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Seminarium Zakładu Biomatematyki i Teorii Gier

Prowadzi: Urszula Foryś


2014-12-03, godz. 14:15, s. 4050
Emad Rezk (PhD student from Egipt)
Angiogenesis model with distributed delay
The aim of this presentation is to show the results of the work done during the stay in Poland. The model proposed by Bodnar et al. (2013) was change to consider distributed delays instead of discrete delays. Results concerning existence of global positive solutions and stability of steady states will be presented.
2014-11-26, godz. 14:15, s. 4050
Błażej Miasojedow (Uniwersytet Warszawski)
UKRYTE MODELE MARKOWA, PRZYKŁADY I ALGORYTMY - kontynuacja
Przedstawimy ogólny schemat ukrytych modeli Markow I parę przykładów zastosowań. W miarę naszych możliwości postaramy się skupić na zastosowaniach w biologii I chemii. Zrobimy przegląd algorytmów obliczeniowych I symulacyjnych, ze szczególnym uwzględnieniem MCMC dla ukrytych procesów z czasem ciągłym, które wydaja się najważniejsze w modelowaniu kinetyki reakcji chemicznych (i dla których mamy dość świeże wyniki własne).
2014-11-19, godz. 14:15, s. 4050
Wojciech Niemiro (Uniwersytet Warszawski)
UKRYTE MODELE MARKOWA, PRZYKŁADY I ALGORYTMY
Przedstawimy ogólny schemat ukrytych modeli Markow I parę przykładów zastosowań. W miarę naszych możliwości postaramy się skupić na zastosowaniach w biologii I chemii. Zrobimy przegląd algorytmów obliczeniowych I symulacyjnych, ze szczególnym uwzględnieniem MCMC dla ukrytych procesów z czasem ciągłym, które wydaja się najważniejsze w modelowaniu kinetyki reakcji chemicznych (i dla których mamy dość świeże wyniki własne).
2014-11-05, godz. 14:15, s. 4050
Krzysztof R. Apt (Uniwersytet Warszawski)
Potentials and their uses in strategic games
The idea of a potential was introduced in the influential paper of
Monderer and Shapley in 1994. It is a technique that allows one to conclude that a strategic game has a Nash equilibrium.  We shall discuss the main aspects of this paper and subsequently illustrate other uses of potentials in our recent work on coordination games on graphs.
In each such game the players are the nodes of a graph. Each node selects a colour from a set of colours (privately) available for it. The payoff to a node is the number of neighbours who chose the same colour.  These games capture the idea of coordination in a local setting strategies. We shall focus on the issue of existence of Nash equilibria, k-equilibria and strong equilibria.
This is a joint work with Mona Rahn, Guido Schaefer and Sunil Simon.
2014-10-29, godz. 14:15, s. 4050
Małgorzata Półtorak ((dawna absolwentka MISMAP-u) )
FUNCTIONING IN CLOSE RELATIONSHIPS: MATHEMATICAL MODEL
I will present modified Felmlee & Greenberg (1999) model of the dynamics of relationships between two persons. Using various parameters of the model we are able to explain various types of relationships and give some suggestions to make the relationship better.
2014-10-22, godz. 14:15, s. 4050
Je-Chiang Tsai (Chung Cheng University, Taiwan )
Curvature dependence of propagating velocity for a simplied calcium model
It is known that the relation between curvature and wave speed plays a key role in the propagation of two-dimensional waves in an excitable model. For typical excitable models (e.g., the FitzHugh-Nagumo (FHN) model), such a relation is believed to obey the linear eikonal equation, which states that the relation between the normal velocity and the local curvature is approximately linear. In this talk, we show that for a caricature model of intracellular calcium dynamics, although its temporal dynamics can be investigated by analogy with the FHN model, the curvature relation does not obey the linear eikonal equation even in the limiting case. Hence this caricature calcium model may be an unexpected excitable system, whose wave propagation properties cannot be always understood by analogy with the FHN model.
2014-10-15, godz. 14:15, s. 4050
Wojciech Borkowski (Instytut Studiów Społecznych im. prof. B. Zajonca, UW )
Dynamiczna Psychologia Społeczna
W istniejącym już od ponad 20 lat "Ośrodku badania układów złożonych i nowych technologii" w Instytucie Studiów Społecznych UW, zajmujemy się zastosowaniami teorii układów złożonych i informatyki w psychologii społecznej  (dział ten bywa nazywany "dynamiczną psychologią społeczną"), ale także w obszarach stanowiących dla niej tło - psychologii "ja", dynamice sieci neuronowych czy, z drugiej strony, procesów ewolucyjnych. Jednym z głównych wątków badawczych i teoretycznych były dla nas zawsze procesy wpływu społecznego, który uznajemy za podstawowy mechanizm sterujący uwspólnianiem informacji adaptacyjnej w ludzkich społeczeństwach, choć w tej chwili praktyka badań znacznie się rozszerzyła w kierunku informatyki i ekonomi.
Poza różnorodnymi metodami badań psychologicznych i socjologicznych centralną metodą badawczych jest modelowanie komputerowe dynamiki procesów psychicznych, społecznych i ewolucyjnych za pomocą symulacji komputerowych, jako że narzędzie to pozwala badać zjawiska emergentne, powszechne w tych systemach. Stosujemy szeroką gamę modeli: od automatów komórkowych, algorytmów genetycznych, sztucznych sieci neuronowych i innych modeli sieciowych, po utworzone na ich bazie "minimalistyczne modele agentowe" będące naszą główną specjalnością, oraz rozbudowane - "realistyczne" modele agentowe.
Kilka dosyć aktualnych linków rozszerzających temat:
-   Dynamical Minimalism: Why Less is More in Psychology,   Andrzej Nowak http://www.ncbi.nlm.nih.gov/pubmed/15223518, http://psr.sagepub.com/content/8/2/183.full.pdf+html
-   Culture Change: The Perspective of Dynamical Minimalism, Andrzej Nowak, Wouter de Raad, and Wojciech Borkowski rozdział  http://global.oup.com/academic/product/advances-in-culture-and-psychology-9780199840694
-   Why Simulate? To Develop a Mental Model, Andrzej Nowak, Agnieszka Rychwalska and Wojciech Borkowski http://jasss.soc.surrey.ac.uk/16/3/12.html
 -  UKŁADY ZŁOŻONE W NAUKACH SPOŁECZNYCH Wybrane zagadnienia (książka) http://scholar.com.pl/sklep.php?md=products&id_p=2029
2014-06-04, godz. 14:15, s. 4050
Paweł Ciosmak (Uniwersytet Warszawski)
Numerical methods for the dynamics of the populations with structure
In many situations the description of the dynamics of a population requires taking into account its structure. In the case of one real structural parameter, like the age or size of an individual, nonlinear first order hyperbolic equations are often applied. We will present the escalator boxcar train method, which can be used to solve these equations, and discuss its convergence.
2014-05-14, godz. 14:15, s. 4050
María Vela Pérez (Service de Physique de l'Etat Condensé, CEA-Saclay, 91191 Gifsur-Yvette, France)
From Individual to Collective Dynamics in Argentine Ants
Social insects are an important example of complex collective behavior. In particular, ant colonies develop different tasks as foraging, building and allocation [1]. While they search for food they deposit a pheromone that it is considered as a crucial element in the mechanism for finding minimal paths. The experimental observations suggest that the model should include the presence of pheromone and the persistence (tendency to follow straight paths in the absence of other effects). In our study, based on the experimental data described in [2], we develop a model in the plane to describe the behavior of Argentine Ants when foraging in the plane.
Following the ideas explained in [3] we consider ants as random walkers. We treat them as pure random walkers when they detect an amount of pheromone that is below a certain threshold. The idea is that ants, once out of the nest, start foraging for food and do it following a random walk with the probability distribution for the change in direction that is fitted, from experimental data, to a distribution with fat tails. Once the ant detects an amount of pheromone concentration above the threshold, the motion changes to a reinforced random walk where a component of the change in the ant's direction is proportional to the gradient of the amount of pheromone.
[1] B. Holldobler and K. Wilson, The ants, Berlin: Springer, 1990
[2] A. Perna, et al. (2012) Individual rules for trail pattern formation in
Argentine ants (Linepithema humile). PLOS Comput Biol 8(7):e1002592.
[3] M. Vela-Perez, et al. (2013), Ant foraging and geodesic paths in labyrinths: Analytical and computational results, J. Theo. Biol. 320, 100-112.
2014-05-07, godz. 14:15, s. 4050
Emad Attia (doktorant z Egiptu)
On the distance between adjacent zeros of solutions of first order differential equation with distributed delays
The results of many publications that estimated the upper bound of the distance between adjacent zeros of any solution of first order delay differential equation will be displayed. We show  some fundamental results  for the lower bound of of the distance between adjacent zeros of any solution of first order delay differential equation. New estimations of the upper bound of  the distance between successive zeros of any solution of a first order differential equation with distributed will be discussed.
2014-04-30, godz. 14:15, s. 4050
Urszula Foryś (Uniwersytet Warszawski)
The survey on Lyapunov functions on the basis of the article by Sze-Bi Hsu c.d.

2014-04-09, godz. 14:15, s. 4050
Agnieszka Bartłomiejczyk (Politechnika Gdańska) Henryk Leszczyński (Uniwersytet Gdański)
Asymptotyka rozwiązań w modelach populacyjnych o strukturze cech fizjologicznych z dyfuzją
Badamy model ewolucyjny uwzględniający śmiertelność, wzrost, narodziny i dyfuzję z warunkami brzegowymi Fellera. Rozpoczynamy rozważania od zasady maksimum, następnie analizujemy rozwiązania rozwijane w szereg Fouriera w przestrzeni Hilberta, która wyznaczona jest przez warunki Fellera. Struktura tych rozwinięć pozwala na przewidywanie asymptotycznych własności, w szczególności tzw. profilu.
2014-04-02, godz. 14:15, s. 4050
Agnieszka Dziekańska (alumni of University College Dublin, Systems Biology, Ireland)
Computational analysis of multi-stable signalling biochemical networks
In this talk, two novel approaches to investigate systemic properties of signalling biochemical networks will be introduced.

