In this paper we present a nonstandard numerical scheme for solving a reaction-diffusion-chemotaxis mathematical model of migration and invasion of cancer cells. The model is approximated by a system of nonlinear algebraic equations, whose solution is non-negative. The role of the production rate of cancer cells on their invasive ability is analyzed by numerical experiments.

We propose a new model of computation based on Nonstandard Analysis. Intuitively, the role of ‘algorithm’ is played by a new notion of finite procedure, called Ω-invariance and inspired by physics, from Nonstandard Analysis. Moreover, the role of ‘proof’ is taken up by the Transfer Principle from Nonstandard Analysis. We obtain a number of results in Constructive Reverse Mathematics to illustra...

We analyze a class of predator-prey models with BeddingtonDeAngelis type functional response. The models incorporate the mutual interference between predators, which stabilizes predator-prey interactions even when only a linear intrinsic growth rate of the prey population is considered. Positive and elementary stable nonstandard (PESN) finite-difference methods, having the same qualitative feat...

Predator-prey systems with linear and logistic intrinsic growth rate of the prey are analyzed. The models incorporate the mutual interference between predators into the functional response which stabilizes predatorprey interactions in the system. Positive and elementary stable nonstandard (PESN) finite-difference methods, having the same qualitative features as the corresponding continuous pred...

Discrete-time Lotka-Volterra competition models are obtained by applying nonstandard finite difference (NSFD) schemes to the continuous-time counterparts of the model. The NSFD methods are noncanonical symplectic numerical schemes when applying to the predator-prey model x = x − xy and y = −y + xy. The local dynamics of the discrete-time model are analyzed and compared with the continuous model...

The optimal rate of convergence of the wave equation in both the energy and the L-norms using continuous Galerkin method is well known. We exploit this technique and design a fully discrete scheme consisting of coupling the nonstandard finite difference method in the time and the continuous Galerkin method in the space variables. We show that, for sufficiently smooth solution, the maximal error...

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 predato...