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Bifurcation manifolds in predator-prey models computed by Gröbner basis method. (English) Zbl 1451.92253

Summary: 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.

MSC:

92D25 Population dynamics (general)
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
34C23 Bifurcation theory for ordinary differential equations
37G10 Bifurcations of singular points in dynamical systems
37M20 Computational methods for bifurcation problems in dynamical systems
65P30 Numerical bifurcation problems
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