We develop and test a relatively simple enhancement of the classical model reduction method applied to a class of chemical networks with mass conservation properties. Both the methods, being (i) the standard quasi-steady-state approximation method, and (ii) the novel so-called delayed quasi-steady-state approximation method, firstly proposed by Vejchodský (2014), are extensively presented. Both theoretical and numerical issues related to the setting of delays are discussed. Namely, for one slightly...

In this paper, we consider solutions to the following chemotaxis system with general sensitivity $$\left\{\begin{array}{c}\tau u_t=\Delta u-\nabla ·\left(u\nabla \chi \left(v\right)\right)\text{in}\Omega \times (0,\infty ),\hfill \\ \eta v_t=\Delta v-v+u\text{in}\Omega \times (0,\infty ),\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }=\frac{\partial u}{\partial \nu }=0\text{on}\partial \Omega \times (0,\infty ).}\hfill \end{array}\right.$$ Here, $\tau$ and $\eta$ are positive constants, $\chi$ is a smooth function on $(0,\infty )$ satisfying ${\chi }^{\text{'}}(·)>0$ and $\Omega$ is a bounded domain of ${𝐑}^n$ ($n\ge 2$). It is well known that the chemotaxis system with direct sensitivity ($\chi \left(v\right)={\chi }_{0}v$, ${\chi }_0>0$) has blowup solutions in the case where $n\ge 2$. On the other hand, in the case where $\chi \left(v\right)={\chi }_{0}logv$ with $0<{\chi }_0\ll 1$, any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness of...

In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, a sequence of period-doubling bifurcations for small-scale oscillations precedes the transition into the spiking regime. For a wide range of values of the timescale separation parameter, this scenario is recovered numerically. Its relation to the singularly perturbed integrable system is discussed.

This paper extends the volume filling chemotaxis model [18, 26] by taking into account the cell population interactions. The extended chemotaxis models have nonlinear diffusion and chemotactic sensitivity depending on cell population density, which is a modification of the classical Keller-Segel model in which the diffusion and chemotactic sensitivity are constants (linear). The existence and boundedness of global solutions of these models are discussed and...

Many doctors believe that a patient will survive a heart attack unless a succeeding attack occurs in a week. Treating heart attacks as failures in Bernoulli trials we reduce the lifetime after a heart attack to the waiting time for the first failure followed by a success run shorter than a given k. In order to test the "true" critical period of the lifetime we need its distribution. The probability mass function and cumulative distribution function of the waiting time are expressed in explicit and...

Seeds of sunflowers are often modelled by $n⟼{\varphi }_{\theta }\left(n\right)=\sqrt{n}{e}^{2i\pi n\theta }$ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance $2\pi \theta$ for $\theta$ the golden ratio. We associate to such a map ${\varphi }_{\theta }$ a geodesic path ${\gamma }_{\theta }:{ℝ}_{\&gt;0}⟶{\mathrm{PSL}}_2\left(\mathbb{Z}\right)\setminus ℍ$ of the modular curve and use it for local descriptions of the image ${\varphi }_{\theta }\left(\mathbb{N}\right)$ of the phyllotactic map ${\varphi }_{\theta }$.

A pharmacodynamic model introduced earlier in the literature for in silico prediction of rifampicin-induced CYP3A4 enzyme production is described and some aspects of the involved curve-fitting based parameter estimation are discussed. Validation with our own laboratory data shows that the quality of the fit is particularly sensitive with respect to an unknown parameter representing the concentration of the nuclear receptor PXR (pregnane X receptor). A detailed analysis of the influence of that parameter...

This paper contains some results concerning self-similar radial solutions for some system of chemotaxis. This kind of solutions describe asymptotic profiles of arbitrary solutions with small mass. Our approach is based on a fixed point analysis for an appropriate integral operator acting on a suitably defined convex subset of some cone in the space of bounded and continuous functions.

* The author was supported by NSF Grant No. DMS 9706883.Let P be a bi-variate algebraic polynomial of degree n with the real senior part, and Y = {yj }1,n an n-element collection of pairwise noncolinear unit vectors on the real plane. It is proved that there exists a rigid rotation Y^φ of Y by an angle φ = φ(P, Y ) ∈ [0, π/n] such that P equals the sum of n plane wave polynomials, that propagate in the directions ∈ Y^φ .

We consider the following singularly perturbed elliptic problem$$\begin{array}{cc}\hfill {ϵ}^2\Delta u-u+f\left(u\right)& =0,u\&gt;0\text{in}\Omega ,\hfill \\ \hfill ϵ\frac{\partial u}{\partial \nu }+\lambda u& =0\text{on}\partial \Omega ,\hfill \end{array}$$where $f$ satisfies some growth conditions, $0\le \lambda \le +\infty$, and $\Omega \subset {ℝ}^N$ ($N\&gt;1$) is a smooth and bounded domain. The cases $\lambda =0$ (Neumann problem) and $\lambda =+\infty$ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant ${\lambda }_{*}\&gt;1$ such that, as $ϵ\to 0$, the least energy solution has a spike near the boundary if $\lambda \le {\lambda }_{*}$, and has an interior spike near the innermost part of the domain if $\lambda \&gt;{\lambda }_{*}$. Central to our study is the corresponding problem...

We develop the analogy between self-gravitating Brownian particles and bacterial populations. In the high friction limit, the self-gravitating Brownian gas is described by the Smoluchowski-Poisson system. These equations can develop a self-similar collapse leading to a finite time singularity. Coincidentally, the Smoluchowski-Poisson system corresponds to a simplified version of the Keller-Segel model of bacterial populations. In this biological context, it describes the chemotactic aggregation...

Influenza has been responsible for human suffering and economic burden worldwide. Isolation is one of the most effective means to control the disease spread. In this work, we incorporate isolation into a two-strain model of influenza. We find that whether strains of influenza die out or coexist, or only one of them persists, it depends on the basic reproductive number of each influenza strain, cross-immunity between strains, and isolation rate. We propose criteria that may be useful for controlling...

We propose and analyze a nonlinear mathematical model of hematopoiesis, describing the dynamics of stem cell population subject to impulsive perturbations. This is a system of two age-structured partial differential equations with impulses. By integrating these equations over the age, we obtain a system of two nonlinear impulsive differential equations with several discrete delays. This system describes the evolution of the total hematopoietic stem cell populations with impulses. We first examine...

The blow-up of solutions for a parabolic equation with nonlocal exponential nonlinearity is studied.