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Robert Szczelina

Publications and source records attributed to Robert Szczelina.

5 recordsLinked to original sources
arXiv2026

Instant cost and delayed reward. Demographic eco-evolutionary game dynamics under the impact of the delay resulting from the offspring maturation time

In this paper, we extend the demographic eco-evolutionary game approach, based on explicit birth and death dynamics instead of abstract "fitness" interpreted as an abstract "Malthusian parameter", by the introduction of the delay resulting from the juvenile maturation time. This leads to the application of the Delay Differential Equations (DDE). We show that delay seriously affects the resulting dynamics and may lead to the loss of stability of equilibria when critical delay is exceeded. We provide theoretical tools for the assessment of the critical delays and the parameter values when this may happen. Our results emphasize the importance of the mechanisms of density dependence. We analyze the impact of three different suppression modes based on: adult mortality, juvenile recruitment survival after the maturation period (without delay), and juvenile recruitment at birth (with the delay). The last mode leads to extreme patterns such as bifurcations, complex cycles, and chaotic dynamics. However, surprisingly, this mode leads to extension of the duration of the temporary transient metastable states known as "ghost attractors". In addition, we also focus on the problem of resilience of the analyzed systems against external periodic perturbations and feedback-driven factors such as additional predator pressure.

q-bio.PE↗
arXiv2025

Sharkovskii theorem for infinite dimensional dynamical systems

We present an adaptation of a relatively simple topological argument to show the existence of many periodic orbits in an infinite dimensional dynamical system, provided that the system is close to a one-dimensional map in a certain sense. Namely, we prove a Sharkovskii-type theorem: if the system has a periodic orbit of basic period $m$, then it must have all periodic orbits of periods $n \triangleright m$, for $n$ preceding $m$ in Sharkovskii ordering. The assumptions of the theorem can be verified with computer assistance, and we demonstrate the application of such an argument in the case of Delay Differential Equations (DDEs): we consider the Rössler ODE system perturbed by a delayed term and we show that it retains periodic orbits of all natural periods for fixed values of parameters.

math.DS↗
arXiv2024

Stable periodic orbits for delay differential equations with unimodal feedback

We consider delay differential equations of the form $ y'(t)=-ay(t)+bf(y(t-1)) $ with positive parameters $a,b$ and a unimodal $f:[0,\infty)\to [0,1]$. It is assumed that the nonlinear $f$ is close to a function $g:[0,\infty)\to [0,1]$ with $g(ξ)=0$ for all $ξ>1$. The fact $g(ξ)=0$ for all $ξ>1$ allows to construct stable periodic orbits for the equation $x'(t)=-cx(t)+dg(x(t-1))$ with some parameters $d>c>0$. Then it is shown that the equation $ y'(t)=-ay(t)+bf(y(t-1)) $ also has a stable periodic orbit provided $a,b,f$ are sufficiently close to $c,d,g$ in a certain sense. The examples include $f(ξ)=\frac{ξ^k}{1+ξ^n}$ for parameters $k>0$ and $n>0$ together with the discontinuous $g(ξ)=ξ^k$ for $ξ\in[0,1)$, and $g(ξ)=0$ for $ξ>1$. The case $k=1$ is the famous Mackey--Glass equation, the case $k>1$ appears in population models with Allee effect, and the case $k\in(0,1)$ arises in some economic growth models. The obtained stable periodic orbits may have complicated structures.

math.DS↗
arXiv2023

High-order Lohner-type algorithm for rigorous computation of Poincaré maps in systems of Delay Differential Equations with several delays

We present a Lohner-type algorithm for rigorous integration of systems of Delay Differential Equations (DDEs) with multiple delays and its application in computation of Poincaré maps to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase-space and it exploits the smoothing of solutions occurring in DDEs to produces enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behavior, for example, existence of (apparently) unstable periodic orbits in Mackey-Glass Equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system).

math.DS↗