The first approach focuses on the analysis of information processing in signalling networks performed by the application of information theory.  Molecular components of any signalling network are constantly subject to intracellular fluctuations in gene expression. The study applies the concept of Shannon’s entropy and channel capacity to investigate how fluctuations in the level of network components affect information processing capacity in biochemical networks. A sensitivity analysis of channel capacity could be applied to detect the nodes whose perturbations do not translate to significant changes in channel capacity and therefore do not play an important role in information transduction across the network. However, such a detailed analysis can be performed only in presence of high quality mathematical models of signalling networks. Lack of such models poses a main obstacle in current applications of IT in Systems Biology.

The second approach concentrates on computation of steady-states of biochemical networks by the application of the concept of the biochemical landscape. The biochemical landscape quantifies the propensity of the system to settle in any of the possible steady-states. The study introduces two methods of computation of the biochemical landscape. The first method applies the concept of the quasi-potential in order to compute the gradient of trajectory of a dynamic system as it evolves to its steady-state. The second method applies stochastic simulations (Gibbs sampling) for the purpose of deriving the probability densities that correspond to steady-states of the system. Calculation of landscapes for gene regulatory or signalling networks can determine the relative stability of steady-states to fluctuations present in the network. The concept of an attractor positioned on the landscape surface could represent the discrete phenotypes of many human diseases such as cancer. It could also provide an explanation of the origin and development of the diseases and influence the current method of drug discovery.
2014-03-26, godz. 14:15, s. 4050
Krzysztof Fujarewicz (Politechnika Śląska)
Structural sensitivity analysis of mathematical models in biology
Sensitivity analysis plays very important and useful role during modeling and analysis of dynamical systems. It answers the question how changes in model parameters affects the model solution. The answer to this question can be useful in solving of many tasks, such as: estimation of model parameters, design of experiments, or the optimization of the structure of the model. Typically, the sensitivity functions with respect to model parameters are calculated but it is possible to perform the sensitivity analysis w.r.t. initial conditions or signals stimulating the system. During the presentation the so-called structural sensitivity analysis will be presented. It assumes that the system is presented in a structural form: as a block diagram. It simplifies the rules for the adjoint system creation and may be treated as a special case of so called automatic differentiation. The adjoint sensitivity analysis for systems of ordinary differential equation (ODE), delayed differential equations (DDE) and cellular automata used for solving partial differential (PDE) will be formulated. Results of application of the approach to parameter estimation and gradient-based optimization for various models will be presented.
2014-03-12, godz. 14:15, s. 4050
Marek Bodnar (Uniwersytet Warszawski)
General model of a cascade of reactions with time delays: global stability analysis c.d.
I discuss a general model 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 will are presented. Then the abstract result is applied for the Hes1 gene expression model.
2014-03-05, godz. 14:15, s. 4050
Marek Bodnar (Uniwersytet Warszawski)
General model of a cascade of reactions with time delays: global stability analysis
I discuss a general model 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 will are presented. Then the abstract result is applied for the Hes1 gene expression model.
2014-02-26, godz. 14:15, s. 4050 (ZMIANA SALI)
Jacek Waniewski (IBIB PAN)
Matematyczne modele transportu w czasie dializy otrzewnowej c.d.

2014-02-19, godz. 14:15, s. 4050 (ZMIANA SALI)
Jacek Waniewski (IBIB PAN)
Matematyczne modele transportu w czasie dializy otrzewnowej
Modele opisują procesy wymiany płynu i substancji pomiędzy tkanką a:
1) krwią przepływającą przez naczynia włosowate zawarte w tkance, oraz 2) otoczeniem tkanki (np. płynem dializacyjnym w jamie otrzewnowej lub żelem nałożonym na powierzchnię skóry). Modele używane w zastosowaniach klinicznych mają charakter modeli kompartmentowych, natomiast fizjologię tych procesów można opisać modelami opartymi o równania różniczkowe cząstkowe typu równań dyfuzja-konwekcja-reakcja. Pokażemy, że liniowe wersje tych równań pozwalają na wyprowadzenie ciekawych zależności dostarczających interpretacji danych klinicznych i fizjologicznych. Uwzględnienie nieliniowych interakcji charakterystycznych dla żywej tkanki wymaga prowadzenia badań numerycznych.
2014-01-22, godz. 14:15, s. 5840
Maciej Cytowski, Zuzanna Szymańska (ICM UW)
Large Scale Parallel Simulations of 3-D Cell Colony Dynamics
Biological processes are inherentlyvery complex and involve many unknown relationships and mechanisms at different scales. Despite many efforts, one still cannot explain all the observed phenomena and, if necessary, make any desirable changes in the dynamics. Recently, it has become apparent that the opportunity lies in complementing the traditional, heuristic experimental approach with mathematical modelling and computer simulations. Achieving a realistic simulation scale is still a huge challenge, however it is necessary to understand and control complex biological processes. In this paper we present a novel high performance computational approach allowing simulations of 3D cell colony dynamics in realistic, previously unavailable scale. Due to the high parallel scalability we are able to simulate cell colonies composed of 109 cells, which allows for instance to describe tumor growth in its early clinical stage.
2014-01-15, godz. 14:15, s. 5840
Jan Poleszczuk (MiSDoMP i IBIB PAN)
Pulse wave propagation models
During the seminar I will present the state of the art of mathematical methods utilized in the modeling of the pulse wave propagation (PWP) through the arterial tree.
In the spatially distributed approach, the whole arterial tree is divided into segments that are assumed to be straight compliant vessels (each segment may have different characteristics), some of which bifurcate into two subsequent smaller vessels. A typical blood vessel segment is modeled as an axisymmetric compliant cylinder with wall assumed to be impermeable (or permeable only to a small extent). The assumption about the axisymmetry allows to reduce the continuous flow into one spatial dimension, i.e. position along the vessel. The relation between pressure p and flow q for each vascular segment is derived from conservation of mass and the momentum equations by assuming fully developed incompressible Newtonian flow in a straight vessel.
In other approach, each arterial segment is lumped and spatial information about the flow is lost. This approach allows to express the system as the electrical circuit analog, with capacitors and resistors as the main components. Obviously, these models are simpler than the distributed ones and the mathematical complications are kept to a minimum. They can also yield useful insight into the behavior of the system under investigation.
2014-01-08, godz. 14:15, s. 5840
Paweł Zwoleński (IM PAN)
Phenotypic evolution of hermaphrodites
We consider finite, phenotype-structured population of hermaphrodites, and build an individual based model which describes interactions between the individuals. The model contains such elements as mating of individuals (random or assortative), inheritance of phenotypic traits including mutations, intra-specific competition and mortality. Here offspring’s phenotype depends on parential traits. We consider the limit passage with the number of individuals to infinity, what leads us to continuous distribution of phenotypic traits in the population. The model is described by evolution equation in the space of measures, which contains nonlinear operators. The first of the operators is in charge of mating of individuals and inheritance, the other corresponds to the competition.
The limiting version of the model for random mating is an evolutionary equation, containing bilinear operator. The particular case of the equation is Tjon-Wu equation which appears in the description of the energy distribution of colliding particles. In the case of random mating, under suitable conditions we prove the asymptotic stability result: distribution of the phenotypic traits in the population converges to a stationary distribution. As a by-product we obtain relatively easy proof of Lasota-Traple theorem concerning asymptotic stability of Tjon-Wu equation. Moreover, we show applications of our theorem to some biologically reasonable situations of phenotypic inheritance.
2013-12-11, godz. 14:15, s. 5840
Beata Zduniak (SGGW Warszawa)
A VENTRICULAR TACHYCARDIA AND AV NODAL DOUBLE RESPONSE TACHYCARDIA IN A MODIFIED VAN DER POL EQUATION
A modified van der Pol equation is a mathematical model used to recreate physiological behaviour in the conducting heart's system. The use of certain values for coupled terms allows to simulate circulating re-entry waves, which play an important role in generating a pathological heart rate. The existence of re-entry may entail serious disorders, like auricular fibrillation or tachycardia.
2013-12-04, godz. 14:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Global stability for some types of delayed logistic equations - continuation
This time we will focus on the main goal that is the study of simple epidemic model of vector-borne diseases proposed by Cooke which belong to the class of delayed logistic equations.
2013-11-20, godz. 14:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Global stability for some types of delayed logistic equations
We will focus on proving global stability for three different types of delayed logistic equation. Our main goal will be to study simple epidemic model of vector-borne diseases proposed by Cooke which belong to the class of delayed logistic equations.
2013-11-06, godz. 14:15, s. 5840
Marek Bodnar (Uniwersytet Warszawski)
Global stability of steady steady state of delay differential equations in neural network model
We prove that a strong attractor of a discrete map implies global stability of a corresponding system of delay differential equations. We apply this result to a delayed Hopfield's model. We prove also that every attractor one-dimentional map is a strong attractor and we present an example that this is not true in dimension higher than one.
2013-10-23, godz. 14:15, s. 5840
Magdalena Bogdańska (Uniwersytet Warszawski)
Delay effects in the response of low grade gliomas to radiotherapy: a mathematical model and its therapeutical implications
Low grade gliomas (LGGs) are a group of primary brain tumors, which are highly infiltrative and generally incurable but have median survival time of more than 5 years because of low proliferation. Management of LGGs has historically been controversial because these patients are typically young, with few, if any, neurological symptoms. Recently Pallud et al. studied patients with LGGs treated with first-line radiation therapy and found the counter-intuitive result that tumors with a fast response to the therapy had a worse prognosis than those responding late. We construct a mathematical model describing the basic facts of glioma progression. Radiation therapy included in our mathematical model captures the essentials of the dynamics and explains the relationship between proliferation, response to the therapy and prognosis. It can also provide an explanation to the observations of Pallud et al. and it can be used to explore different radiation regimes. Using the model we propose radiation fractionation schemes that might be therapeutically useful by helping to evaluate the tumor malignancy. It could help the oncologists in making the best possible decision on when and how act on the tumor.
2013-10-16, godz. 14:15, s. 5840
Simon Angus ((Monash University) )
he Similarity of Human Interest Amongst the Nations
Abstract: Are Australians more like Americans, British, New Zealanders or Indonesians? What shapes human interests more: contemporary events or events from long ago? This talk aims to answer these questions with a novel data source and new statistical technologies. Whilst several attempts have been made to get at cultural similarity amongst peoples of Earth, thus far all have relied on survey data. In contrast, we take a 'big data' approach to the question and utilise Google Trends data -- aggregate search volume data by Google across 38 nations -- to construct a model of the similarity of human interest amongst nations. We use this model to produce synthetic similarities for out of sample ties enabling a hierarchical interest similarity presentation of the major and minor divisions in international human interest.
2013-10-09, godz. 14:15, s. 5840
Simon Angus (Monash University)
"Challenges in silico: modelling delays, death and repair in EMT6/Ro tumor cells under a variety of multi-dose irradiation protocols"
Abstract: In silico (computational) techniques offer the potential to investigate efficiently many aspects of tumour development and progression. In particular, a calibrated, dynamic, in silico tumour model could be used to probe the combinatorially extensive, and largely unexplored, irradiation protocol space (dose size and timing sequence) in a facile way, with the potential to discover large gains in efficacy within a given total dose envelope, meriting further clinical investigation. However, to do this, the tumour model must present realistic delay, death and repair dynamics under multi-dose irradiation. Given that the exact mechanism of repair, cell cycle delay and death is not perfectly understood, the calibration approach itself allows for the testing of various theoretical assumptions. Our study, building on our previous work with single-dose irradiation (Angus & Piotrowska, 2013), finds that delay--death--repair dynamics are well represented by a reciprocal repair function (Fowler, 1999 & 2002) which includes an unrepairable cell fraction (Carabe-Fernandez, 2001).
2013-06-05, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Traveling waves in reaction-diffusion equations with time delay
I will talk about traveling waves in reaction diffusion systems with delay. On the basis on some simple equation I will show problems that arise when looking for traveling fronts in equation with delay. In fact, the system for the shape of wave front can be delay differential system and finding an appropriate solution is more difficult han is the case without delay and it is necessary to use Fixed Point Theorem or some interaction scheme.
2013-05-15, godz. 16:15, s. 5820
Prof. Urszula Ledzewicz (USA)
Timing and Dosage of Anti-Cancer Drug Therapies: An Optimal Control Approach
A systematic study of cancer treatment requires that we take into account not only the cancerous cells and their growth, but also various aspects of the tumor microenvironment. Its elemets include various types of cancer cells, sensitive and resistant to the treatment, healthy cells, tumor vasculature, immune system and more.  We shell discuss mathematical models that describe the dynamics of tumor growth in relation to its supproting vasculature under reciprocal angiogenic signaling. For such models, in addition to standard direct treatments that kill cancer cells, one can introduce the second indirect treatment by means of angiogenic inhibitors that target the tumor vasculature. A second indirect approach is to consider the interactions between the tumor and the immune system, inclusing tumor surveillance. Here a classical, minimally parametrized model of tumor-immune interactions will be discussed in connection with optimizing the treatment that combined traditional chemotherapy with   a stimulatory effect of the immune system. We will also outline some future work on a more complex model wich encompasses more elements of the microenvironment  and multi-target therapies. This will naturally lead to the topic of metronomic chemotherapy, a new direction in cancer treatment where the various aspects of the tumor microenvironment being targeted by one cytotoxic drug applied accord to a properly chosen "metronomic" protocol which involves between 20% and 80% MTD. It has been observed in experimental studies that such treatment has lower toxicity, lowe drug resistance and in addition exhibits anti-agiogenic and immune system stimulatory effects. The challenges concerning the modeling and mathematical analysis for metronomic chemotherapy will be addressed.
2013-05-08, godz. 16:15, s. 5820
Marzena Dołbniak (Gliwice)
Protocols of anti-angiogenic and combined anticancer therapy
In brief introduction information about negative results obtained after single anti-angiogenic treatment and reasons for use anti-angiogenic factors to normalization of chaotic and unregulated tumor vessels are presented. I discussed several simple ordinary differential equation (ODE) models of tumor growth taking into account the development of its vascular network. Different biological aspects are considered from the simplest model of Hahnfeldt et al. proposed in 1999 [4]; by modification included influence of vessel density and ”pruning” effect (d’Onofrio-Gandolfi [3]); or separation endothelial cells for mature and immature (Benzekry et al. [2]); to the model in which separation of sensitive and resistance cancer cells after chemotherapy occur (Świerniak [5]). Some of these models can be used in clinical oncology to optimize anti-angiogenic and cytostatic drugs delivery so as to ensure maximum efficiency.

