##
**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.

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)

### MSC:

92D25 | Population dynamics (general) |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

35B35 | Stability in context of PDEs |

### Keywords:

cell cycle; age and size structured model; global stability; cell population dynamics; proliferating phase; resting phase; maturation; delay differential equation; asymptotic distribution; time-dependent transport equation; asymptotically stable steady-state density; time- independent transport equation; constrictive Markov operators
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\textit{M. C. Mackey} and \textit{R. Rudnicki}, J. Math. Biol. 33, No. 1, 89--109 (1994; Zbl 0833.92014)

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### References:

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