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Jacek Banasiak

Publications and source records attributed to Jacek Banasiak.

At least 19 recordsLinked to original sources
arXiv2026

Network transport with nonlinear dynamics at the nodes

In this paper, we consider a network transport model in which agents moving along the edges can contribute to dynamics at nodes or bypass them. The model takes the form of a system of first-order partial differential equations coupled with a system of ordinary differential equations, and can describe a range of phenomena, from diseases in metapopulations to migratory systems with delays, to cell differentiation processes, providing a unified platform that includes network transport and delay systems as particular cases. We prove the well-posedness of the model in $ L^p$ spaces, $ 1\leq p<\infty$, study long-term asymptotics, and illustrate the theory using an SIS disease in a metapopulation consisting of several sites where the disease develops, connected by routes along which the population can migrate, as an example.

math.AP↗
arXiv2026

Fragmentation-coagulation processes with advection or diffusion in space

In this paper, we consider a continuous fragmentation--coagulation model in which the reacting particles can be transported in physical space through either advection or diffusion. We prove new results on the generation of $C_0$-semigroups with parameter and use them to show that the Abstract Cauchy Problem associated with a more general version of the advection/diffusion--fragmentation problem generates a positive $C_0$-semigroup in spaces $L_1(\mathbb R_+, X_x, (1+m^r)dm),$ where $m$ is the particle mass, $X_x$ is either the space of integrable or continuous functions with respect to the spatial variable, and the weight exponent $r$ is sufficiently large. These results enable us to prove the classical solvability of a wide range of advection/diffusion--fragmentation--coagulation equations with unbounded coagulation kernels.

math.AP↗
arXiv2024

Mating versus alternative blood sources as determinants to mosquito abundance and population resilience

A deterministic nonlinear ordinary differential equation model for mosquito dynamics in which the mosquitoes can quest for blood either within a human population or within non-human/vertebrate populations is derived and studied. The model captures both the mosquito's aquatic and terrestrial forms and includes a mechanism to investigate the impact of mating on mosquito dynamics. The model uses a restricted form of homogeneous mixing based on the idea that the mosquito has a blood-feeding habit by accounting for the mosquitoes' blood-feeding preferences as well as its gonotrophic cycle. This characterization allows us to compartmentalise the total mosquito population into distinct compartments according to the spatial location of the mosquito (breeding site, resting places and questing places) as well as blood-fed status. Issues of overcrowding and intraspecific competition both within the aquatic and the terrestrial stages of the mosquito's life forms are addressed and considered in the model. Results show that the inclusion of mating induces bi-stability; a phenomenon whereby locally stable trivial and non-trivial equilibria co-exist with an unstable non-zero equilibrium. The local nature of the stable equilibria is demonstrated by numerically showing that the long-term state of the system is sensitive to initial conditions. The bi-stability state is analogous to the phenomenon of the Allee effect that has been reported in population biology. The model's results, including the derivation of the threshold parameter of the system, are comprehensively tested via numerical simulations. The output of our model has direct application to mosquito control strategies, for it clearly shows key points in the mosquito's developmental pathway that can be targeted for control purposes.

q-bio.PE↗
arXiv2023

Multiscale malaria models and their uniform in-time asymptotic analysis

In this paper, we show that an extension of the classical Tikhonov--Fenichel asymptotic procedure applied to multiscale models of vector-borne diseases, with time scales determined by the dynamics of human and vector populations, yields a simplified model approximating the original one in a consistent, and uniform for large times, way. Furthermore, we construct a higher-order approximation based on the classical Chapman-Enskog procedure of kinetic theory and show, in particular, that it is equivalent to the dynamics on the first-order approximation of the slow manifold in the Fenichel theory.

q-bio.PE↗
arXiv2022

Growth-fragmentation equations with McKendrick--von Foerster boundary condition

The paper concerns the well-posedness and long-term asymptotics of growth--fragmentation equation with unbounded fragmentation rates and McKendrick--von Foerster boundary conditions. We provide three different methods of proving that there is a strongly continuous semigroup solution to the problem and show that it is a compact perturbation of the corresponding semigroup with a homogeneous boundary condition. This allows for transferring the results on the spectral gap available for the later semigroup to the one considered in the paper. We also provide sufficient and necessary conditions for the irreducibility of the semigroup needed to prove that it has asynchronous exponential growth. We conclude the paper by deriving an explicit solution to a special class of growth--fragmentation problems with McKendrick--von Foerster boundary conditions and by finding its Perron eigenpair that determines its long-term behaviour.