I am aware that there is a big gap between the simulated and the real world and it this why I try to focus on several questions. The first is how modifications of the basic model improves the fit between the simulated therapy protocol and the real clinical results. The second question is how the dynamics of this model will look like after implementing protocols already used in medicine. Simple models of continuous and periodic protocols of combined therapy are implemented. Discussion on the dynamics of the models and their complexity is presented.
2013-04-24, godz. 16:15, s. 5820
Łukasz Płociniczak (Politechnika Wrocławska)
A New Mathematical Model of Corneal Topography

Sight is the most important sense that we posses. It is crucial to

understand the mechanics of vision in order to treat various diseases

that may occur and disturb regular seeing. The eye's main part

responsible for about two-thirds of refractive power is the cornea.

Cornea is transparent, shell-like structure situated in the frontal part

of the eye. It is one of the most sensitive parts of human body and its

various irregularities can cause many seeing disorders. Precise

knowledge of corneal shape is very important and accurate

mathematical models are necessary to fully understand biomechanics

of cornea. We present a new model of corneal geometry based on a

nonlinear membrane equation. We establish existence of solution and

provide some estimates. When fitting with data we use its simplied

form and find that mean error is of order of a few percent. Also, we

are concerned with determining some unknown parameters when the

solution is known (usually with a noise). This is one example of so

called Inverse Problems. They are usually more difficult to solve and

analyze than direct ones. Moreover, they often are ill-posed, that is,

not necessarily have unique solution which is continuous with respect

to the initial data. We propose some regularization methods and apply

them to the real corneal data. We struggle with two types of

different inverse problems. The first one concerns constant parameter

case. It turns out that determination of these constants is a nonlinear

problem in two unknowns to which solution we develop an iteration

scheme and prove rates of convergence. The second problem is linear

one and concerns the case when one of unknown parameters is not

necessarily constant.
We propose some regularization method based on classical ones. In

the end we obtain a stable method of determining unknown

parameters of our differential equation from the knowledge of

corneal shape. These parameters may have direct relation to some

biomechanical properties of the eye and can be used to provide some

insights of corneal structure.

2013-04-17, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
News from IMBM: modelling of prostate cancer immunotherapy
Abstract: During my visit in IMBM in January I have discussed the idea of new project with the team of Prof. Zvia Agur. We have decided to focus on the modelling of prostate cancer immunotherapy. I will present the model which was published in PlosOne recently. It is the system of several ODEs, however most of them are linear. It occurs that for one boost of immunotherapy this large system has simple, in fact one-dimensional dynamics. I will also present the analysis for impulsive treatment and give conditions sufficient to cure. At the end I will focus on the proposed changes in the model equations that are of our interest now.
2013-04-10, godz. 16:15, s. 5820
Michal Komorowski (IPPT Homing+)
The concept of information at the interface between statistics and systems biology

The purpose of the talk is to discuss intricacies of quantifying

information in some problems arising at the interface between statistic

and biology.  Behaviour generated by interacting molecules, cells or

tissues is not random but focuses on sustaining life processes.

Sustaining life, however, is to the large extend based on information

storage, transmission and processing with examples ranging from the

iconic DNA double-helix through signal transduction pathways to

proteomic and metabolic networks. Understanding how information is

processed in living cells and organism is therefore one of the crucial

elements to understand how life is sustained.

The concept of information however is also inherent to statistics with

Fisher and Shannon as most well known concepts. It has been

originally associated with a precision  parameters can be estimated in

an experiment. Alternative approach by C. Shannon arose on the basic

of communication theory and was later  assimilated by statistical

sciences.

In the talk I will discuss relationships between such objects as

posterior distribution, Shannon Information, Fisher Information and

Jeffrey's prior. Using examples of  experimental design techniques and

signal transduction pathways I will demonstrate how statistical theory

can be used to better understand biological systems at the cellular

level.

2013-03-20, godz. 16:15, s. 5820
Andrea Quartarone (University of Messina, Italy) and Tatiana V. Ryabukha (MIMUW and NANU, Ukraine)
Stability of the solutions of the mesoscopic equation that corresponds to the replicator equation.
The replicator equation is a deterministic nonlinear equation arising in evolutionary game theory describing the evolving lifeforms in terms of frequencies of strategies. It is related to a mean field approach and therefore it has a macroscopic character: the description is referred to the frequencies (densities) of agents playing the corresponding strategies.  However, the macroscopic approach is not sufficient to describe the dynamics of complex living systems by reducing  the complexity of the overall systems. In some applications to consider the agents as discrete interacting units is important in order to capture the complexity of (biological) phenomena.

We propose a class of kinetic type equations that describe the replicator  dynamics at the mesoscopic level. Under suitable assumptions we show the asymptotic (exponential) stability of the solutions to such kinetic equations in the case when the corresponding macroscopic equation is asymptotically stable. To obtain the mesoscopic model corresponding to the replicator equation we follow the techniques developed by N. Bellomo with coautors applying tools of the kinetic theory of active particles for complex living systems.

In perspective, the obtained results could be used for analysing the asymptotic behaviour in time of mathematical model which describes tumour-immune system competition.
2013-03-13, godz. 16:15, s. 5820
Jan Poleszczuk (MISDoMP Uniwersytet Warszawski)
Cellular senescence — a key player in radiotherapy and radiation induced bystander effect?
I'm going to present a novel biological hypothesis which combines two hitherto distinct phenomena: cellular senescence and radiation induced bystander effects. Hypothesis gives a new insight into the principles governing the tumor response to ionizing radiation and provides elegant explanation for the various types of bystander effects. Basic analysis based on a mathematical models reveals that the radiation-dose survival curves, and hence the tumor cure probabilities are highly dependent on the amount of cellular senescence triggered by the ionizing radiation or by the bystander signals.
2013-03-06, godz. 16:15, s. 5820
Agnieszka Wiszniewska-Matyszkiel (Uniwersytet Warszawski)
Applications of large dynamic games to modelling markets and exploitation of ecosystems

2013-02-27, godz. 16:15, s. 5820
Agnieszka Wiszniewska-Matyszkiel (Uniwersytet Warszawski)
Applications of large dynamic games to modelling markets and exploitation of ecosystems
I'm going to present my results on large games -- mainly games with a continuum of players and their economic applications. Games with a continuum of players may be regarded as an equivalent of continuum -- microscale models of physics. They were developed to model situation in which the number of interacting agents, called players (human beings, animals or even particles),is large enough to make the influence of a single player on some global variables negligible.

