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 C0-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.
Highlights
Nonlinear compartment models [1], which in the autonomous case generate semidynamical systems on a simplex, have been used in many areas of science, in life sciences, to study transmissions among different compartments of a system
While for smooth systems the unstable hyperbolic disease-free equilibrium (DFE) cannot attract an open subset in the interior of the simplex and the endemic equilibrium (EE) attracts all interior points of the simplex, we show in Section 4.2 that there are systems nearby w.r.t
For time-dependent parameters, given by λ(t) = (α(t), β(t)) in (14), rate-induced tipping can occur, and artifacts of this tipping phenomenon can be seen in the original smooth system leading to wave-like convergence to the EE, where the waves stay longer near the DFE if there is tipping to the DFE in a nearby idealized system
Summary
Nonlinear compartment models [1], which in the autonomous case generate semidynamical systems on a simplex, have been used in many areas of science, in life sciences, to study transmissions among different compartments of a system. The main aim of this article is to show that rate-induced tipping caused by basin instability can occur in nonlinear compartment models, which are continuous but non-smooth near a boundary equilibrium point attracting an open interior set, and which have another locally asymptotically stable equilibrium in the interior. In the case of epidemiology, in such idealized systems epidemic behaviour, where the disease dies out, and endemic behaviour, where the disease remains, do coexist, but can even interchange during time in dependence on the measures undertaken to contain the disease Artifacts of this behaviour in idealized systems can be seen in the original system, and even in the autonomous case the dynamics of the idealized systems may explain the long-wave-like convergence to the locally asymptotically stable equilibrium in the interior. We discuss that these dynamics occur generically after a two-parameter bifurcation of an equilibrium at a corner of the simplex
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