Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability
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.
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Stochastic Dynamical Systems in Biology: Numerical Methods and Applications” where work on this paper was undertaken. This work was supported by EPSRC grant no EP/K032208/1. This work was partially supported by a grant from the Simons Foundation. Tomáš Vejchodský would like to acknowledge the institutional support RVO 67985840. Radek Erban would also like to thank the Royal Society for a University Research Fellowship.
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