The main theoretical results from the point of view of mathematics, are various decomposition theorems allowing to simplify the procedure of finding an equilibrium in dynamic games: the original problem, joining dynamic optimization and fixed point procedure in a space of functions is reduced to a sequence of static optimization and fixed point procedures applied to static, one stage problems, coupled by some reccurential or differential equation.

I will also present economic applications of such games to exploitation of a common ecosystem and to modelling financial markets, mainly stock exchange.
2013-01-23, godz. WYJĄTKOWO o 16:30, s. 5820
Martin Parisot (MIMUW ERCIM)
Intermediate Modeling between Kinetic Equations and Hydrodynamic Limits: Application to the Spitzer-Harm regime
This work is devoted to the study of a problem resulting from plasma physics: heat transfer of electrons in a plasma close to Maxwellian equilibrium. A formal derivation from the Vlasov equations is proposed. A hierarchy of intermediate models between the kinetic equations and the hydrodynamic limit is described. In particular, a new system
hydrodynamics, integro-differential in nature, is proposed. The system Schurtz and Nicolai appears as a simplification of the system resulting from the diversion. The local existence and uniqueness of the solution of the nonstationary system are established in a simplified framework. The last part is devoted to the implementation of a specific numerical scheme for solving these models. We propose a finite volume approach can be effective on unstructured grids. The accuracy of this scheme to capture specific effects such as kinetic, which may not be reproduced by the asymptotic Spitzer-Harm model.The consistency of this pattern with that of the Spitzer-Harm equation is highlighted, paving the way for a strategy of
coupling between the two models.
2012-12-12, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Constructing Lyapunov functionals IV

2012-12-05, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Constructing Lyapunov functionals III

2012-11-28, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Constructing Lyapunov functionals II

2012-11-21, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Constructing Lyapunov functionals
During the talk we will recall/summarise methods allowing to construct Lyapunov functionals for ODEs. We will start from linear ODEs and then switch to some non-linear examples.
2012-11-07, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Linear Chain Trick, distributed delays, and stability of steady states
The seminar is devoted to the theory of delay differential equations with distributed delays or in other words with memory. The Linear Chain Trick is discussed and I show under which condition an equation with distributed delays is equivalent to the system of ordinary differential equations. The stability of steady states will be discussed. I show that the equation with distributed delays is more stable than with a discrete delay in the sense that steady states are stable for a wider range of parameters.
2012-10-24, godz. 16:15, s. 5820
Jan Karbowski (MIMUW i IBIB PAN)
Translating neural activity into behavior
How neural activity in the brain generates a particular behavior is not well understood. The focus of the talk will be on modeling neural circuit responsible for forward and backward locomotion in the nematode C. elegans. In particular, I will discuss an optimal pattern of connections
between neurons that gives the best match to the locomotory data for this animal.
2012-10-17, godz. 16:15, s. 5820
Piotr Szopa (Uniwersytet Warszawski)
Cellular calcium dynamics
During the talk I want to describe key aspects of cellular
calcium dynamics e.g. calcium channels, pumps and exchangers
responsible for calcium flows and the role of mitochondria and endoplasmic reticulum in calcium homeostasis.
I want say a few words about modeling those mechanisms from
microscopic and macroscopic point of view.
2012-10-10, godz. 16:15, s. 5820
Ishtiaq Ali (Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan, and MIMUW)
Numerical Solutions of Delay Differential Equations Using Spectral Methods
In this talk I shall present an efficient numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. We develop a spectral approach for the pantograph type-delay differential equations. A Legendre spectral-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. The exponential convergence was obtained theoretically which was confirmed by some numerical examples, provided the data in the given pantograph delay differential equation are smooth. We also discuss briefly the case of nonlinear and system of pantograph-type delay
differential equations.

2012-05-16, godz. 16:15, s. 5820
Monika Piotrowska (Uniwersytet Warszawski)
Noise-sustained fluctuations in stochastic dynamics with a delay

2012-05-09, godz. 16:15, s. 5820
Jan Poleszczuk (MISDOMP)
The impact of time delays on synchronizability in stochastic systems

2012-04-25, godz. 16:15, s. 5820
Bogdan Kaźmierczak (IPPT)
Spatially extended model of kinase-receptor interaction in cells
We consider a spatially-extended model describing mutual
phosphorylation of cytosolic kinase molecules and membrane receptors. The analyzed regulatory subsystem is a part of signal transduction mechanisms characteristic for immune receptors and Src family kinases. It couples differential equations defined in a domain and on its boundary via
nonlinear Robin boundary conditions. Assuming a spherically symmetric framework, our approach is to consider an auxiliary problem in which a uniform Dirichlet boundary condition is imposed on the external boundary of the spherical shell instead of the Robin boundary condition. This method allows us to find the stationary spherically-symmetric solutions, both stable and unstable.

2012-04-18, godz. 16:15, s. 5820
Marina Dolfin (University of Messina)
MODELLING Th1-Th2 CELL BALANCE DURING T CELL MEDIATED IMMUNE RESPONSE
Experimental observations show the relevance of Th1-Th2 cell balance in hypersensitive reactions. We propose a theoretical model of T cell mediated immune response focusing on Th1-Th2 cell balance, in the mathematical framework of the theory of reacting fluid mixtures with proliferative events. In our model the proliferative events, i.e. events which are not mass preserving, are the clonal expansions of Th1 and Th2 cells. Smooth approximate solutions of the resulting PDE's system are analyzed by using a double-scale approach enlightening some features regarding the multi-scale complexity of the phenomenon under observation.
2012-04-04, godz. 16:15, s. 5820
Jurij Kozicki (UMCS)
Markov Dynamics in a Spatial Ecological Model with Dispersion and Competition Part II but hopefully self-contained

2012-03-28, godz. 16:15, s. 5820
Beata Zduniak (Uniwersytet Warszawski)
INFLUENCE OF DELAYED FEEDBACK ON MODIFIED VAN DER POL OSCILLATOR
Modified van der Pol equation can be used to describe the activity of elements of electrical conduction system of human heart: sinoatrial node (SA), atrioventicular node (AV), and His Purkinje system. This model has a number of interesting properties allowing do reconstruct phenomena observed in physiological experiments as well as in Holter
electrocardiographic recordings. The aim of the presented work was totake into consideration the influence of feedback and delay which occur in normal heart action mode as well as in pathological modes. Usage of certain values for feedback and delay parameters in this model allows to simulate the heart action in situation when an accessory conducting
pathway is present (Wolff - Parkinson - White syndrome).
2012-03-21, godz. 16:15, s. 5820
Jacek Banasiak (KwaZulu-Natal, Durban)
Classical solutions of fragmentation-coagulation equations with unbounded coagulation rates
So far fragmentation-coagulation equations with strong coagulation have been analysed using weak compactness techniques which only yielded weak solutions. Recently we have proved that the fragmentation operator is sectorial which, combined with some interpolation techniques, allowed for proving classical solvability of such equations for large classes of fragmentation and coagulation rates.
2012-03-14, godz. 16:15, s. 5820
Jurij Kozicki (UMCS)
Markov Dynamics in a Spatial Ecological Model with Dispersion and Competition
The evolution of an individual-based spatial ecological model with dispersion and competition is studied. In the model, an infinite number of individuals, point particles in R^d, reproduce themselves, compete, and die at random. These events are described by a Markov generator, which determines the evolution of states understood as probability measures on the space of particle configurations. The main result is a statement that the corresponding correlation functions evolve in a scale of Banach spaces and remain sub-Poissonian, and hence no clustering occurs, if the
dispersion is subordinate to the competition.
2012-03-07, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
On some SIRC epidemic model and vaccination strategies
We consider a model of SIRC type, taking into account four groups: susceptible, infected, removed and carriers. We study the dynamics in two interesting cases: when a positive steady state does not exist and when it exists. Efficient vaccination strategies are proposed which are based on
impulsive modelling.

2012-02-29, godz. 16:15, s. 5820
Tatyana Ryabukha (Institute of Mathematics, National Academy of Sciences of Ukraine and MIMUW)
Equilibrium solutions for a microscopic model of population dynamics
Mirosław Lachowicz (MIMUW)
Introduction to Tatiana's talk


In the talk, the problem of existence and uniqueness of equilibrium solutions to a microscopic model of population dynamics will be discussed. The model is a modified Liouville equation corresponding to a Markov jump process. The condidtions guarateeing uniqueness or non-uniqueness of equilibrium solutions will be proposed.

