Qualitative analysis of a coupled reaction-diffusion model in biology with time delays
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.
time-delayed ordinary differential equations
linear parabolic partial differential equations
Goodwin model for biochemical control of genes
coupled system of delay-differential and reaction-diffusion equations
scientific article; zbMATH DE number 3934827