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2017 | OriginalPaper | Chapter

Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability

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Abstract

Theoretical results regarding two-dimensional ordinary-differential equations (ODEs) with second-degree polynomial right-hand sides are summarized, with an emphasis on limit cycles, limit cycle bifurcations, and multistability. The results are then used for construction of two reaction systems, which are at the deterministic level described by two-dimensional third-degree kinetic ODEs. The first system displays a homoclinic bifurcation, and a coexistence of a stable critical point and a stable limit cycle in the phase plane. The second system displays a multiple limit cycle bifurcation, and a coexistence of two stable limit cycles. The deterministic solutions (obtained by solving the kinetic ODEs) and stochastic solutions [noisy time-series generating by the Gillespie algorithm, and the underlying probability distributions obtained by solving the chemical master equation (CME)] of the constructed systems are compared, and the observed differences highlighted. The constructed systems are proposed as test problems for statistical methods, which are designed to detect and classify properties of given noisy time-series arising from biological applications.

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Appendix
Available only for authorised users
Footnotes
1
Let us note that the limit cycles corresponding to (3) are highly sensitive to changes in the parameters (4). Thus, during numerical simulations, parameters (4) should not be rounded-off. One can also design bicyclic systems which are less parameter sensitive, see Appendix 2.
 
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Metadata
Title
Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability
Authors
Tomislav Plesa
Tomáš Vejchodský
Radek Erban
Copyright Year
2017
DOI
https://doi.org/10.1007/978-3-319-62627-7_1

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