Qualitative analysis of a coupled reaction-diffusion model in biology with time delays
type: Item
instance of: scholarly article
title: Qualitative analysis of a coupled reaction-diffusion model in biology with time delays
zbMATH Open document ID: 0583.92012
DOI: 10.1016/0022-247X(85)90156-8
author: C. V. Pao
author: Joseph M. Mahaffy
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: This paper analyzes in terms of existence and stability a coupled system of delay-differential and reaction-diffusion equations. Such systems have been recently considered in the study of biological processes. Thus, a cell can be considered as a heterogeneous place with well-mixed sites where substances are produced (the nucleus, the cell membrane are such sites) and sites where these substances are only subject to displacements. Until recently, these systems were treated as only differential when emphasizing the behavior for large times or as reaction-diffusion when looking at nonlinearities. The paper in question considers a cell as consisting of 3 compartments, 2 being well-mixed and the third one with a diffusion process in a 1- dimensional setting. There are 6 equations, two for each compartment, governing the evolution w.r. to time and space of 2 species (mRNA and a repressor protein). The evolution in the different compartments is connected through membrane exchanges (nucleus/cytoplasm, membrane cell/cytoplasm). The system is too complicated to be quoted here. Let us indicate that there are delays in the equations, particularly in a nonlinear term f. In proving existence, the authors use the method of lower-upper solutions; a special flavour due to the delay is that both lower and upper solutions are coupled and such a coupled sequence is proved to converge towards a solution under very mild conditions on f. After that, they prove that under a condition on f depending only on reaction rates, the stationary solution is globally asymptotically stable, independently on the delays. The proof of this is also based on the determination of a family of lower and upper solutions around the stationary solution, approaching it exponentially. Other contributions along this line have been done by the present authors [see J. Math. Biol. 20, 39-57 (1984; Zbl 0577.92010)] and more recently by S. Busenberg and the second author in a 3-dimensional setting.
Mathematics Subject Classification ID: 92Cxx
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 3934827
zbMATH Keywords: biochemical reaction
zbMATH Keywords: time-delayed ordinary differential equations
zbMATH Keywords: linear parabolic partial differential equations
zbMATH Keywords: three-compartment model
zbMATH Keywords: Goodwin model for biochemical control of genes
zbMATH Keywords: negative feedback mechanism
zbMATH Keywords: coupled system of delay-differential and reaction-diffusion equations
zbMATH Keywords: mRNA
zbMATH Keywords: repressor protein
zbMATH Keywords: membrane exchanges
zbMATH Keywords: nucleus
zbMATH Keywords: cytoplasm
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: stationary solution
reviewed by: Ovide Arino
MaRDI profile type: Publication
cites work: A global stability criterion for simple control loops
cites work: Global asymptotic stability of certain models for protein synthesis and repression
cites work: Models of genetic control by repression with time delays and spatial effects
cites work: Reaction diffusion equations with nonlinear boundary conditions
cites work: A Boltzmann-type equation in the kinetic theory of vehicular traffic
cites work: Q5533735
cites work: Q5540565
cites work: Q3895289
full work available at URL: 0022-247x(85)90156-8
OpenAlex ID: W2031207886
description: scientific article; zbMATH DE number 3934827