2012-02-22, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Chinese reminiscences

2012-01-18, godz. 16:15, s. 5820
Jędrzej Jabłoński (Uniwersytet Warszawski)
Modeling mortality of a size-structured population using optimization approach to predator's behaviour (case of fish and Daphnia)
Existing models of fish predation are either descriptive or lack realistic assumptions on predator's motion. Presented model fills this gap. The pure optimization approach gives answers not only about prey mortality, but also about predator's trajectory and speed.
2011-12-14, godz. 16:15, s. 5820
Tatiana Ryabukha (Institute of Mathematics, National Academy of Sciences of Ukraine and MIMUW)
The Liouville Equation for a Stochastic Particle System
2011-12-07, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Friction dominated dynamics of interacting particles locally close to a crystallographic lattice towards conclusion but of course self contained
2011-11-30, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Friction dominated dynamics of interacting particles locally close to a crystallographic lattice. Self-contained continuation
2011-11-23, godz. 16:15, s. 5820
Karol Wawrzyniak (ICM UW & National Centre for Nuclear Research)
On Phenomenology, Dynamics and some Applications of the Minority Game


Minority game (MG) was designed as a microscopic model of adaptive behavior observed in multi-agent systems where being in minority is profitable. It was shown that the MG exhibits different modes of behavior, depending on the game parameters: the random, cooperation, and herd. The latter case is characterized by small strategy space compared to the overall number of agents. In the first part of our seminar we will incorporate the utility function to study phenomenology of MGs in their herd regime. We will prove that the utility is bounded and the number of states is finite. We can represent the game as a Markov process and we can substantially reduce the number of states and calculate their probabilities. Then such interesting features of an important extensive random variable called an aggregated demand, like its strong inhomogeneity and presence of patterns in time, can be easily interpreted. In the second part, the generalization of the minority game to more than one market will be considered. We will find that if the payoff function allows for strong fluctuation of the utility then market occupancies can differ significantly, with preference given to this market where the fluctuation occurred first. There exists a critical size of agent population above which agents on bigger market behave collectively. In this regime, there always exists a history of decisions for which all agents on a bigger market react identically. We developed a model that explains these phenomena. In the last part of the seminar we will focus on applications. We will present an improved version of the MG predictor. We will apply it to both the synthetic and real time series using data from financial markets. The statistical analysis of these data will be performed and the theory how to optimize predictor's parameters will be presented.

2011-11-16, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Friction dominated dynamics of interacting particles locally close to a crystallographic lattice
We study a system of particles, in general d-dimensional space, that interact by means of pair potential and adjust their positions according to the gradient flow dynamics induced by the total energy of the system. We consider the case when the range of the interaction is of the same order as the mean interparticle distance. It is also assumed that particles, locally, are located close to some crystallographic lattice. An appropriate system of equations that describes the evolution of macroscopic deformation of the crystallographic lattice, as well as the system that describe the evolution of the main crystallographic directions is derived. Well posedness of the derived system is studied as well as the stability of the particle system. Same examples of potentials that yield stable and unstable systems are given.
2011-10-19, godz. 16:15, s. 5820
Roman Cherniha, (Institute of Mathematics of NASU, National University 'Kyiv-Mohyla Academy ')
Mathematical model for fluid-glucose-albumin transport in dialysis and its steady-state solutions
Mathematical description of fluid and solute transport between blood and
dialysis fluid in the peritoneal cavity has not been formulated fully yet,
in spite of the well known basic physical laws for such transport. Recent
mathematical, theoretical and numerical studies introduced new concepts on
peritoneal transport and yielded better results for the transport of fluid
and osmotic agent [1, 2, 3]. However, the problem of a combined
description of osmotic ultra filtration to the peritoneal cavity,
absorption of osmotic agent from the peritoneal cavity and leak of
macromolecules (proteins, e.g., albumin) from blood to the peritoneal
cavity has not been addressed yet. Therefore, we present here a new
extended model for these phenomena
and investigate its mathematical structure [4]. The model is based on a
three-component nonlinear system of two-dimensional partial differential
equations with the relevant boundary and initial conditions.
The non-constant steady-state solutions of the model are studied.
The restrictions on the parameters arising in the model were established
with the aim to obtain exact formulae for non-constant steady-state
solutions. As the result, exact formulae for the density of fluid
flux from blood to tissue and the volumetric flux across the tissue were
constructed and two linear autonomous ordinary differential equations for
glucose and albumin concentrations were derived. The analytical results
were checked for their applicability for the description of fluid-
glucose-albumin transport in peritoneal dialysis.

References

[1] Cherniha, R., Waniewski, J.: Exact solutions of a mathematical model
for fluid transport in peritoneal dialysis. Ukrainian Math. J., 57,
1112{1119 (2005)
[2] R. Cherniha, V.Dutka, J.Stachowska-Pietka and J.Waniewski.
Fluid transport in peritoneal dialysis: a mathematical model and numerical
solutions. //Mathematical Modeling of Biological Systems, Vol.I. Ed. by
A.Deutsch et al., Birkhaeuser, P.291-298, 2007
[3] Waniewski J, Dutka V, Stachowska-Pietka J, Cherniha R:
Distributed modeling of glucose-induced osmotic flow.
Adv Perit Dial 2007;23:2-6.
[4] Cherniha, R., Waniewski, J.:New mathematical model for
fluid-glucose-albumin transportin peritoneal dialysis. ArXiv:1110.1283v1 5
oct.2011





2011-10-12, godz. 16:15, s. 5820
Maria Vela-Perez (IE University, Segovia)
Geodesic paths in simple graphs for some social insects
Social insects are an important example of complex collective behavior.
In particular, ant colonies develop diff erent tasks as foraging, building
and allocation [1]. While they search for food they deposit a pheromone
that it is considered as a crucial element in the mechanism for finding
minimal paths. The experimental observations suggest that the model should
include the presence of pheromone and the persistence (tendency to follow
straight paths in the absence of other effects).

In our study [2], we will consider ants as random walkers where the
probability to move in one or another direction is influenced by the
concentration of pheromone near them (reinforced random walks). We are
mainly interested not in an individual random walker but rather on a large
number of random walkers, their collective behavior, and the possibility
for them to aggregate forming geodesic paths between two points in some
simple networks.

We investigate the behavior of ants in a two node network and in a three
node network (with and without directionality constraint). Our analytical
and computational results show that in order for the ants to follow
shortest paths between nest and food, it is necessary to superimpose to
the ants' random walkthe chemotactic reinforcement. It is also needed a
certain degree of persistence so that ants tend to move preferably without
changing their direction much. Another important fact is the number of
ants, since we will show that the speed for finding minimal paths
increases very fast with it.

Bibliography

[1] B. Holldobler and K. Wilson. The ants, Berlin: Springer, 1990
[2] M. Vela-Perez, M. A. Fontelos and J. J. L. Velazquez.
Ant foraging and minimal paths in simple graphs, submitted for publication
2011-06-01, godz. 16:15, s. 5820
Tatiana Ryabukha (Institute of Mathematics, National Academy of Sciences of Ukraine and MIMUW)
Non-equilibrium cluster expansions in the theory of many-particle dynamical systems
This talk deals with analytical methods in statistical mechanics. It's
goal is to give a deductive presentation of the statistical mechanics of
nonequilibrium systems based on BBGKY hierarchy studying - description of
the evolution of states - as well as to treat the basic principles of
interplay of statistical mechanic systems with the classical mechanics. A
compact pesentation was prefered wherever possible. The matherial is made
as understandable as possible by inclusion of the mathematical steps and a
detailed presentation of some intermediate culculations.

2011-05-25, godz. 16:15, s. 5820
Adam Bobrowski (Politechnika Lubelska)
From diffusions on graphs to Markov chains via asymptotic state lumping
We show that finite-state Markov chains may be approximated by fast
diffusions on finite graphs with semi-permeable membranes on vertices.
The approximation involves a singular perturbation with singularity in
both operator and boundary/transmission conditions. The result is
motivated by recent models of synaptic depression.
2011-05-18, godz. 16:15, s. 5820
Tadeusz Płatkowski (Uniwersytet Warszawski)
Game dynamics for players with complex personalities
We consider populations of individuals who are engaged in n--person public
good games or in two--person non symmetric or symmetric social dilemma
games. The players imitate the most attractive strategies, and the choice
is motivated not only by their payoffs, but also by their popularity in the
population. The aggregated parameter which determines the influence of
these two factors on the strategy choice of the players is identified with
the sensitivity to reinforcements parameter in the Hernstein's Matching
Law of mathematical psychology. The idea of of imitating the most
successful, and the copying the most popular strategies leads to
stabilization of cooperation in the populations of individuals in the
considered classes of games. The level of cooperation depends on the
sensitivity to reinforcements. We discuss the existence of equilibria and
their stability for such populations. A unique threshold of the
sensitivity is found, below which the polymorphic equilibria are stable,
and above which they are unstable.
2011-05-04, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
More on linear differential systems with small delays
We'll study the aysymptotic dynamics of linear systems of DDEs with small delays. It can be shown that asymptotically such systems behave as the so-called special solutions and for linear DDEs these solutions coincide with fundamental solutions. Moreover, we are able to calculate the limitof X^{-1}(t)x(f)(t) as t tends to infinity, where X(t) is the fundamental matrix (a special solution matrix) and x(f) is the solution to DDe with the initial function f at t=0. This limit can be expressed in terms of the basis of the generalised eigenspace for the formal adjoint equation.
2011-04-20, godz. 16:15, s. 5820
Wizytówka Biomatematyki i Teorii Gier

Przedstawiamy się cd

1. Tatiana Ryabukha Institute of Mathematics, National Academy of Sciences of Ukraine On solutions of classical and dual BBGKY hierarchies

2. Jacek Miękisz On games, genes, and delays O grach, genach i czasowych opóźnieniach

3. Urszula Foryś Delays in different natural phenomena Opóźnienia w różnych procesach naturalnych

4. Monika Piotrowska About the fascinating delays and cellular automata O fascynujących opóźnieniach i automatach komórkowych

5. Jan Poleszczuk Cancer stem cell hypothesis, tumor angiogenesis and influence of delay on biochemical networks Hipoteza komórek macierzystych nowotworu, angiogeneza oraz wpływ opóźnień na przebieg reakcji chemicznych   

2011-04-06, godz. 16:15, s. 5820
Wizytówka Biomatematyki i Teorii Gier

Przedstawiamy się

1. Tatiana Ryabukha, Institute of Mathematics, National Academy of Sciences of Ukraine, On interplay between statistical mechanics and life sciences in science globalization perspective

2. Mirosław Lachowicz, Markov jumps everywhere, Markowskie skoki wszędzie

3. Tadeusz Płatkowski, Some models of cooperation, O kilku modelach kooperacji

4. Agnieszka Wiszniewska-Matyszkiel, Dynamic games, continuum of players and expectations, Gry dynamiczne, continuum graczy i oczekiwania

5. Jacek Miękisz, On games, genes, and delays, O grach, genach i czasowych opóźnieniach

6. Urszula Foryś, Delays in different natural phenomena, Opóźnienia w różnych procesach naturalnych

7. Monika Piotrowska, About the fascinating delays and cellular automata, O fascynujących opóźnieniach i automatach komórkowych

