Nonlinear Compartment Models with Time-Dependent Parameters

Jochen Merker, Benjamin Kunsch and Gregor Schuldt
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Jochen Merker: Centre for Mathematics and Natural Sciences, Leipzig University of Applied Sciences, PF 30 11 66, 04251 Leipzig, Germany
Benjamin Kunsch: Wilhelm-Ostwald-Gymnasium, Willi-Bredel-Str. 15, 04279 Leipzig, Germany
Gregor Schuldt: Centre for Mathematics and Natural Sciences, Leipzig University of Applied Sciences, PF 30 11 66, 04251 Leipzig, Germany

Mathematics, 2021, vol. 9, issue 14, 1-13

Abstract: A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C 0 -topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.

Keywords: compartment model; time-dependent parameters; rate-induced tipping; non-smooth dynamical system; boundary bifurcation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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