We consider a nonlinear parabolic system modelling chemotaxis $u_t=\nabla ·(\nabla u-u\nabla v)$, $v_t=\Delta v+u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

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...

Mathematical models of tumour spheroids, proposed since the early seventies, have been generally formulated in terms of a single diffusive nutrient which is critical for cell replication and cell viability. Only recently, attempts have been made to incorporate in the models the cell energy metabolism, by considering the interplay between glucose, oxygen and lactate (or pH). By assuming glucose and lactate as the only fuel substrates, we propose a simple model for the cell ATP production which takes...

While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. As shown in our previous paper, the zero solution of their equation may lose...

A mathematical model of the microalgal growth under various light regimes is required for the optimization of design parameters and operating conditions in a photobioreactor. As its modelling framework, bilinear system with single input is chosen in this paper. The earlier theoretical results on bilinear systems are adapted and applied to the special class of the so-called intermittent controls which are characterized by rapid switching of light and dark cycles. Based on such approach, the following...

The well-known bottleneck of systems pharmacology, i. e., systems biology applied to pharmacology, refers to the model parameters determination from experimentally measured datasets. This paper represents the development of our earlier studies devoted to inverse (ill-posed) problems of model parameters identification. The key feature of this research is the introduction of control (or periodic forcing by an input signal being a drug intake) of the nonlinear model of drug-induced enzyme production...

In this paper, we will consider blowup solutions to the so called Keller-Segel system and its simplified form. The Keller-Segel system was introduced to describe how cellular slime molds aggregate, owing to the motion of the cells toward a higher concentration of a chemical substance produced by themselves. We will describe a common conjecture in connection with blowup solutions to the Keller-Segel system, and some results for solutions to simplified versions of the Keller-Segel system, giving the...

A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. General conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.

This paper is concerned with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\left(1\right)}_t+u·1=𝔻1-{\chi }_1·\left(1c\right)+{\mu }_{1}1(1-1-a_{1}2)\hfill & \text{in}\times (0,\infty ),\hfill \\ {\left(2\right)}_t+u·2=𝔻2-{\chi }_2·\left(2c\right)+{\mu }_{2}2(1-a_{2}1-2)\hfill & \text{in}\times (0,\infty ),\hfill \\ c_t+u·c=𝔻c-(\alpha 1+\beta 2)c\hfill & \text{in}\times (0,\infty ),\hfill \\ u_t+(u·)u=𝔻u+\nabla P+(\gamma 1+2)\Phi ,·u=0\hfill & \text{in}\times (0,\infty )\hfill \end{array}\right.\end{array}$$ under homogeneous Neumann boundary conditions and initial conditions, where is a bounded domain in R3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier–Stokes system, we can...