8. Jan Poleszczuk, Cancer stem cell hypothesis, tumor angiogenesis and influence of delay on biochemical networks, Hipoteza komórek macierzystych nowotworu, angiogeneza oraz wpływ opóźnień na przebieg reakcji chemicznych

2011-03-30, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Small delays
2011-03-16, godz. 16:15, s. 5820
Marta Kulik (UW)
Hipoteza Zielonej Ziemi - model komputerowy

Wszechobecność roślin zielonych zainspirowała Hairstona, Smitha i Slobodkina do opracowania ogólnej teorii kontroli populacji lądowych. W Hipotezie Zielonej Ziemi (HSS) zauważyli oni, że do znacznego uszczuplenia populacji roślin dochodzi niezmiernie rzadko, liczebność (biomasa) roślinożerców zatem nie może być limitowana przez zasoby. Z drugiej strony nie ogranicza ich również środowisko abiotyczne, stąd czynnikiem ograniczającym jest drapieżnictwo. Stworzony został komputerowy model ewolucji testujący hipotezę, że względne dostosowanie roślinożerców jest niższe niż roślin (i szczytowych drapieżników), co wyklucza osiągnięcie przez nie wysokiego zagęszczenia, bliskiego pojemności środowiska. Założono istnienie trzech populacji, należących do jednego łańcucha troficznego (populacje roślin, roślinożerców i mięsożerców). Osobniki w każdej z populacji określono za pomocą binarnych sekwencji ich genotypów. Dostosowanie osobników, wyrażające ich zdolność do korzystania z zasobów oraz obrony przed drapieżnikami, oszacowano licząc zgodne allele w odpowiednich genotypach. Rozrodczość oraz śmiertelność w każdym kroku czasowym były funkcją dostosowania i liczebności populacji. Analiza zmienności genetycznej pokazała, że najwyższa różnorodność genetyczna występowała w populacji roślinożerców. Zgodnie z przewidywaniami, dostosowanie roślinożerców oraz poziom wypełnienia ich pojemności środowiska były istotnie niższe niż wartości otrzymywane dla skrajnych poziomów troficznych (roślin i drapieżników). Model powstał we współpracy z prof. Piotrem Dawidowiczem oraz prof. Dariuszem Wrzoskiem.

2011-03-02, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Small time delays
2011-02-23, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Small time delays
2011-02-16, godz. 16:15, s. 5820
Natalia Bielczyk (Uniwersytet Warszawski)
Dynamical Models of Dyadic Interactions with Time Delay
We will discuss a general class of linear models of dyadic interactions with a constant discrete time delay. In such models, changes of stability of stationary points occur for various intervals of parameters which determine the intensity of interactions. Conditions guaranteeing arbitrary number (that is zero, one or more) switches are formulated and the relevant theorems are proved. A systematic analysis of all generic cases is carried out. Possible generalizations to triadic interactions will also be presented. The talk is based on the joint work with U. Foryś and T. Płatkowski.
2011-01-19, godz. 16:15, s. 5820
Michał Matuszak (UMK Toruń)
Coevolution of networks and strategies
Evolution of cooperation, within a framework of Prisoner's Dilemma game, on scale-free graphs will be discussed. We will present coevolutionary stochastic dynamics which lead to scale-free graphs with various exponents. 
2011-01-12, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Still more on small time delays
2011-01-04, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
More on small time delays
2010-12-08, godz. 16:15, s. 5820
Urszula Foryś (Uniwersytet Warszawski)
Equations with small delays
I'll present a review of known results concerning equations with small delays. This talk will be mainly focused on linear and semi-linear delay differential equations with small delays.
2010-11-24, godz. 16:15, s. 5820
Kazimierz Piechór (IPPT PAN)
Mechanochemiczny model waskulogenezy i angiogenezy, liniowa stabilność modelu, fale biegnące
2010-11-03, godz. 16:15, s. 5820
Piotr Pokarowski (Uniwersytet Warszawski)
A minimal protein-like lattice model

I will summarize results of my papers with Andrzej Kolinski. We have designed a minimal model of protein folding that reproduces in a qualitative way the most pronounced features of globular proteins. Using Monte Carlo simulations we have estimated a set of parameters for which the native state is the global minimum of conformational energy. This implies the minimality of our force field. Then we have found such interaction parameters for which the model satisfies the thermodynamic hypothesis and folding transition is maximally cooperative. Contrary to H-P models, long range interactions are consistent with statistical contact potentials within an average protein environment and not with the transfer energies of residues from water. Cooperativity depends on protein architecture: it is the highest for the beta motif and the lowest for the alpha motif.

2010-10-27, godz. 16:15, s. 5820
Mirosław Lachowicz (Uniwersytet Warszawski)
DNA
Few remarks on the possible mathematical description of the process of denaturation of DNA
2010-10-20, godz. 16:15, s. 5820
Michał Matuszak (UMK)
Smooth Conditional Trajectory Evaluation in Dynamical Gaussian Networks
The seminar will present possibilities of applying Gaussian networks in realistic animation of virtual characters and optimal redeployment in multiagent environments. It will be achieved by solving the variational equations for Freidlin-Wentzell action functionals.
2010-05-26, godz. 16:15, s. 5820
Roman Cherniha (NAS of Ukraine and University Kyiv Mohyla Academy)
Symmetries and exact solutions of the diffusive Lotka-Volterra system
Lie and conditional symmetries of the classical Lotka-Volterra sys- tem inthe case of one space variable are completely described and sets of suchsymmetries in explicit form are constructed. These symmetries are used toreduce the classical Lotka-Volterra systems with correctly- specifiedcoefficients to ODE systems and examples of exact solutions in explicitform are found. The properties of the obtained solutions are examined withthe aim to provide the relevant interpretation for population dynamics. 

 

