Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary

The main goal of this paper is to study a stationary problem arising from angiogenesis, including terms of chemotaxis and flux at the boundary of the tumor. We give sufficient conditions on terms of the data of the problems assuring the existence of positive solutions.Ministerio de Educación y Cienci

Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences

We design nonstandard finite difference (NSFD) schemes which are unconditionally dynamically consistent with respect to the positivity property of solutions of cross-diffusion equations in biosciences. This settles a problem that was open for quite some time. The study is done in the setting of three concrete and highly relevant cross-diffusion systems regarding tumor growth, convective predator–prey pursuit and evasion model and reaction–diffusion–chemotaxis model. It is shown that NSFD schemes used for classical reaction–diffusion equations, such as the Fisher equation, for which the solutions enjoy the positivity property, are not appropriate for cross-diffusion systems. The reliable NSFD schemes are therefore obtained by considering a suitable implementation on the crossdiffusive term of Mickens’ rule of nonlocal approximation of nonlinear terms, apart from his rule of complex denominator function of discrete derivatives. We provide numerical experiments that support the theory as well as the power of the NSFD schemes over the standard ones. In the case of the cancer growth model, we demonstrate computationally that our NSFD schemes replicate the property of traveling wave solutions of developing shocks observed in Marchant et al. (2000).South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation : SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.elsevier.com/locate/camwa2015-11-30hb201

Competitive exclusion in a two-especies chemotasis model

We consider a mathematical model for the spatio-temporal evolution of two biological species in a competitive situation. Besides diffusing, both species move toward higher concentrations of a chemical substance which is produced by themselves. The resulting system consists of two parabolic equations with Lotka–Volterra-type kinetic terms and chemotactic cross-diffusion, along with an elliptic equation describing the behavior of the chemical. We study the question in how far the phenomenon of competitive exclusion occurs in such a context. We identify parameter regimes for which indeed one of the species dies out asymptotically, whereas the other reaches its carrying capacity in the large time limit

Existence of global solution and nontrivial steady states for a system modeling chemotaxis

We consider a reaction-diffusion system modeling chemotaxis, which describes the situation of two species of bacteria competing for the same nutrient. We use Moser-Alikakos iteration to prove the global existence of the solution. We also study the existence of nontrivial steady state solutions and their stability

Advanced numerical treatment of chemotaxis driven PDEs in mathematical biology

From the first formulation of chemotaxis-driven partial differential equations (PDEs) by Keller and Segel in the 1970's up to the present, much effort has been expended in modelling complex chemotaxis re- lated processes. The shear complexity of such resulting PDEs crucially limits the postulation of analytical results. In this context the sup- port by numerical tools are of utmost interest and, thus, render the implementation of a numerically well elaborated solver an undoubt- edly important task. In this work I present different iteration strategies (linear/nonlinear, decoupled/monolithic) for chemotaxis-driven PDEs. The discretiza- tion follows the method of lines, where I employ finite elements to resolve the spatial discretization. I extensively study the numerical efficiency of the iteration strategies by applying them on particular chemotaxis models. Moreover, I demonstrate the need of numerical stabilization of chemotaxis-driven PDEs and apply a exible scalar algebraic ux correction. This methodology preserves the positivity of the fully discretized scheme under mild conditions and renders the numerical solution non-oscillatory at a low level of additional compu- tational costs. This work provides a first detailed study of accurate, efficient and exible finite element schemes for chemotaxis-driven PDEs and the implemented numerical framework provides a valuable basis for fu- ture applications of the solvers to more complex models