Global stability in a delayed partial differential equation describing cellular replication. (English) Zbl 0833.92014

The paper considers a general model of cell population dynamics with two phases: the proliferating phase (\(p\)-phase) during which some maturation variable \(m\) grows according to an equation \(dm/dt= V(m (t))\) and the resting phase \(G_0\) (or \(n\)-phase), where there is no growth. The duration of the \(p\)-phase is supposed to last for a fixed period of time \(\tau\), after which each cell divides into two daughter cells. Daughter cells start their life in the \(n\)-phase. They may stay in that phase for an indefinitely long time. At each instant, a fraction \(\beta\) of the resting cells enter the \(p\)-phase; \(\beta\) is a function of the total number of the resting cells \(\overline {N}\).
The model is considered at three levels: at the individual level, cells are classified according to whether they are in the \(p\)-phase or in the \(n\)-phase, also according to their age and the maturation \(m\). At the next level, they are only distinguished by the maturation \((P, N)\). At the upper level, just the total number of (resting) cells is considered \((\overline {N})\).
For the analysis, just resting cells are considered: the dynamics of the proliferating cells follow from theirs. Under simplifying assumptions, a delay differential equation is derived for \(\overline {N}\), whose behavior has been thoroughly studied in the literature. The asymptotic distribution of resting cells with respect to \(m\) (at the intermediate level) is then investigated, that is, the density: \(M= N/\overline {N}\). \(M\) verifies a time-dependent transport equation with deviated arguments. In the case when \(\overline {N} (t)\) tends to a limit as \(t\to +\infty\), fast enough, the transport equation has a time-independent limit. It is shown (Proposition 1) that solutions of both equations with same initial value behave the same asymptotically.
The main result of the paper (theorem 1, section 4) establishes the existence of an asymptotically stable steady-state density for the time- independent transport equation. The proof is done in two steps: (1) A quasi-convergence property is shown, that is, the difference of any two solutions tends to zero; (2) Existence of a steady-state is proved, using a result due to J. Komornik [Tôhoku Math. J., II. Ser. 38, 15-27 (1986; Zbl 0578.47020)] on fixed points of the so-called constrictive Markov operators. As a preparatory step, the authors introduce a change of variable on \(m\) by which the equation is transformed into a similar one with \(V\) changed to \(W(x)= cx\), where \(c\) is a constant.
Comparative remarks with the literature made in section 5 of the paper point to some of its specific features: non eventual compactness is the most remarkable one, in contrast to earlier related work by M. Gyllenberg and H. J. A. M. Heijmans [SIAM J. Math. Anal. 18, 74-88 (1987; Zbl 0634.34064)], or also by O. Arino and M. Kimmel [J. Math. Biol. 27, No. 3, 341-354 (1989; Zbl 0715.92011)]. (The latter reference is not quoted in the paper. There are a few other related works, not many.) The distinguishing trait may be the fact that the growth rate function \(V\) is such that \(V(0) =0\), which induces a singular kernel. Comparison is also made with earlier studies by Mackey and his collaborators, in which nonlinearities of a “wilder” type than here, associated with the same singular growth function, are shown numerically to give rise to complex behavior of some of the solutions.
Reviewer: O.Arino (Pau)


92D25 Population dynamics (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B35 Stability in context of PDEs
Full Text: DOI