2010-05-19, godz. 16:15, s. 5820
Irena Lasiecka, profesor wizytujący w ramach projektu Nowoczesny Uniwersytet programu Kapitał Ludzki Unii Europejskiej (University of Virginia)
Boundary stabilization of a 3-D Navier Stokes flow in the neighborhood of an unstable equilibrium
Połączone seminarium:  Równania Fizyki Matematycznej i Biomatematyka i Teoria Gier
We shall consider a 3-D Navier Stokes flow defined in the neighborhood of an unstable equilibrium. For large Reynolds numbers, the steady state solutions are unstable and cause turbulence in their surroundings. Our goal is to construct a feedback operator, supported on the boundary of the domain, which will produce local exponential stability of such steady-state solution. The construction is based on solvability of optimization problem with "high" gain (unbounded) functional cost. This leads to the analysis of existence and uniqueness of the associated non-standard Riccati equations with boundary controls.
2010-04-28, godz. 16:15, s. 5820
Monika Piotrowska (Uniwersytet Warszawski)
Family of angiogenesis models with discrete delays
We will discuss a family of models with delays describing the process of angiogenesis, that is a physiological process involving the growth of new blood vessels from pre-existing ones. This family includes the well known models of tumour angiogenesis proposed by Hahnfeldt et al. and Ergun et al. and is based on the Gompertz type of the tumour growth. The family of considered models depends on two time delays and a parameter which reflects how strongly the vessels dynamics depends on the ratio between tumour and vessels volume. 
2010-04-21, godz. 16:15, s. 5820
Jacek Miekisz (Uniwersytet Warszawski)
Gry ewolucyjne na grafach
Będziemy rozważać Zamieć Śnieżną na grafach regularnych i z losowym doborem graczy oraz Dylemat Więźniaa bezskalowych grafach losowych Barabasi-Alberty.
2010-04-14, godz. 16:15, s. 5820
Ryszard Rudnicki (US i IMPAN)
O własnościach gatunków semelpatrycznych
Gatunek semelpatryczny, to taki, którego przedstawiciele rozmnażają się raz w życiu i umierają. Będziemy dodatkowo zakładać, że długość  życia osobników n jest stała. Przedstawimy model struktury-wiekowej populacji semelpatrycznej z dyskretnym czasem. Dla n=1 model redukuje się do funkcji jednej zmiennej. W tym przypadku naszkicujemy dowód twierdzenia o globalnej stabilności punktu stałego w oparciu o twierdzenieSzarkowskiego.Dla n>1 model sprowadza się do transformacji w przestrzeni n-wymiarowej. Model ten ma zaskakujące  własności. Między inny dla n parzystych rozwiązanie stacjonarne nigdy nie jest stabilne. Również asymptotyka długoczasowa jest zaskakująca. Konkurencja wewnątrz gatunkowa prowadzi do eliminacji wszystkich roczników za wyjątkiem jednego. Jest to zgodne z obserwacjami biologicznymi np. różnych gatunków owadów.
2010-03-31, godz. 16:15, s. 5820
Jacek Miekisz (Uniwersytet Warszawski)
Gry ewolucyjne na grafach regularnych i losowych
Główne pytanie: jak struktura grafu wpływa na poziom kooperacji to znaczy postać miary stacjonarnej odpowiedniego łańcucha Markowa? Odpowiedź na seminarium.
Będziemy rozważać Zamieć Śnieżną na sieciach regularnych i z losowym doborem graczy oraz Dylemat Więźnia na bezskalowych sieciach Barabasi-Alberty.
2010-03-24, godz. 16:15, s. 5820
Jacek Miękisz (Uniwersytet Warszawski)
Gry ewolucyjne na grafach
Od pewnego czasu wiadomo, że struktura przestrzenna sprzyja zachowaniom kooperacyjnym. Przedstawię wyniki dotyczące poziomu kooperacji w grach typu Dylemat Więźnia, Zamieć Śnieżna oraz Jeleń-Zając.
Pokażę, że poziom kooperacji w stanie stacjonarym odpowiedniego łańcucha Markowa zależy od sposobu doboru graczy. Przedyskutuję zależność długoterminowego zachowania graczy od poziomu szumu i wielkości populacji.
2010-03-17, godz. 16:15, s. 5820
Jarosław Bihun ((Narodowy Uniwersytet im. Jurija Fedkowycza w Czerniowcach))
Averaging in Multifrequency Systems of Differential-Functional Equations
This work is devoted to the development and justification of averaging
circuit for systems of differential equations with slow and fast variables
which pass through resonances in the course of evolution, the resonance
relation for frequencies dependant on retardation of argument in fast
variables is introduced. We build the uniform estimates for oscillation
integrals that are appropriate to the multifrequency systems under the
condition of invariable and variable delay. The case of systems with
linear delay is considered in detail. Under imposed conditions the
asymptotics of estimates is unimprovable. On the basis of obtained
estimates, we prove new
theorems on justification of averaging method for systems with invariable
and variable delay when frequencies depend on time lag or slow variables.
We
propose and justify the averaging circuit for systems with delay when
multipoint or integral boundary conditions are given. The averaging
procedure is also applied to integral boundary conditions.
2010-03-10, godz. 16:15, s. 5820
Krzysztof Mogielski i Tadeusz Płatkowski (MIMUW) (Uniwersytet Warszawski)
A Mechanism of Dynamical Interactions for Two-Person Social Dilemmas
We propose a new mechanism of interactions between game - theoretical
agents in which the weights of the connections between interacting
individuals are dynamical, payoff - dependent variables.
Their evolution depends on the difference between the payoff of the agents
from a given type of encounter and their average payoff. The mechanism is
studied in the framework of two models: agents distributed on a random
graph
and a mean field model. Symmetric and asymmetric connections between the
agents are introduced. Long time behavior of both systems is discussed for
the Prisoner's Dilemma and the Snow Drift games.
2010-03-03, godz. 16:15, s. 5820
Tomasz Lipniacki (IPPT)
Przestrzenna regulacja kaskady kinaz
2010-01-20, godz. 16:15, s. 5820
Michał Ramsza (SGH)
Twierdzenie Markusa-Yamabe w teorii gier ewolucyjnych
Podczas prezentacji zostanie przedstawione twierdzenie Markusa-Yamabe (Markus, Yamabe, 1960) o globalnej asymptotycznej stabilności. Twierdzenie zostanie przedstawione w uproszczonej wersji przystosowanej do zastosowania w teorii gier ewolucyjnych. Jako przykład zastosowania zostaną podane dwa różne dowody twierdzenia o globalnej asymptotycznej stabilności równowagi w symetrycznej jednookresowej grze przetargowej w dynamice testowania i porównań binarnych.
2010-01-13, godz. 16:15, s. 5820
Jakub Kowalski (Uniwersytet Wrocławski)
Ewolucja populacji - model Penna
Przedstawię założenia teoretyczne modelu zarówno z punktu widzenia matematyki, jak i biologii, a także powiem, do jakich wniosków prowadzą symulacje z jego wykorzystaniem.
2010-01-06, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Effects of time delay on stability for simple linear DDEs
We illustrate the influence of time delay on the stability of steady state for some simple linear delay differential equations. We prove that for one DDE with one discrete delay the only stability switch that can occur while delay is increasing is from the stable case to the unstable one. We also justify that in the case of several discrete delays if the steady state is unstable in the case without delays then it remains unstable for positive delays and we give an example showing that for two discrete delays while enlarging them, we can have the following stability switch: stable - unstable - stable - unstable. We also present an example of a system of two linear DDEs with one discrete delay tau, such that the steady state is unstable for tau = 0 and is stable for some tau>0.
2009-12-16, godz. 16:15, s. 5820
Monika Piotrowska (Uniwersytet Warszawski)
O pracy Tembine, Altmana i El-Azouzi czyli o symetrycznych i asymetrycznych opóźnieniach w grach ewolucyjnych.
Zajmiemy się tematyką opóźnień w symetrycznych grach ewolucyjnych, gdzie z każdą czystą strategią powiązane jest dyskretne opóźnienie pojawiające się w równaniach replikatorowych.
2009-12-09, godz. 16:15, s. 5440 (wyjatkowo inna sala)
Benoit Perthame (Pierre & Marie Curie University, Paris)
Adaptive evolution: a population view
Połączone Seminarium Biomatematyka i Teoria Gier i Seminar Ph.D. Programme Mathematical Methods in Natural Sciences Więcej informacji na http://mmns.mimuw.edu.pl/perthame/ Seminarium będzie samozawierające się
2009-12-02, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Stability of the expected value and boundness of the variance of some stochastic time-delay differential equations - conclusion
2009-11-25, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Stability of the expected value and boundness of the variance of some stochastic time-delay differential equations
2009-11-18, godz. 16:15, s. 5820
Daniel Wójcik (Nencki Institute of Experimental Biology)
Simple point process models of neural spike trains
A necessary ingredient for a quantitative theory of neural coding is appropriate spike kinematics: a precise description of spike trains - sequences of standard voltage impulses used for communication between neurons. While summarizing experiments by complete spike time collections is clearly inefficient and probably unnecessary, the most common probabilistic model used in neurophysiology, the inhomogeneous Poisson process, often seems too crude. Recently a more general model, the inhomogeneous Markov interval (IMI) model (Berry & Meister, 1998; Kass & Ventura, 2001), was considered, which takes into account both the current experimental time and the time from the last spike. In my talk I will discuss the nature of neural data to be understood and show how they naturally fall into point process description. I will introduce some basic point process models (Poisson and renewal processes) reaching IMI models in some general framework. Then I will propose a direct method of estimation that is easy to implement, fast, and conceptually simple. The method will be illustrated with an analysis of sample data from the cat's superior colliculus. Zapraszam, Jacek Miękisz
2009-11-04, godz. 16:15, s. 5820
Marek Bodnar (Uniwersytet Warszawski)
Stability of the expected value and boundness of the variance of some stochastic time-delay differential equation and its application to the hemapoietic stem cell regulation system
The results of work of J. Lei and M. Mackey (SIAM J. Appl. Math., 67, 387-407 (2007)) will be presented. We will study the stability of the trivial solution of a stochastic delay differential equation in the presence of an additive and multiplicative white noise. We will show the conditions guaranteeing that the expected value of the solution converges to zero and that the variance is bounded. We will also give a condition for unboudness of the variance. The results will be applied to the hemapoietic stem cell regulation system.
2009-10-28, godz. 16:15, s. 5820
Mirosław Lachowicz (Uniwersytet Warszawski)
Denaturacja DNA i pewne równania różniczkowo-całkowe
Zostanie przedstawiona pewna klasa równań różniczkowo-całkowych i dość desperacka próba opisu zjawiska denaturacji DNA (pękania wiązań wodorowych)pod wpływem temperatury - tzw. topnienie DNA.
2009-10-07, godz. 16:15, s. 5820
Ofer Biham (Racah Insitute of Physics, The Hebrew University, Jerusalem)
Stochastic analysis of toggle switch and toxin-antitoxin modules
Regulation processes in cells are performed by networks of interacting genes, which regulate each other's expression. To analyze the function of genetic networks, we simulate the dynamics of small functional modules using stochastic methods which take into account the effects of fluctuations. In this talk I will consider two genetic modules - the toggle switch (encoded synthetically on plasmids) and the toxin-antitoxin system (I will focus on the hipBA module in E. coli). I will show that in such modules, which include feedback, fluctuations give rise to crucial quantitative and qualitative effects. For the toggle switch I will show how the details of the system architecture affect the bistable state and the switching time. For the hipBA toxin-antitoxin system, I will focus on its role in the phenomenon of bacterial persistence.
2009-06-03, godz. 16:15, s. 5840
Jan Poleszczuk (Uniwersytet Warszawski)
Validity of delayed differential equations in biochemical reactions systems
