Andrzej Lasota

Globally asymptotic properties of proliferating cell populations

This paper presents a general model for the cell division cycle in a population of cells. Three hypotheses are used: (1) There is a substance (mitogen) produced by cells which is necessary for mitosis; (2) The probability of mitosis is a function of mitogen levels; and (3) At mitosis each daughter cell receives exactly one-half of the mitogen present in the mother cell. With these hypotheses we derive expressions for the α and β curves, the distributions of mitogen and cell cycle times, and the correlation coefficients between mother-daughter (ρmd) and sister-sister (ρss) cell cycle times.The distribution of mitogen levels is shown to be given by the solution to an integral equation, and under very mild assumptions we prove that this distribution is globally asymptotically stable. We further show that the limiting logarithmic slopes of α(t) and β(t) are equal and constant, and that ρmd⩽0 while ρss⩾0. These results are in accord with the experimental results in many different cell lines. Further, the transition probability model of the cell cycle is shown to be a simple special case of the model presented here.

The generic property of existence of solutions of differential equations in Banach space

where f: 77 --P E is continuous. A solution z(*) of (I) will be said to be a solution through us = (to , x0) if (2) is satisfied. Examples of continuous f are known for which there is no solution x defined on any neighborhood of t, . Such examples were constructed in the Banach space c,, by Dieudonne [I] and in Hilbert space by Yorke [2]. A new simplified treatment of the example in [2} is presented here in an appendix. The known existence theorems require f to be either Lipschitzean or compact or more generally to be ol-Lipschitzean [3--51. Let X be the set of continuousf: U -+ I?. We show (Theorem 2) that the set of such f is also meager (first category) in X in the scnsc of the Ba&e Category Theorem. Since preprints of the earliest version of this paper were first circulated in June 19’70, two interesting papers extending some of the ideas have appeared Costello [14] generalizes some ofthe partial differential equation results in [lo], and Vidossich [15] studies generic properties concerning fixed points, with . . apphcatrons to ordinary differential equations.

Asymptotic periodicity of the iterates of Markov operators

On dit que P:L 1 →L 1 est un operateur de Markov si: 1) Pf≥0 pour f≥0 et 2) ∥Pf∥=∥f∥ si f≥0. On montre que tout operateur de Markov P a une certaine decomposition spectrale si, pout tout f∈L 1 avec f≥0 et ∥f∥=1, P n f→#7B-F quand n→∞, ou #7B-F est un sous ensemble fortement compact de L 1

Minimizing therapeutically induced anemia

A model for erythroid production based on a continuous maturationproliferation scheme is developed. The model includes a simple control mechanism operating at the proliferating cell level, and analytic solutions for the time dependent response of the model are derived. Using this model, the response of the erythron to a massive depletion of the proliferating cell compartment (due for example to cytostatic drugs or radiation) is calculated. It is demonstrated that a therapeutic measure designed to decrease the erythroid precursor maturation velocity may considerably ameliorate the deleterious effects of proliferating cell destruction. One way to decrease the erythroid cell maturation rate would be by having the patient breathe in an oxygen enriched atmosphere.

Existence of solutions of two-point boundary value problems for nonlinear systems

Let I‘.” denote the inner product and 1 x / will denote the Euclidean norm. For a matrix A we will say A is positive dejnite and write A > 0 if x . (Ax) > 0 for all x (X f 0) in Rd. We say that A > 0 if either A > 0 or A is identically 0. We assume in our theorems either that A,, A, >, 0 or that A,, A, > 0. Our two-point boundary conditions are analogs of the boundary conditions studied in dimension d = 1 by Keller [9] and Bebernes and Gaines [2], (and the other conditions of our Theorem 3 generalize their conditions). On the other hand, problems (I .I), (1.3), and (1.4) include (by letting A, = A, = 0) the classical two-point boundary value problems (1.1) (1.2)

Asymptotic behavior for differential equations which cannot be locally linearized

For functions ƒ which are continuous and locally Lipschitz the authors define a multi-valued differential Df and prove that if all solutions of the multi-valued differential equation u′ ϵ Df(u) approach zero as t → ∞, then all solutions x(·) of x′ = ƒ(x) with small |x(0)| approach zero exponentially as t → ∞. If ƒ is continuously differentiable, then Df coincides with the (single-valued) Frechet differential of ƒ. Other results on the asymptotic behavior of solutions of perturbed, multi-valued differential equations are presented.

The statistical dynamics of recurrent biological events

In this paper we develop a general modeling framework within which many models for systems which produce events at irregular times through a combination of probabilistic and deterministic dynamics can be comprehended. We state and prove new sufficient conditions for the global asymptotic behaviour of the density evolution in these systems, and apply our results to many previously published models for the cell division cycle. In addition, we develop a new interpretation for the statistics of action potential production in excitable cells.

Noise-Induced Global Asymptotic Stability

We prove analytically that additive and parametric (multiplicative) Gaussian distributed white noise, interpreted in either the Itô or Stratonovich formalism, induces global asymptotic stability in two prototypical dynamical systems designated as supercritical (the Landau equation) and subcritical, respectively. In both systems without noise, variation of a parameter leads to a switching between a single, globally stable steady state and multiple, locally stable steady states. With additive noise this switching is mirrored in the behavior of the extrema of probability densities at the same value of the parameter. However, parametric noise causes a noise-amplitude-dependent shift (postponement) in the parameter value at which the switching occurs. It is shown analytically that the density converges to a Dirac delta function when the solution of the Fokker-Planck equation is no longer normalizable.

Stochastic perturbation of dynamical systems: The weak convergence of measures

In this paper we consider the asymptotic behaviour of randomly perturbed discrete dynamical systems. We treat this problem by examining the evolution of the corresponding sequences of distributions. We show that an average contractive property is sufficient to ensure the weak convergence of the sequence of distributions to a unique stationary measure.

From fractals to stochastic differential equations

New sufficient conditions for asymptotic stability of Markov operators on metric spaces are presented. These criterion are applied to iterated function systems and stochastic differential equations with Poisson type perturbations.