**Marek Bodnar**(Uniwersytet Warszawski):

*General model of a cascade of reactions with time delays: global stability analysis*

The considered problem consists of a cascade of reactions with discrete as well as distributed delays, which arose in the context of Hes1 gene expression. For the abstract general model sufficient conditions for global stability are presented. The method is based on comparison of the behaviour of the system of delay differential equations with an appropriate discrete system. Then the abstract result is applied to the Hes1 model and the condition for global stability of the steady state is given.

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**Marek Czachor**(Politechnika Gdańska):

*Relativity of arithmetics as a fundamental symmetry of physics*

Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters `plus’ or `times’ one has certain freedom of interpreting this operation. This leads to some freedom in denitions of derivatives, integrals and, thus, practically all equations occurring in natural sciences. A change of realization of arithmetics, without altering the remaining structures of a given equation, plays the same role as a symmetry transformation. An appropriate construction of arithmetics turns out to be particularly important for dynamical systems in fractal space-times. Simple examples from classical and quantum, relativistic and nonrelativistic physics are discussed.

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**Urszula Foryś**(Uniwersytet Warszawski):

*Analysis of cancer-immune system interactions model*

We consider simplified model of cancer-immune system interactions for non-immunogenic tumours. The model is described as a system of three ODEs. We study asymptotic behaviour of this system, including existence of steady states, local stability of these states and possibility of bifurcations. We present some results concerning global stability for the states reflecting healthy organism as well as presence of tumour cells in the organism.

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**Ewa Girejko**(Politechnika Białostocka):

*On mathematical model of Bats' Roost Searching Strategies*

On the basis of cavity roosting bats’ behavior living in Bialowiea Forest a mathematical model of their searching strategies is presented. We present a dynamical system describing a way of roost finding, appropriate for a certain species of bats, which consists of two nonlinear difference recursive equations of a special form. In the paper, stability of stationary solutions of the considered system is examined. Stationary solutions in a biological interpretation mean points, where are tree cavities with habitat conditions suitable for a certain species of bats. Attractors are tree cavities that are settled by animals, while repulsers are cavities without settlement. Presented results are illustrated by computer simulations.

Co-authors: Robert Jankowski and Ewa Schmeidel

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**Piotr Hachuła**(Centrum Wiedzy Logistycznej, Instytut Logistyki i Magazynowania w Poznaniu):

*The analysis of demand and inventory steering model and its application*

The demand and inventory steering model introduced by Ma-Feng has been analysed in regard to its application in real business case. It has been shown that the model can describe specific situation of a product with a~time-limited sale and its sale stimulation. The case is modelled by a discrete dynamical system — three first order recurence equations showing dependence between changes of stock, demand and deliveries over time. Stability analysis conducted with numerical methods and biffurcation diagram is shown.

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**Marta Kornafel**(Uniwersytet Ekonomiczny w Krakowie):

*Optimal investment problem in multidimensional capital accumulation model*

We will present the solution of the optimal investment problem in multidimensional model of capital accumulation. The ageing process of capital may be irregular function, so we rely our analysis on viscosity approach. The stability result for viscosity solutions will be presented.

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**Tadeusz Kosztołowicz**(UJK w Kielcach):

*Modelling subdiffusion: from difference equations to fractional differential equations*

A discrete model of random walk appears to be a useful tool in modelling subdiffusion or normal diffusion. We base the model of subdiffusion on a random walk model in a system with both discrete time and space variables. The particle’s random walk is then described by a set of difference equations which can be solved by means of the generating function method. Using the generating function obtained for these equations we pass from discrete to continuous time and space variables by means of the procedure presented in this contribution. Finally we get the subdiffusion differential equation with fractional time derivative.

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**Karolina Kropielnicka**(Uniwersytet Gdański):

*A numerical approach to two sex age-structured populations models in a space of Radon measures*

The two sex age-structured populations model in a space of Radon measures is roughly speaking a system of transport equations with nonlocal boundary conditions considered in a certain space of measures equipped with flat metric. A numerical method described in this talk is based on a splitting technique, where we deal with two semigroups. One is defined by the transport operator while the second one is obtained from the nonlocal boundary condition. This setting allows us to approximate the solution of the underlaying problem as a sum of Dirac delta functions. These are results of a joint work with Piotr Gwiazda from University of Warsaw.

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**Sergey Leble**(Politechnika Gdańska):

*The method of dynamic projection operators in the theory of hyperbolic systems of partial differential equations with variable coefficients*

We consider a generalization of the projecting operators method for the case of Cauchy problem for systems of 1D evolution differential equations of first order with variable coefficients. It is supposed that the coefficients dependence on the only variable x is weak, that is described by a small parameter introduction. Such problem corresponds, for example, to the case of wave propagation in a weakly inhomogeneous medium. As an example, we specify the problem to adiabatic acoustics. For the Cauchy problem, to fix unidirectional modes, the projection operators are constructed. The method of successive approximations (perturbation theory) is developed and based on pseudodifferential operators theory. The application of these projection operators allows to obtain approximate evolution equations corresponding to the separated directed waves.

