We study the existence and some properties of travelling waves in partially
degenerate reaction-diffusion systems. Such systems may for example describe intracellular
calcium dynamics in the presence of immobile buffers. In order to prove the wave existence,
we first consider the non degenerate case and then pass to the limit as some of the diffusion
coefficient converge to zero. The passage to the limit is based on a priori estimates of
solutions independent of the values of the diffusion coefficients....
In this work we study a nonlocal reaction-diffusion equation arising in population
dynamics. The integral term in the nonlinearity describes nonlocal stimulation of
reproduction. We prove existence of travelling wave solutions by the Leray-Schauder method
using topological degree for Fredholm and proper operators and special a priori estimates
of solutions in weighted Hölder spaces.
We obtain solvability conditions in
(ℝ) for a
sixth order partial differential equation which is the linearized Cahn-Hilliard problem
using the results derived for a Schrödinger type operator without Fredholm property in our
preceding article [18].
In this paper we propose a mathematical model to describe the evolution of leukemia
in the bone marrow. The model is based on a reaction-diffusion system of equations in a porous
medium. We show the existence of two stationary solutions, one of them corresponds to the normal
case and another one to the pathological case. The leukemic state appears as a result of a bifurcation
when the normal state loses its stability. The critical conditions of leukemia development
are determined by the proliferation...
Plant growth occurs due to cell proliferation in the meristem. We model the case of
apical meristem specific for branch growth and the case of basal meristem specific for
bulbous plants and grass. In the case of apical growth, our model allows us to describe
the variety of plant forms and lifetimes, endogenous rhythms and apical domination. In the
case of basal growth, the spatial structure, which corresponds to the appearance of
leaves, results...
Intra-specific competition in population dynamics can be described by integro-differential
equations where the integral term corresponds to nonlocal consumption of resources by individuals
of the same population. Already the single integro-differential equation can show the
emergence of nonhomogeneous in space stationary structures and can be used to model the process
of speciation, in particular, the emergence of biological species during evolution [S. Genieys et al., Math. Model. Nat. Phenom....
We study a reaction-diffusion equation
with an integral term describing nonlocal consumption of resources
in population dynamics. We show that a homogeneous equilibrium can
lose its stability resulting in appearance of stationary spatial
structures. They can be related to the emergence of biological
species due to the intra-specific competition and random
mutations.
Various types of travelling waves are observed.
The paper is devoted to solvability conditions for linear elliptic problems with non-Fredholm operators. We show that the operator becomes normally solvable with a finite-dimensional kernel on properly chosen subspaces. In the particular case of a scalar equation we obtain necessary and sufficient solvability conditions. These results are used to apply the implicit function theorem for a nonlinear elliptic problem; we demonstrate the persistence of travelling wave solutions to spatially periodic...
In this work, we introduce a new software created to study hematopoiesis at the cell
population level with the individually based approach. It can be used as an interface between theoretical
works on population dynamics and experimental observations. We show that this software
can be useful to study some features of normal hematopoiesis as well as some blood diseases such
as myelogenous leukemia. It is also possible to simulate cell communication and the formation of
cell colonies in the bone marrow.
...
Evolution of cell populations can be described with dissipative particle dynamics, where
each cell moves according to the balance of forces acting on it, or with partial
differential equations, where cell population is considered as a continuous medium. We
compare these two approaches for some model examples
A hybrid model of red blood cell production, where cells are considered as discrete
objects while intra-cellular proteins and extra-cellular biochemical substances are
described with continuous models, is proposed. Spatial organization and regulation of red
blood cell production (erythropoiesis) are investigated. Normal erythropoiesis is
simulated in two dimensions, and the influence on the output of the model of some
parameters involved in cell...
We study the influence of natural convection on stability of reaction fronts in
porous media. The model consists of the heat equation, of the equation for the depth of
conversion and of the equations of motion under the Darcy law. Linear stability analysis of
the problem is fulfilled, the stability boundary is found. Direct numerical simulations are
performed and compared with the linear stability analysis.
Propagation of polymerization fronts with liquid monomer and liquid polymer is considered
and the influence of vibrations on critical conditions of convective instability is
studied. The model includes the heat equation, the equation for the concentration and the
Navier-Stokes equations considered under the Boussinesq approximation. Linear stability
analysis of the problem is fulfilled, and the convective instability boundary is found
depending on...
The paper is devoted to mathematical modelling of erythropoiesis,
production of red blood cells in the bone marrow.
We discuss intra-cellular regulatory networks which determine
self-renewal and differentiation of erythroid progenitors.
In the case of excessive self-renewal, immature cells can fill
the bone marrow resulting in the development of leukemia.
We introduce a parameter characterizing the strength of mutation.
Depending on its value, leukemia will or will not develop.
The simplest...
In this work we study the inflammatory process resulting in the development of
atherosclerosis. We develop a one- and two-dimensional models based on reaction-diffusion
systems to describe the set up of a chronic inflammatory response in the intima of an
artery vessel wall. The concentration of the oxidized low density lipoproteins (ox-LDL) in
the intima is the critical parameter of the model. Low ox-LDL concentrations do not lead
to a chronic inflammatory reaction. Intermediate ox-LDL concentrations...
Various particle methods are widely used to model dynamics of complex media. In this work
molecular dynamics and dissipative particles dynamics are applied to model blood flows
composed of plasma and erythrocytes. The properties of the homogeneous particle fluid are
studied. Capillary flows with erythrocytes are investigated.
Composition gradients in miscible liquids can create volume forces resulting in various
interfacial phenomena. Experimental observations of these phenomena are related to some difficulties
because they are transient, sufficiently weak and can be hidden by gravity driven flows.
As a consequence, the question about their existence and about adequate mathematical models is
not yet completely elucidated. In this work we present some experimental evidences of interfacial
phenomena in miscible liquids...
Marangoni convection caused by a photochemical reaction of the type A B in a deep liquid layer is studied. Linear stability analysis is performed and the conditions
for Marangoni convection to occur are obtained. It is shown that increasing the rate of the
direct reaction, for example, by increasing the light intensity, destabilizes the steady state
and causes convective motion of the fluid, whereas increasing the rate of the inverse reaction
stabilizes the steady state. A weakly nonlinear analysis...
Propagation of chemical waves at very low temperatures, observed
experimentally [V.V. Barelko , Advances in Chem. Phys. 74 (1988), 339-384.] at velocities of order 10 cm/s, is due to
a very non- standard physical mechanism. The energy liberated by
the chemical reaction induces destruction of the material, thereby
facilitating the reaction, a process very different from standard
combustion. In this work we present recent experimental results and develop a
new mathematical model which takes into...
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