259