math.AP↗
arXiv2022

On spectral gaps of growth-fragmentation semigroups in higher moment spaces

We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces $L^{1}(\mathbb{R}_{+};\ x^{α}dx)$ and $L^{1}(\mathbb{R} _{+};\ \left( 1+x\right) ^{α}dx)$ for unbounded total fragmentation rates and continuous growth rates $r(.)$\ such that $\int_{0}^{+\infty } \frac{1}{r(τ)}dτ=+\infty .\ $The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that $α>\widehat{α}$ for a suitable threshold $\widehat{ α}\geq 1$ that depends on the moment space we consider. A systematic functional analytic construction is provided. Various examples of fragmentation kernels illustrating the theory are given and an open problem is mentioned.

math.FA↗
arXiv2021

Telegraph type systems on networks and port-Hamiltonians. II. Graph realizability

Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of $2\times 2$ hyperbolic equations on a metric graph $Γ$. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of $Γ$ and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of $Γ$.

math.DS↗
arXiv2021

Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness

The paper is concerned with a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff's type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems for the associated Riemann invariants and provide a semigroup theoretic proof of its well-posedness. A number of examples showing the relation of our results with recent research is also provided.

math.AP↗
arXiv2020

Growth--fragmentation--coagulation equations with unbounded coagulation kernels

In this paper we prove the global in time solvability of the continuous growth--fragmentation--coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth--fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic.

math.AP↗
arXiv2019

Global solutions of continuous coagulation-fragmentation equations with unbounded coefficients

In this paper we prove the existence of global classical solutions to continuous coagulation-fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation-fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation--fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.

math.AP↗
arXiv2018

Analysis and Simulations of the Discrete Fragmentation Equation with Decay

Fragmentation--coagulation processes, in which aggregates can break up or get together, often occur together with decay processes in which the components can be removed from the aggregates by a chemical reaction, evaporation, dissolution, or death. In this paper we consider the discrete decay--fragmentation equation and prove the existence and uniqueness of physically meaningful solutions to this equation using the theory of semigroups of operators. In particular, we find conditions under which the solution semigroup is analytic, compact and has the asynchronous exponential growth property. The theoretical analysis is illustrated by a number of numerical simulations.

math.DS↗
arXiv2018

The Discrete Unbounded Coagulation-Fragmentation Equation with Growth, Decay and Sedimentation

In this paper we study the discrete coagulation--fragmentation models with growth, decay and sedimentation. We demonstrate the existence and uniqueness of classical global solutions provided the linear processes are sufficiently strong. This paper extends several previous results both by considering a more general model and and also signnificantly weakening the assumptions. Theoretical conclusions are supported by numerical simulations.

math.DS↗
arXiv2015

Asymptotic state lumping in transport and diffusion problems on networks

One of the aims of systems biology is to build multiple layered and multiple scale models of living systems which can efficiently describe phenomena occurring at various level of resolution. Such models should consist of layers of various microsystems interconnected by a network of pathways, to form a macrosystem in a consistent way; that is, the observable characteristics of the macrosystem should be, at least asymptotically, derivable by aggregation of the appropriate features of the microsystems forming it, and from the properties of the network. In this paper we consider a general macromodel describing a population consisting of several interacting with each other subgroups, with the rules of interactions given by a system of ordinary differential equations, and we construct two different micromodels whose aggregated dynamics is approximately the same as that of the original macromodel. The micromodels offer a more detailed description of the original macromodel's dynamics by considering an internal structure of each subgroup. Here, each subgroup is represented by an edge of a graph with diffusion or transport occurring along it, while the interactions between the edges are described by interface conditions at the nodes joining them. We prove that with an appropriate scaling of such models, roughly speaking, with fast diffusion or transport combined with slow exchange at the nodes, the solutions of the micromodels are close to the solution to the macromodel.

math.DS↗
arXiv2015

Semigroup approach to diffusion and transport problems on networks

Models describing transport and diffusion processes occurring along the edges of a graph and interlinked by its vertices have been recently receiving a considerable attention. In this paper we generalize such models and consider a network of transport or diffusion operators defined on one dimensional domains and connected through boundary conditions linking the end-points of these domains in an arbitrary way (not necessarily as the edges of a graph are connected). We prove the existence of $C_0$-semigroups solving such problems and provide conditions fully characterizing when they are positive.

math.DS↗
arXiv2012

On a macroscopic limit of a kinetic model of alignment

In the present paper the macroscopic limits of the kinetic model for inter-acting entities (individuals, organisms, cells) are studied. The kinetic model is one-dimensional and entities are characterized by their position and orientation (+/-) with swarming interaction controlled by the sensitivity parameter. The macroscopic limits of the model are considered for solutions close either to the diffusive (isotropic) or to the aligned (swarming) equilibrium states for various sensitivity parameters. In the former case the classical linear difusion equation results whereas in the latter a traveling wave solution does both in the zeroth (`Euler') and frst (`Navier-Stokes') order of approximation.

math.AP↗