arXiv subjects

Ryszard Rudnicki

Publications and source records attributed to Ryszard Rudnicki.

15 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

Beyond classical Hamilton's Rule. State distribution asymmetry and the dynamics of altruism

This paper analyzes the relationships between demographic and state-based evolutionary games and Hamilton's rule. It is shown that the classical Hamilton's rule (counterfactual method), combined with demographic payoffs, leads to easily testable models. It works well when the roles of donor and receiver are randomly drawn during each interaction event. This is illustrated by the alarm call example. However, we can imagine situations in which role-switching results from external mechanism, such as, fluxes of individuals between the border and the interior of the habitat, when only border individuals may spot the threat and warn their neighbors. To cover these cases, a new model is extended to the case with explicit dynamics of the role distributions among carriers of different strategies, driven by some general mechanisms. It is shown that even in the case when fluxes between roles are driven by neutral mechanisms (acting in the same way on all strategies), differences in mortality in the focal interaction lead to different distributions of roles for different strategies. This leads to a more complex rule for cooperation than the classical Hamilton's rule. In addition to the cost and benefit components, the new rule contains a third component weighted by the difference in proportions of the donors among carriers of both strategies. Depending on the sign, this component can be termed the "survival surplus", when the donors survival have greater survival than receivers, or the "sacrifice cost" (when it decreases the benefit), when the receiver's survival exceeds that of the helping donor. When we allow different role-switching rates for different strategies, cooperators can win even in the case when the assortment mechanism is inefficient (i.e., the probability of receiving help for noncooperators is slightly greater than for cooperators), which is impossible in classical Hamilton's rule.

q-bio.PE↗

arXiv2024

With Andrzej Lasota there and back again

The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota's results on the author's research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differential equations.

math.PR↗

arXiv2024

Transformation semigroups and their applications

In this chapter we present transformation semigroups and their applications. We begin with Klein's approach to geometry based on invariants of transformation groups. Then we present symmetry groups in chemistry and in classical mechanics. Next we introduce one-parameter semigroups of transformations and their applications in ergodic theory. Our main subject are one-parameter semigroups of operators, in particular stochastic semigroups. We present general results on their existence and long-time behaviour. We also give examples of one-parameter semigroups related to Markov chains, diffusion and processes with jumps. We focus on the applications of semigroups of operators in biology. Among other things, we study models of: DNA evolution; growth of erythrocyte population; gene expression; cell cycle; the movement of bacteria and insects. We also consider models with stochastic noise and different population models.

math.FA↗

arXiv2024

Ergodic and chaotic properties of some biological models

In this note we present two types of biological models which have interesting ergodic and chaotic properties. The first type are one-dimensional transformations, like a logistic map, which are used to describe the change in population size in successive generations. We study ergodic properties of such transformations using Frobenius--Perron operators. The second type are some structured populations models, for example a space-structured model, or a model of maturity-distribution of precursors of blood cells. These models are described by partial differential equations, which generate semiflows on the space of functions. We construct strong mixing invariant measures for these semiflows using stochastic precesses. From properties of invariant measures we deduce some chaotic properties of semiflows such as the existence of dense trajectories and strong instability of all trajectories.

math.DS↗

arXiv2022

Asymptotic properties of a general model of immune status

We consider a model of dynamics of the immune system. The model is based on three factors: occasional boosting and continuous waning of immunity and a general description of the period between subsequent boosting events. The antibody concentration changes according to a non-Markovian process. The density of the distribution of this concentration satisfies some partial differential equation with an integral boundary condition. We check that this system generates a stochastic semigroup and we study the long-time behaviour of this semigroup. In particular we prove a theorem on its asymptotic stability.

math.PR↗

arXiv2021

Cell cycle length and long-time behaviour of an age-size model

We consider an age-size structured cell population model based on the cell cycle length. The model is described by a first order partial differential equation with initial-boundary conditions. Using the theory of semigroups of positive operators we establish new criteria for an asynchronous exponential growth of solutions to such equations. We discuss the question of exponential size growth of cells. We study in detail a constant size growth model and a model with target size division. We also present versions of the model when the population is heterogeneous.

