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Nonlinear Compartment Models with Time-Dependent Parameters

Author

Listed:
  • 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
    These authors contributed equally to this work.)

  • Gregor Schuldt

    (Centre for Mathematics and Natural Sciences, Leipzig University of Applied Sciences, PF 30 11 66, 04251 Leipzig, Germany
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Jochen Merker & Benjamin Kunsch & Gregor Schuldt, 2021. "Nonlinear Compartment Models with Time-Dependent Parameters," Mathematics, MDPI, vol. 9(14), pages 1-13, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:14:p:1657-:d:594357
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    References listed on IDEAS

    as
    1. Igor Volkov & Jayanth R. Banavar & Stephen P. Hubbell & Amos Maritan, 2003. "Neutral theory and relative species abundance in ecology," Nature, Nature, vol. 424(6952), pages 1035-1037, August.
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