We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the ’attitude areas’, which depend on the distance between the agents’ opinions. The position of the agents on the network evolves based on the distance between the agents’ opinions. The goal is to study radicalisation, polarisation and fragmentation of the population while changing its open-mindedness and the radius of interaction.

A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

We propose a numerical method for the simulation of a quasi-linear parabolic biofilm model that exhibits three non-linear diffusion effects: (i) a power law degeneracy, (ii) a super diffusion singularity and (iii) non-linear cross-diffusion. The method is based on a spatial Finite Volume discretisation in which cross-diffusion terms are formally treated as convection terms. Time-integration of the resulting semi-discretised system is carried out using an error-controlled, time-adaptive, embedded Rosenbrock–Wanner method. We compare several variants of the method and two variants of the model to investigate how details such as the choice cross-diffusion coefficients, and specific variants of the time integrator affect simulation time.

An overview of recently obtained authors’ results on traveling wave solutions of someclasses of PDEs is presented. The main aim is to describe all possible travelling wavesolutions of the equations. The analysis was conducted using the methods of qualitativeand bifurcation analysis in order to study the phase-parameter space of the correspondingwave systems of ODEs. In the first part we analyze the wave dynamic modes of populationsdescribed by the “growth - taxis - diffusion" polynomial models. It is shown that“suitable" nonlinear taxis can affect the wave front sets and generate non-monotone waves,such as trains and pulses, which represent the exact solutions of the model system.Parametric critical points whose neighborhood displays the full spectrum of possible modelwave regimes are identified; the wave mode systematization is given in the form ofbifurcation diagrams. In the second part we study a modified version of theFitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assumethat this propagation is (at least, partially) caused by the cross-diffusion connectionbetween the potential and recovery variables. We show that the cross-diffusion version ofthe model, besides giving rise to the typical fast travelling wave solution exhibited inthe original “diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slowtraveling wave solution. We analyze all possible traveling wave solutions of the model andshow that there exists a threshold of the cross-diffusion coefficient (for a given speedof propagation), which bounds the area where “normal" impulse propagation is possible. Inthe third part we describe all possible wave solutions for a class of PDEs withcross-diffusion, which fall in a general class of the classical Keller-Segel modelsdescribing chemotaxis. Conditions for existence of front-impulse, impulse-front, andfront-front traveling wave solutions are formulated. In particular, we show that anon-isolated singular point in the ODE wave system implies existence of free-boundaryfronts.