Periodic patterns in simple biological differential delay models

A. Ivanov; S. Shelyag

Home/Archive/arXiv

arXiv · 2509.25396

Periodic patterns in simple biological differential delay models

A. Ivanov·S. Shelyag

View original source ↗Read source document ↗Export metadata ↓

Abstract

Periodic patterns in dynamical behaviours of biological models described by simple form differential delay equations are studied. Mathematical models are given by a class of scalar delay differential equations with a multiplicative time periodic mixed coefficient and a nonlinear delayed negative feedback. The dynamics is studied analytically with supportive numerical simulation and justification of the theoretical outcomes. The principal nonlinearity involving the state variable is of the negative feedback type, the periodic multiplicative coefficient can change its sign, leading to equations with mixed positive-negative feedback. The existence of slowly oscillating periodic solutions of two different types is established. The theoretical analysis and derivation are based on the reduction of dynamics in the delay equations to that of interval maps. The theoretical outcomes are verified and supported by comprehensive numerical computations. The differential delay equations considered are generalisations of some well-known autonomous models from biological applications.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

A. Ivanov, S. Shelyag. 2025-09-29. Periodic patterns in simple biological differential delay models. https://arxiv.org/abs/2509.25396

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Equidistribution of saddle periodic points for Hénon-like maps

We prove that under a natural assumption on the dynamical degrees, the saddle periodic points of a Hénon-like map in any dimension equidistribute with respect to the equilibrium measure. Our work is a generalization of the results of Bedford-Lyubich-Smillie, Dujardin, and Dinh-Sibony along with improvements of their techniques. We also investigate some fine properties of Green currents associated with the map.

math.DS↗

arXiv2026

A flux-based approach for analyzing the disguised toric locus of reaction networks

Dynamical systems with polynomial right-hand sides are very important in various applications, e.g., in biochemistry and population dynamics. The mathematical study of these dynamical systems is challenging due to the possibility of multistability, oscillations, and chaotic dynamics. One important tool for this study is the concept of reaction systems, which are dynamical systems generated by reaction networks for some choices of parameter values. Among these, disguised toric systems are remarkably stable: they have a unique attracting fixed point, and cannot give rise to oscillations or chaotic dynamics. The computation of the set of parameter values for which a network gives rise to disguised toric systems (i.e., the disguised toric locus of the network) is an important but difficult task. We introduce new ideas based on network fluxes for studying the disguised toric locus. We prove, under mild assumptions, that the disguised toric locus of any network $G$ is a contractible manifold with boundary, and introduce an associated graph $G^{\max}$ that characterizes its interior. These theoretical tools allow us, for the first time, to compute the full disguised toric locus for many networks of interest.

math.DS↗

arXiv2026

On dissonance and orthogonal projections of self-conformal measures

Let $μ$ be a self-conformal measure on $\mathbb{R}^d$. We establish conditions for $μ$ under which $\dim(μ*ν) = \min\lbrace d,\dimμ+\dimν\rbrace$ holds when $ν$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $μ$ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. In addition, we show that $\dim μ\circπ^{-1} = \min\{ k, \dim μ\}$ for every ortohogonal projection $π:\mathbb{R}^d\to\mathbb{R}^k$, $0<k<d$, when either $d=2$ and $μ$ is not self-similar and not supported on a line, or $d\geq 3$ and $μ$ is totally non-linear and not supported on a smooth hypersurface.

math.DS↗