Theoretical framework for linear stability of an anomalous sub-diffusive activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on anomaly exponents of various species. In addition to monotonous instability, known from normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly exponents for both species the type of unstable modes is determined by the ratio of the reactants' diffusion coefficients. When the ratio exceeds its normal...

Relaxation oscillations are limit cycles with two clearly different time scales. In this article the spatio-temporal dynamics of a standard prey-predator system in the parameter region of relaxation oscillation is investigated. Both prey and predator population are distributed irregularly at a relatively high average level between a maximal and a minimal value. However, the slowly developing complex pattern exhibits a feature of “inverse excitability”: Both populations show collapses which occur...

The aim of this paper is to present some approaches to tumour growth modelling using the logistic equation. As the first approach the well-known ordinary differential equation is used to model the EAT in mice. For the same kind of tumour, a logistic equation with time delay is also used. As the second approach, a logistic equation with diffusion is proposed. In this case a delay argument in the reaction term is also considered. Some mathematical properties of the presented models are studied in...

We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $\epsilon cₜ=k_c\Delta c-f\left(c\right)c+g(a,c)$, x ∈ Ω, t > 0, for $\Omega \subset {ℝ}^N$, completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^p$-estimates.

In this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction...

We study a class of parabolic-ODE systems modeling tumor growth, its mathematical modeling and the global in time existence of the solution obtained by the method of Lyapunov functions.

The paper is devoted to mathematical modelling and numerical computations of a nonstationary free boundary problem. The model is based on processes of molecular diffusion of some products of chemical decomposition of a solid organic substance concentrated in bottom sediments. It takes into account non-stationary multi-component and multi-stage chemical decomposition of organic substances and the processes of sorption desorption under aerobic and anaerobic conditions. Such a model allows one to...

We consider the early carcinogenesis model originally proposed as a deterministic reaction-diffusion system. The model has been conceived to explore the spatial effects stemming from growth regulation of pre-cancerous cells by diffusing growth factor molecules. The model exhibited Turing instability producing transient spatial spikes in cell density, which might be considered a model counterpart of emerging foci of malignant cells. However, the process...

We study the unsaturated flow of an incompressible liquid carrying a bacterial population through a porous medium contaminated with some pollutant. The biomass grows feeding on the pollutant and affecting at the same time all the physics of the flow. We formulate a mathematical model in a one-dimensional setting and we prove an existence theorem for it. The so-called fluid media scaling approach, often used in the literature, is discussed and its limitations are pointed out on the basis of a specific...

We present a model coupling the fire propagation equations in a bidimensional domain representing the surface, and the air movement equations in a three dimensional domain representing an air layer. As the air layer thickness is small compared with its length, an asymptotic analysis gives a three dimensional convective model governed by a bidimensional equation verified by a stream function. We also present the numerical simulations of these equations.

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...

Bacillus subtilis swarms rapidly over the surface of a synthetic medium creating remarkable hyperbranched dendritic communities. Models to reproduce such effects have been proposed under the form of parabolic Partial Differential Equations representing the dynamics of the active cells (both motile and multiplying), the passive cells (non-motile and non-growing) and nutrient concentration. We test the numerical behavior of such models and compare...

In [A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministic markov process in infinite dimensions. Adv. Appl. Probab. 44 (2012) 749–773], the authors addressed the question of averaging for a slow-fast Piecewise Deterministic Markov Process (PDMP) in infinite dimensions. In the present paper, we carry on and complete this work by the mathematical analysis of the fluctuations of the slow-fast system around the averaged limit. A central limit theorem is derived and the associated...

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form $\partial u_{\epsilon }^{\omega }/\partial t+1/{ϵ}_3𝒞\left(T_3(x/{\epsilon }_3){\omega }_3\right)·\nabla u_{\epsilon }^{\omega }-div\left(\alpha \left(T_1(x/{\epsilon }_1){\omega }_1,T_2(x/{\epsilon }_2){\omega }_2,t\right)\nabla u_{\epsilon }^{\omega }\right)=f$. It is shown, under certain structure assumptions on the random vector field $𝒞\left({\omega }_3\right)$ and the random map $\alpha ({\omega }_1,{\omega }_2,t)$, that the sequence $\left\{u_{ϵ}^{\omega }\right\}$ of solutions converges in the sense of G-convergence of parabolic operators to the solution $u$ of the homogenized problem $\partial u/\partial t-div\left(ℬ\right(t\left)\nabla u\right)=f$.