Abstract
The introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics, ranging from periodic solutions through to spatio-temporal chaos. In this paper, we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria, periodic orbits and travelling waves. Importantly, we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function. Using this as a computational tool to determine stability, we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback. Direct numerical simulations are used to confirm our bifurcation analysis, and to probe some of the rich behaviour possible for mixed feedback.
Highlights
Delayed differential equations (DDEs) arise naturally as models of dynamical systems where memory effects are important
In contrast to models without delay the analysis of DDEs is notoriously hard and has generated considerable activity in the mathematics community. This is directly attributable to the fact that the solution space for DDEs is infinite dimensional despite only a finite number of dynamical variables appearing in a model
Direct numerical simulations are used to confirm our predictions, and establish that such wave phenomenon are robust in the sense that they persist for smooth caricatures of the feedback nonlinearities
Summary
Delayed differential equations (DDEs) arise naturally as models of dynamical systems where memory effects are important. Since threshold models are relatively common in the applied biological sciences, arising for example in models of calcium release [8], neural tissue [9] and gene networks [10], it is worthwhile developing a more comprehensive treatment of delayed threshold models Since both positive and negative feedback are seen as important enhancers of the properties of biological systems [11] generic examples from both these classes, as well as a mixture of the two, should be considered. We discuss natural extensions of the work in this paper
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