In the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using a expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy . An numerical example is also given to illustrate the effectiveness of the algorithm.

We consider a diffuse interface model for tumour growth consisting of a Cahn–Hilliard equation with source terms coupled to a reaction–diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn–Hilliard equation leads to new difficulties when one aims to derive a priori estimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.

We consider a theoretical model for the spread of avian influenza in a poultry population. An avian influenza epidemic model incorporating spatial spread as a diffusive process is discussed, where the infected individuals are restricted from moving to prevent spatial transmission but infection occurs when susceptible individuals come into contact with infected individuals or the virus is contracted from the contaminated environment (e.g. through water or food). The infection is assumed to spread radially and isotropically. After a stability and phase plane analysis of the equivalent system of ordinary differential equations, it is shown that an analytical solution can be obtained in the form of a travelling wave. We outline the methodology for finding such analytical solutions using a travelling wave coordinate when the wave is assumed to move at constant speed. Numerical simulations also produce the travelling wave solution, and a comparison is made with some predictions based on empirical data reported in the literature.

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h1/2). As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

We consider the early carcinogenesis model originally proposed as a deterministicreaction-diffusion system. The model has been conceived to explore the spatial effectsstemming from growth regulation of pre-cancerous cells by diffusing growth factormolecules. The model exhibited Turing instability producing transient spatial spikes incell density, which might be considered a model counterpart of emerging foci of malignantcells. However, the process of diffusion of growth factor molecules is by its nature astochastic random walk. An interesting question emerges to what extent the dynamics of thedeterministic diffusion model approximates the stochastic process generated by the model.We address this question using simulations with a new software tool called sbioPN (spatialbiological Petri Nets). The conclusion is that whereas single-realization dynamics of thestochastic process is very different from the behavior of the reaction diffusion system,it is becoming more similar when averaged over a large number of realizations. The degreeof similarity depends on model parameters. Interestingly, despite the differences, typicalrealizations of the stochastic process include spikes of cell density, which however arespread more uniformly and are less dependent of initial conditions than those produced bythe reaction-diffusion system.

Cancer spread is a dynamical process occurring not only in time but also in space which, for solid tumors at least, can be modeled quantitatively by reaction and diffusion equations with a bistable behavior: tumor cell colonization happens in a portion of tissue and propagates, but in some cases the process is stopped. Such a cancer proliferation/extintion dynamics is obtained in many mathematical models as a limit of complicated interacting biological fields. In this article we present a very basic model of cancer proliferation adopting the bistable equation for a single tumor cell dynamics. The reaction-diffusion theory is numerically and analytically studied and then extended in order to take into account dispersal effects in cancer progression in analogy with ecological models based on the porous medium equation. Possible implications of this approach for explanation and prediction of tumor development on the lines of existing studies on brain cancer progression are discussed. The potential role of continuum models in connecting the two predominant interpretative theories about cancer, once formalized in appropriate mathematical terms, is discussed.

We investigate in this paper the dependence relation between the space–time periodic coefficients A, q and μ of the reaction–diffusion equation and the spreading speed of the solutions of the Cauchy problem associated with compactly supported initial data. We prove in particular that (1) taking the spatial or temporal average of μ decreases the minimal speed, (2) if μ is not constant with respect to x, then increasing the amplitude of the diffusion matrix A does not necessarily increase the minimal speed and (3) if A = IN, μ is a constant, then the introduction of a space periodic drift term q = ∇Q decreases the minimal speed. In order to prove these results, we use a variational characterisation of the spreading speed that involves a family of periodic principal eigenvalues associated with the linearisation of the equation near zero. We are thus back to the investigation of the dependence relation between this family of eigenvalues and the coefficients.

Bacillus subtilis swarms rapidly over the surface of a synthetic mediumcreating remarkable hyperbranched dendritic communities. Models to reproduce such effectshave been proposed under the form of parabolic Partial Differential Equations representingthe 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 ofsuch models and compare them to relevant experimental data together with a criticalanalysis of the validity of the models based on recent observations of the swarmingbacteria which show that nutrients are not limitating but distinct subpopulations growingat different rates are likely present.

Many models in biology and ecology can be described by reaction-diffusionequations wit time delay. One of important solutions for these type of equationsis the traveling wave solution that shows the phenomenon of wave propagation.The existence of traveling wave fronts has been proved for large class ofequations, in particular, the monotone systems, such as the cooperativesystems and some competition systems. However, the problem on the uniqueness oftraveling wave (for a fixed wave speed) remains unsolved. In this paper, we showthat, for a class of monotone diffusion systems with time delayed reaction term,the mono-stable traveling wave font is unique whenever it exists.

In this paper we explore a new model of field carcinogenesis, inspired by lungcancer precursor lesions, which includes dynamics of a spatially distributed population ofpre-cancerous cells c(t, x), constantly supplied by an influx μ of mutated normal cells. Cellproliferation is controlled by growth factor molecules bound to cells, b(t, x). Free growthfactor molecules g(t, x) are produced by precancerous cells and may diffuse before they becomebound to other cells. The purpose of modelling is to investigate the existence of solutions,which correspond to formation of multiple spatially isolated lesions of pre-cancerous cellsor, mathematically, to stable spike solutions. These multiple lesions are consistent with thefield theory of carcinogenesis. In a previous model published by these authors, the influxof mutated cells was equal to zero, μ = 0, which corresponded to a single pre-malignantcolony of cells. In that model, stable patterns appeared only if some of the growth factorwas supplied from outside, arguably, a biologically tenuous hypothesis. In the present model,when μ > 0, that hypothesis is no more required, which makes this model more realistic.We present a range of results, both mathematical and computational, which taken togetherallow understanding the dynamics of this model. The equilibrium solutions in the currentmodel result from the balance between new premalignant colonies being initiated and theold ones dying out.

We prove uniqueness of the invariant measure and the exponential convergence to equilibriumfor a stochastic dissipative system whose drift is perturbed by a bounded function.