math.AP↗

arXiv2020

Dynamics of antibody levels: asymptotic properties

We study properties of a piecewise deterministic Markov process modeling the changes in concentration of specific antibodies. The evolution of densities of the process is described by a stochastic semigroup. The long-time behaviour of this semigroup is studied. In particular we prove theorems on its asymptotic stability.

math.PR↗

arXiv2019

Replicator dynamics for the game theoretic selection models based on state

The paper contains the attempt to integration of the classical evolutionary game theory based on replicator dynamics and the state based approach of Houston and Mcnamara. In the new approach, individuals have different heritable strategies, however the individuals carrying the same strategy can differ on the state, role or situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths) can lead to the change of state of the competing individuals. The special cases of the new framework of stage structured models where the state changes describe developmental steps or aging are derived. New approach is illustrated by the example of Owner-Intruder game with explicit dynamics of the role changes. New model is the generalization of the demographic version of the Hawk-Dove game, the difference is that opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Intruders check random nest sites, and play the Hawk-Dove game with the Owner if they are occupied. Interesting feedback mechanism is produced by fluxes of individuals between subpopulations. Owners produce newborns which become Intruders, since they should find a free nest site to reproduce.

q-bio.PE↗

arXiv2019

Invariant density & time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

This paper deals with collisionless transport equations in bounded open domains $Ω\subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial Ω$, orthogonally invariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $% L^{1}(Ω\times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion of irreducibility of $% \left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ and we show that, under very natural assumptions, if an invariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ is \emph{sweeping} in the sense that, for any density $φ$, the total mass of $ U_{\mathsf{H}}(t)φ$ concentrates near suitable sets of zero measure as $ t\rightarrow +\infty .$ We show also a general weak compactness theorem of interest for the existence of invariant densities. This theorem is based on several results on smoothness and transversality of the dynamical flow associated to $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}.$

math.AP↗

arXiv2018

Applications of stochastic semigroups to cell cycle models

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.

math.PR↗

arXiv2017

Does a population with the highest turnover coefficient win competition?

We consider a discrete time competition model. Populations compete for common limited resources but they have different fertilities and mortalities rates. We compare dynamical properties of this model with its continuous counterpart. We give sufficient conditions for competitive exclusion and the existence of periodic solutions related to the classical logistic, Beverton-Holt and Ricker models.

math.DS↗

arXiv2015

Piecewise deterministic Markov processes in biological models

We present a short introduction into the framework of piecewise deterministic Markov processes. We illustrate the abstract mathematical setting with a series of examples related to dispersal of biological systems, cell cycle models, gene expression, physiologically structured populations, as well as neural activity. General results concerning asymptotic properties of stochastic semigroups induced by such Markov processes are applied to specific examples.

math.PR↗

arXiv2015

On a stochastic gene expression with pre-mRNA, mRNA and protein contribution

In this paper we develop a model of stochastic gene expression, which is an extension of the model investigated in the paper [T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol. 238 (2006) 348-367]. In our model, stochastic effects still originate from random uctuations in gene activity status, but we precede mRNA production by the formation of pre-mRNA, which enriches classical transcription phase. We obtain a stochastically regulated system of ordinary differential equations (ODEs) describing evolution of pre-mRNA, mRNA and protein levels. We perform mathematical analysis of a long-time behaviour of this stochastic process, identified as a piece-wise deterministic Markov process (PDMP). We check exact results using numerical simulations for the distributions of all three types of particles. Moreover, we investigate the deterministic (adiabatic) limit state of the process, when depending on parameters it can exhibit two specific types of behavior: bistability and the existence of the limit cycle. The latter one is not present when only two kinds of gene expression products are considered.

math.PR↗

arXiv2013

Model of phenotypic evolution in hermaphroditic populations

We consider an individual based model of phenotypic evolution in hermaphroditic populations which includes random and assortative mating of individuals. By increasing the number of individuals to infinity we obtain a nonlinear transport equation, which describes the evolution of distribution of phenotypic traits. Existence of an one-dimensional attractor is proved and the formula for the density of phenotypic traits in the limiting (asymptotic) population is derived in some particular case.

math.PR↗