Mathematical Models and Methods in Applied Sciences

The papers published in a special issue devoted to chemotactic cross-diffusion in complex frameworks are presented. A brief discussion of their contents aims at opening some perspectives for possible future research.

This paper deals with a micro–macro derivation of a variety of cross-diffusion models for a large system of active particles. Some of the models at the macroscopic scale can be viewed as developments of the classical Keller–Segel model. The first part of the presentation focuses on a survey and a critical analysis of some phenomenological models known in the literature. The second part is devoted to the design of the micro–macro general framework, where methods of the kinetic theory are used to model the dynamics of the system including the case of coupling with a fluid. The third part deals with the derivation of macroscopic models from the underlying description, delivered within a general framework of the kinetic theory.

We consider the coupled chemotaxis Navier–Stokes model with logistic source terms:

This work considers the chemotaxis-growth system

In striking contrast to the corresponding three-dimensional two-component chemo-taxis-growth system to which the global existence or blow-up of classical solutions largely remains open when $\mu >0$ is small, it is shown that whenever $\mu >0,$ $\tau >0$ and $\delta >0$, for any given non-negative and suitably smooth initial data $(u_0,v_0,w_0)$ satisfying $u_0\not\equiv 0$, the system (⋆) admits a unique global classical solution that is uniformly-in-time bounded, which rules out the possibility of blow-up of solutions in finite time or in infinite time.

Moreover, under the fully explicit condition $\mu >\frac{1}{8{\delta }^2}$ the solution $(u,v,w)$ exponentially converges to the constant stationary solution $(1,\frac{1}{\delta },\frac{1}{\delta })$ in the norm of $L^{\infty }(\mathrm{\Omega })$ as $t\to \infty$.

We analyze predator–prey models in which the movement of predator searching for prey is the superposition of random dispersal and taxis directed toward the gradient of concentration of some chemical released by prey (e.g. pheromone), Model II, or released from damaged or injured prey due to predation (e.g. blood), Model I. The logistic O.D.E. describing the dynamics of prey population is coupled to a fully parabolic chemotaxis system describing the dispersion of chemoattractant and predator’s behavior. Global-in-time solutions are proved in any space dimension and stability of homogeneous steady states is shown by linearization for a range of parameters. For space dimension $N\le 2$ the basin of attraction of such a steady state is characterized by means of nonlinear analysis under some structural assumptions. In contrast to Model II, Model I possesses spatially inhomogeneous steady states at least in the case $N=1$.

We investigate a PDE–ODE system describing cancer cell invasion in a tissue network. The model is an extension of the multiscale setting in [G. Meral, C. Stinner and C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dynam. Syst. Ser. B 20 (2015) 189–213] and [C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE–ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal. 46 (2014) 1969–2007], by considering two subpopulations of tumor cells interacting mutually and with the surrounding tissue. According to the go-or-grow hypothesis, these subpopulations consist of moving and proliferating cells, respectively. The mathematical setting also accommodates the effects of some therapy approaches. We prove the global existence of weak solutions to this model and perform numerical simulations to illustrate its behavior for different therapy strategies.