A distributed-delay model for the El Niño Southern Oscillation with a minimum delay: a case study of the shifted linear chain trick

Abstract

When modelling a system of interest with a delay differential equation, a distributed delay may be the appropriate modelling choice when the delayed response occurs over a significant range of times rather than with a single constant delay. In systems with a minimum physical transit or processing time, however, the delay kernel should respect a fixed and positive minimum delay. A shifted Erlang kernel is a convenient choice, because the shifted linear chain trick allows one to replace the convolution associated with this type of kernel by a finite auxiliary-variable representation with one constant delay. As we demonstrate with a case study of the Ghil-Zaliapin-Thompson (GZT) model of the El Niño Southern Oscillation with seasonal forcing and distributed delayed oceanic feedback, this reduction makes it possible to perform a bifurcation analysis with numerical continuation techniques for any width of the Erlang distribution. Specifically, we present the bifurcation and resonance structure of the distributed-delay GZT model in the plane of forcing strength and delay for different widths of the distribution. To ensure a like-for-like comparison and determine the influence of delay distribution, we present these results in terms of the effective delay and, moreover, rescale the feedback strength to account for its attenuation with increasing width of the distribution.