[1] Brunovský, P.: NOtes on chaos in the cell population partial differential equation. Nonlin. Anal. 7, 167–176 (1983) · Zbl 0523.35017
[2] Brunovský, P. and Komornik, J.: Ergodicity and exactness of the shift on C[0, and the semiflow of a first order partial differential equation. J. Math. Anal. Applic. 104, 235–245 (1984) · Zbl 0602.35070
[3] Burns, F. J. and Tannock, I. F.: On the existence of a G 0 phase in the cell cycle. Cell Tissue Kinet. 19, 321–334 (1970)
[4] Crabb, R., Losson, J., Mackey, M. C.: Solution multistability in differential delay equations. Proc. Inter. Conf. Nonlin. Anal. (Tampa Bay) (in press). · Zbl 0846.34069
[5] Diekmann, O., Heijmans, H. J. A. M., Thieme, H. R.: On the stability of the cell size distribution. J. Math. Biol. 19, 227–248 (1984) · Zbl 0543.92021
[6] Gyllenberg, M., Heijmans, H. J. A. M.: An abstract delay differential equation modelling size dependent cell growth and division. SIAM J. Math. Anal. 18, 74–88 (1987) · Zbl 0634.34064
[7] Hale, J.: Theory of Functional Differential Equations. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0352.34001
[8] Komornik, J.: Asymptotic periodicity of the iterates of Markov operators. Tôhoku Math. J. 38, 15–27 (1986) · Zbl 0592.47025
[9] Lasota, A.: Stable and chaotic solutions of a first order partial differential equation. Nonlin. Anal. 5, 1181–1193 (1981) · Zbl 0523.35015
[10] Lasota, A., Mackey, M. C.: Globally asymptotic properties of proliferating cell populations. J. Math. Biol. 19, 43–62 (1984) · Zbl 0529.92011
[11] Lasota, A., Loskot, K., Mackey, M. C.: Stability properties of proliferatively coupled cell replication models. Acta Biotheor. 39, 1–14 (1991)
[12] Loskot, K.: Turbulent solutions of a first order partial differential equation. J. Diff. Eqn. 58, 1–14 (1985) · Zbl 0588.35012
[13] Losson, J., Mackey, M.C., Longtin, A.: Solution multistability in first order nonlinear differential delay equations. Chaos 3, 167–176 (1993) · Zbl 1055.34510
[14] Mackey, M. C.: Unified hypothesis for the origin of a plastic anemia and periodic hematopoiesis. Blood 51, 941–956 (1978)
[15] Mackey, M. C.: Dynamic haematological disorders of stem cell origin. In: Biophysical and Biochemical Information Transfer in Recognition, pp. 373–409 (eds) Vassileva-Popova, J. G., Jensen, E. V. New York: Plenum Press 1979
[16] Mackey, M. C., Dörmer, P. (1981). Enigmatic hemopoiesis. In: Biomathematics and Cell Kinetics, pp. 87–103 (ed.) Rotenberg, M., North Holland, Elsevier 1981
[17] Mackey, M. C., Dörmer, P.: Continuous maturation of proliferating erythroid precursors. Cell Tissue Kinet. 15, 381–392 (1982)
[18] Mackey, M. C., Milton, J. G.: Feedback, delays and the origin of blood cell dynamics. Comm. Theor. Biol. 1, 299–327 (1990)
[19] Metz, J. A. J., Diekmann, O. (eds.): The Dynamics of Physiologically Structured Populations. Berlin, Heidelberg, New York: Springer 1986 · Zbl 0614.92014
[20] Rey, A., Mackey, M. C.: Bifurcations and travelling waves in a delayed partial differential equation. Chaos 2, 231–244 (1992) · Zbl 1055.92512
[21] Rey, A., Mackey, M. C.: Multistability and boundary layer development in a transport equation with retarded arguments. Can. Appl. Math. Quar. 1, 1–21 (1993) · Zbl 0783.92028
[22] Rudnicki, R.: Invariant measures for the flow of a first order partial differential equation. Ergod. Th. & Dynam. Sys. 5, 437–443 (1985) · Zbl 0566.28013
[23] Rudnicki, R.: An abstract Wiener measure invariant under a partial differential equation. Bull. Pol. Acad. Sci. (Math.) 35, 289–295 (1987) · Zbl 0643.28014
[24] Rudnicki, R.: Strong ergodic properties of a first order partial differential equation. J. Math. Anal. Applic. 132, 14–26 (1988) · Zbl 0673.35012
[25] Smith, J. A., Martin, L.: Do cells cycle? Proc. Natl. Acad. Sci. U.S.A. 70, 1263–1267 (1973)
[26] Walther, H. O.: An invariant manifold of slowly oscillating solutions for \(\dot x(t) = - \mu x(t) + f(x(t - 1))\) . J. Reine Angew. Math. 414, 67–112 (1991) · Zbl 0708.34036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.