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**Zbigniew Peradzyński**(WAT Warszawa):

*Mathematical modeling of calcium induced calcium influx waves in cells and tissues*

We propose a mathematical theory of fast calcium waves of CICI type. According to the suggestion of L. F. Jaffe [1] , these waves are supported by the influx of calcium from the intercellular space by the stress activated ion channels located in the cell membrane. The local stretching of the membrane is evoked by a thin cross-linked actin network, the cortex, attached to the cell membrane. Myosin motors in this network are responsible for the appearance of contractile forces, depending on the calcium concentration. The thickness of the cortex is of the order of 100 nm, which is very small in comparison with the size of typical cells (10-20 ). Cells are also equipped with the systems of pumps pumping out the excess of calcium. The competition between these two processes and the diffusion lead to the appearance of the travelling waves. The model is based on a system of reaction diffusion system for calcium and buffer proteins coupled with the mechanical equations for the traction forces produced by the cortex. The important feature of t the system is the dynamic boundary condition which is responsible for the influx of calcium. It is interesting that the theory leads to homoclinic travelling waves (as observed in reality) without postulating additional equation for so called recovery variable as it is usually done in the theory of calcium induced calcium released waves (where the calcium is released from the internal stores located in the cell).

[1] L.F. Jaffe, „Stretch-activated calcium channels relay fast calcium waves propagated by calcium-induced calcium influx”, Biol. Cell 99, 175-184 (2007).

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**Monika J. Piotrowska**(Uniwersyte Warszawski):

*Influence of the distributed time delays on the stability in the family of agniogenesis models*

We propose a family of angiogenesis models, that is a generalisation of Hahnfeldt *et al.* model. Considered family of models is a system of two differential equations with distributed time delays. The global existence and the uniqueness of the solutions are proved. Moreover, the stability of the unique positive steady state is examined in the case when delay distributions are Erlang or piecewise distributions. Theorems guaranteeing the existence of stability switches and occurrence of the Hopf bifurcation are proved. Theoretical results are illustrated by numerical analysis performed for the parameters estimated by Hahnfeldt *et al.* (*Cancer Res.*, 1999).

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**Lucjan Sapa**(AGH Kraków):

*Quaternionic quantum wave and the Klein-Gordon equations in Cauchy-Navier elastic solid*

We show that a quaternionic quantum fild theory can be rigorously derived from the classical balance equations in isotropic crystal, where the energy and momentum transport are described by the Cauchy-Navier equations. We find a mathematical quaternionic formula for coupling between compression and torsion of the diplacement. The formula allows for a spatially localized wave function that is equivalent to the particle. A quantum wave equation that generates the Klein-Gordon equation is derived. We show the self-consistent classical interpretation of wave phenomena.

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**Robert Stańczy**(Uniwersytet Wrocławski): Nonlocal BVPs

We prove some existence and multiplicity results for a class of nonlocal BVPs involving the Dirichlet Laplace operator.

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**Agnieszka Świerczewska-Gwiazda**(Uniwersytet Warszawski):

*Multi-scale polymeric flows*

We will concentrate on a class of mathematical models for polymeric fluids, which involves the coupling of the Navier-Stokes equations for a viscous, incompressible, constant-density fluid with a parabolic-hyperbolic integro-differential equation describing the evolution of the polymer distribution function in the solvent, and a parabolic integro-differential equation for the evolution of the monomer density function in the solvent. The viscosity coefficient, appearing in the balance of linear momentum equation in the Navier–Stokes system, includes dependence on the shear-rate as well as on the weight-averaged polymer chain length. The system of partial differential equations under consideration captures the impact of polymerization and depolymerization effects on the viscosity of the fluid. We discuss the existence of global-in-time, large-data weak solutions under fairly general hypotheses.

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**Paweł Wojda**(Politechnika Gdańska):

*The finite difference methods of computation of X-rays propagation through a system of many lenses*

The propagation of X-ray waves through an optical system consisting of many X-ray refractive lenses and X-ray focusing is considered. For solving the problem of electromagnetic wave propagation, a finite-difference method for the paraxial wave equation is suggested and applied. The error of simulation is estimated mathematically and investigated. It is found out that very detailed difference mesh is necessary for reliable and accurate computation of propagation of X-ray waves through a system of many lenses. The reasons of necessity of very detailed difference mesh is that after the wave passes through a system of many lenses the electric field becomes a quickly oscillating function of coordinates perpendicular to the optical axis and very detailed difference mesh is necessary to digitize such a wave field. To avoid this diculty, we introduce the equation for a complex phase function instead of the equation for an electric field. Equation for complex phase is nonlinear equation, in contrast to the paraxial wave equation. It is shown that equation for a phase function allows to considerably reduce the detail of difference mesh without loss in reliability and precision of simulations of X-rays propagation through the system of many lenses and the X-rays focusing. The simulation error of the suggested method is estimated and the examples of computation result are presented.

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