219 research outputs found
Existence, positivity and stability for a nonlinear model of cellular proliferation
In this paper, we investigate a system of two nonlinear partial differential
equations, arising from a model of cellular proliferation which describes the
production of blood cells in the bone marrow. Due to cellular replication, the
two partial differential equations exhibit a retardation of the maturation
variable and a temporal delay depending on this maturity. We show that this
model has a unique solution which is global under a classical Lipschitz
condition. We also obtain the positivity of the solutions and the local and
global stability of the trivial equilibrium
Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases
Hematopoiesis is a complex biological process that leads to the production
and regulation of blood cells. It is based upon differentiation of stem cells
under the action of growth factors. A mathematical approach of this process is
proposed to carry out explanation on some blood diseases, characterized by
oscillations in circulating blood cells. A system of three differential
equations with delay, corresponding to the cell cycle duration, is analyzed.
The existence of a Hopf bifurcation for a positive steady-state is obtained
through the study of an exponential polynomial characteristic equation with
delay-dependent coefficients. Numerical simulations show that long period
oscillations can be obtained in this model, corresponding to a destabilization
of the feedback regulation between blood cells and growth factors. This
stresses the localization of periodic hematological diseases in the feedback
loop
Functional differential equations with unbounded delay in extrapolation spaces
International audienceWe study the existence, regularity and stability of solutions for nonlinear partial neutral functional differential equations with unbounded de-lay and a Hille-Yosida operator on a Banach space X. We consider two non-linear perturbations: the first one is a function taking its values in X and the second one is a function belonging to a space larger than X, an extrapolated space. We use the extrapolation techniques to prove the existence and regu-larity of solutions and we establish a linearization principle for the stability of the equilibria of our equation
Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics
We study a mathematical model describing the dynamics of a pluripotent stem
cell population involved in the blood production process in the bone marrow.
This model is a differential equation with a time delay. The delay describes
the cell cycle duration and is uniformly distributed on an interval. We obtain
stability conditions independent of the delay. We also show that the
distributed delay can destabilize the entire system. In particularly, it is
shown that Hopf bifurcations can occur
A Dynamic P53-MDM2 Model with Time Delay
Specific activator and repressor transcription factors which bind to specific
regulator DNA sequences, play an important role in gene activity control.
Interactions between genes coding such transcription factors should explain the
different stable or sometimes oscillatory gene activities characteristic for
different tissues. Starting with the model P53-MDM2 described into [6] and the
process described into [5] we developed a new model of this interaction.
Choosing the delay as a bifurcation parameter we study the direction and
stability of the bifurcating periodic solutions. Some numerical examples are
finally given for justifying the theoretical results.Comment: 16 pages, 12 figure
Age-structured model of hematopoiesis dynamics with growth factor-dependent coefficients
International audienceWe propose and analyze an age-structured partial differential model for hematopoietic stem cell dynamics, in which proliferation, differentiation and apoptosis are regulated by growth factor concentrations. By integrating the age-structured system over the age and using the characteristics method, we reduce it to a delay differential system. We investigate the existence and stability of the steady states of the reduced delay differential system. By constructing a Lyapunov function, the trivial steady state, describing cell's dying out, is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state, the most biologically meaningful one, is analyzed using the characteristic equation. This study may be helpful in understanding the uncontrolled proliferation of blood cells in some hematological disorders
Eigenelements of a General Aggregation-Fragmentation Model
We consider a linear integro-differential equation which arises to describe
both aggregation-fragmentation processes and cell division. We prove the
existence of a solution (\lb,\U,\phi) to the related eigenproblem. Such
eigenelements are useful to study the long time asymptotic behaviour of
solutions as well as the steady states when the equation is coupled with an
ODE. Our study concerns a non-constant transport term that can vanish at
since it seems to be relevant to describe some biological processes like
proteins aggregation. Non lower-bounded transport terms bring difficulties to
find estimates. All the work of this paper is to solve this problem
using weighted-norms
Modelling and Asymptotic Stability of a Growth Factor-Dependent Stem Cells Dynamics Model with Distributed Delay
21 pagesInternational audienceUnder the action of growth factors, proliferating and nonproliferating hematopoietic stem cells differentiate and divide, so as to produce blood cells. Growth factors act at different levels in the differentiation process, and we consider their action on the mortality rate (apoptosis) of the proliferating cell population. We propose a mathematical model describing the evolution of a hematopoietic stem cell population under the action of growth factors. It consists of a system of two age-structured evolution equations modelling the dynamics of the stem cell population coupled with a delay differential equation describing the evolution of the growth factor concentration. We first reduce our system of three differential equations to a system of two nonlinear differential equations with two delays and a distributed delay. We investigate some positivity and boundedness properties of the solutions, as well as the existence of steady states. We then analyze the asymptotic stability of the two steady states by studying the characteristic equation with delay-dependent coefficients obtained while linearizing our system. We obtain necessary and sufficient conditions for the global stability of the steady state describing the cell population's dying out, using a Lyapunov function, and we prove the existence of periodic solutions about the other steady state through the existence of a Hopf bifurcation
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