It is well known that the time evolution of spatially homogeneous mixture composition consisting of molecules from N different species that can react through M chemical channels can be deterministically described by some set of ordinary differential equations. The method of generating stochastic simulations of such systems was developed by D.T. Gillespie. There is a high correspondence between quantitative results obtained by these two methods. Recently, to incorporate the fact that some reactions take some time or to reduce complexity of systems there were introduced delays in reactions. Therefore, ordinary differential equations have been reformulated as delay differential equations. To incorporate delay in the stochastic algorithm, some modifications of the Gillespie algorithm were introduced. We investigate validity of deterministic descriptions of two delayed reactions presented in some papers with the usage of stochastic algorithm. Comparison of deterministic and stochastic result revealed that in both delayed reactions the natural formulation of DDEs for those reactions brings assumptions which are not consistent with reactionsdescriptions. Therefore, we propose also another deterministic descriptions for those reactions from which the general idea of formulating deterministic descriptions for delayed reactions can be seen.
2009-05-20, godz. 16:15, s. 5840
Agnieszka Wiszniewska-Matyszkiel (Uniwersytet Warszawski)
Evolutionary games with continuous strategy spaces
2009-05-13, godz. 16:15, s. 5840
Agnieszka Wiszniewska-Matyszkiel (Uniwersytet Warszawski)
Evolutionary games with continuous strategy spaces
2009-05-06, godz. 16:15, s. 5840
Agnieszka Wiszniewska-Matyszkiel (Uniwersytet Warszawski)
Evolutionary games with continuous strategy spaces
We shall consider evolutionary games based on games with continuous strategy spaces. Properties of replicator and adaptive dynamics will be described. The sequence of presentations will be based on papers of Oechsler & Riedel, Cressman, Hofbauer & Cressman and Doebeli, Hauert & Killingback.
2009-04-29, godz. 16:15, s. 5840
Jan Zakrzewski (Uniwersytet Warszawski)
Stochastic fluctuations in selection model based on evolutionary game theory
2009-04-22, godz. 16:15, s. 5840
Jan Zakrzewski (Uniwersytet Warszawski)
Stochastic fluctuations in selection model based on evolutionary game theory
We will investigate frequency-dependent selection model (with one locus, two alleles, two phenotypes and finite population) based on evolutionary game theory. Next we will include stochastic fluctuations in the model, and convert deterministic equation into the master equation. The master equation will be approximated with Fokker-Planck equation, which we will use to predict mean and variance of both allele and phenotypic frequency. Based on Y. Tao, R. Cressman, B. Zhang, and X. Zheng Theoretical and Population Biology 74 (2008) 263-272
2009-04-08, godz. 16:15, s. 5840
Monika Piotrowska (Uniwersytet Warszawski)
Co powinniśmy wiedzieć o Stochastycznych Równaniach Różniczkowych z Opóźnieniem II
2009-04-01, godz. 16:15, s. 5840
Monika Piotrowska (Uniwersytet Warszawski)
Co powinniśmy wiedzieć o Stochastycznych Równaniach Różniczkowych z Opóźnieniem
Celem wystapienia jest zaprezentowanie podstawowych definicji i twierdzen (bez dowodow) teorii stochastycznych rownaniach rozniczkowych z opoznieniami czasowymi.
2009-03-18, godz. 16:15, s. 5840
Jan Karbowski, Institute of Biocybernetics and Biomedical Engineering,Polish Academy of Sciences
COMPUTATIONAL NEURODYNAMICS - HOW NERVOUS SYSTEM GENERATES MOTION?
The ambitious objective of computational neuroscience is to find general rules of brain organization and function. By organization we usually mean the pattern of connections between neurons, and by function the dynamics of neural networks in some context. In this talk, I will focus on undulatory locomotion of a tiny worm called C. elegans. This animal is a good system for biological modeling because we know a great deal about its cellular and molecular organization. I will present some results concerning C. elegans biomechanics, its neural structure, and oscillatory neural dynamics leading to worm locomotion.
2009-03-11, godz. 16:15, s. 5840
Tadeusz Płatkowski (Uniwersytet Warszawski)
Equilibria in populations of agents with complex personality profiles for 2-player games
We propose a theory of evolution of social systems which generalizes the standard proportional fitness rule of the evolutionary game theory, and a formalism of social interactions in which the actors' choice behavior is controlled by parameters which describe their ability to control the relation between a sequence of stimulus and a sequence of responses. The formalism is applied to describe the dynamics of two-person one-shot games played in infinite populations. In particular it predicts the non-zero level of cooperation in the long run for the Prisoner's Dilemma games.
2009-03-04, godz. 16:15, s. 5840
Marek Bodnar (Uniwersytet Warszawski)
Small delay approximation of stochastic delay differential equation
2009-02-25, godz. 16:15, s. 5840
Marek Bodnar (Uniwersytet Warszawski)
Small delay approximation of stochastic delay differential equation
2009-02-18, godz. 16:15, s. 5840
Marek Bodnar (Uniwersytet Warszawski)
Small delay approximation of stochastic delay differential equation
We will review the article by Steve Guillouzic, Ivan L'Heureux, and Andre Longtin Phys. Rev E 59: 3970 (1999).
2009-01-14, godz. 16:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Wpływ szumu na bifurkacje Hopfa w równaniu różniczkowym z opóżnieniem
Rozpatrzymy specyficzne równanie różniczkowe z opóźnieniem dyskretnym, w którym przy zmianie opóźnienia występuje bifurkacja Hopfa. Następnie do tego równania zostanie wprowadzony element stochastyczny na zasadzie procesu Ornsteina-Uhlenbecka (z białym szumem Gaussowskim). Omówimy wyniki analizy numerycznej, a w przypadku asymptotycznym otrzymamy równoważne zagadnienie różnicowe.
2008-12-17, godz. 16:15, s. 5840
Mirosław Lachowicz (Uniwersytet Warszawski)
Stochastyczne równania różniczkowe typu Lotki-Volterry
Omowione zostana uklady rownan rozniczkowych typu Lotki-Volterry ze stochastycznym zaburzeniem. Pokazany zostanie wplyw stochastycznego zaburzenia na zachowanie rozwiazan.
2008-12-10, godz. 16:15, s. 5840
Mirosław Lachowicz (Uniwersytet Warszawski)
Stochastyczne równania różniczkowe z opóżnieniem
Miesiąc temu, 10 listopada, zmarł Kiyosi Ito, twórca teorii stochastycznych równań różniczkowych http://www-groups.dcs.st-and.ac.uk/~history/Biographies/ Ito.html. Podczas seminarium omówione zostaną układy równań różniczkowych typu Lotki-Volterry z opóźnieniem i ze stochastycznym zaburzeniem. Pokazany zostanie wpływ stochastycznego zaburzenia na zachowanie rozwiązań.
2008-12-03, godz. 16:15, s. 5840
Jacek Miękisz (Uniwersytet Warszawski)
Small-delay expansions in simple stochastic models with time delay
2008-11-26, godz. 16:15, s. 5840
Jacek Banasiak (University of KwaZulu-Natal i Politechnika Łódzka)
O liczbie cząsteczek w procesach koagulacji i fragmentacji
Choc w procesach fragmentacji tworza sie przez rozpad nowe czasteczki, matematyczna analiza tych procesow prowadzona jest w przestrzeni kontrolujacej ewolucje masy. Jest to spowodowane tym, ze w tej przestrzeni operator fragmentacji jest dyssypatywny, podczas gdy liczba czasteczek moze wybuchowo rosnac do nieskonczonosci. Z drugiej strony, mozliwosc kontrolowania liczby czasteczek w procesie fragmentacji ma istotne znaczenie przy analizie pelnego rownania fragmentacji i koagulacji. W trakcie wykladu omowimy zastosowanie teorii operatorow o dodatnich rezolwentach do uzyskania oszacowan liczby czasteczek produkowanych w trakcie fragmentacji, zatrzymamy sie na kilku patologiach modelu, i wskazemy mozliwosci zastosowania tych wynikow do kontroli momentow rozwiazan rownania koagulacji i fragmentacji.
2008-11-19, godz. 16:15, s. 5840
Jacek Miękisz (Uniwersytet Warszawski)
Stochastic dynamics with time delay - methematical problems and physical results
We will discuss Master, Fokker-Planck and Langevin equations with time delay in the context of gene expression.
2008-11-05, godz. 16:15, s. 5840
Jacek Miękisz (Uniwersytet Warszawski)
Stochastic models of gene expression with time delay
We will present an elementary introduction to stochastic models of gene expression. Master, Fokker-Planck, and Langevin equations will be discussed. We will review a simple model of protein production which can be completely solved, that is one can obtain analytical expressions for the expected value and the variance of the number of protein molecules (Thattai and Oudenaarden, PNAS, 8614-8619, 2001). Then we will discuss some simplified models with time delay and report on work in progress.
2008-10-22, godz. 14:15 (!), s. 5840
Malgorzata Kubalinska (Politechnika Lubelska)
Model typu Fishera-Wrighta ze zmienna wielkoscia populacji i mutacjami w postaci procesu punktowego
Koalescencja Kingmana jest jednym z wazniejszych pojec genetyki populacyjnej. Jednak przybliza ona tylko proces dokadnej koalescencji (proces tworzenia drzew genealogicznych w modelu Fishera-Wrighta). Przedstawimy model populacji zbudowany w oparciu o proces dokladnej koalescencji z mutacjami opisanymi za pomoc modelu nieskoczenie wielu miejsc. Zbadamy, widziane jako funkcja wielkosci populacji, asymptotyczne zachowanie rozkladow i momentow pary procesow punktowych zwiazanych z czteroelementowa probka wylosowana z takiej populacji (pierwszy element tej pary to roznica symetryczna, tzw. niezgranie, miedzy mutacjami pierwszego i drugiego elementu probki, a drugi to roznica symetryczna miedzy mutacjami trzeciego i czwartego elementu). Sprawdzimy na ile model z czasem dyskretnym rozni sie od modelu z czasem ciaglym (zbudowanym w oparciu o koalescencje Kingmana z mutacjami modelowanymi za pomoca procesu punktowego).
2008-10-15, godz. 16:15, s. 5840
Jacek Miękisz (Uniwersytet Warszawski)
Dyskretne opóźnienia
2008-10-08, godz. 16:15, s. 5840
Zebranie organizacyjne ZBiTG
2008-10-01, godz. 16:15, s. 2180
Mark Chaplain (University of Dundee)
Multiscale mathematical modelling of cancer growth
2008-06-04, godz. 16:15, s. 5840
Christian Maes ((Katholieke Universiteit Leuven))
Large deviations in non-reversible Markov processes
We show differences in the structure of dynamical fluctuations for Markov processes between the reversible and non-reversible case. The rate function for the occupation statistics is connected with the entropy production, and the study of current fluctuations involves characterizations of the largest eigenvalue of some non-normal matrices. This is a joint work with Karel Netocny.
2008-05-28, godz. 16:15-18.00, s. 5840
Zuzanna Szymanska (ICM UW)
Mathematical modelling of the influence of heat shock proteins on cancer invasion of tissue
Tumour cell invasion is crucial for cancer metastasis, which is the main cause of cancer mortality. An important group of proteins involved in cancer invasion are the Heat Shock Proteins (HSPs). According to experimental data, inhibition of one of these proteins, Hsp90, slows down cancer cells while they are invading tissue. To test different biological hypotheses regarding how precisely Hsp90 influences tumour invasion, we use a model of solid tumour growth which accounts for the interactions between Hsp90 dynamics and the migration of cancer cells and, alternatively, between Hsp90 dynamics and the synthesis of matrix degrading enzymes (MDEs). The model consists of a system of reaction-diffusion-taxis partial differential equations describing interactions between cancer cells, MDE, and the host tissue (ECM). Using numerical simulations we investigate the effects of the administration of Hsp90 inhibitors on the dynamics of tumour invasion.
2008-05-14, godz. 16:15-18.00, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Delays again
2008-04-23, godz. 16:15, s. 5840
Delays, delays
2008-04-16, godz. 16:15, s. 5840
Delays, informal discussion goes on
2008-04-09, godz. 16:15, s. 5840
Tomasz Lipniacki (IPPT)
Interplay between bistability and stochasticity in T-cell signaling
The stochastic dynamics of T-cell receptor (TCR) signaling are studied using a mathematical model intended to capture kinetic proofreading (sensitivity to ligand-receptor binding kinetics) and negative and positive feedback regulation. The model incorporates protein-protein interactions and reproduces several experimental observations about the behavior of TCR signaling. Analysis of the model indicates that TCR signaling dynamics are marked by significant stochastic fluctuations and bistability which is caused by the competition between the positive and negative feedbacks. Stochastic fluctuations are such that single-cell trajectories differ qualitatively from the trajectory predicted by the deterministic approximation of the dynamics. Moreover, because of the bistability, the average of single-cell trajectories differs markedly from the deterministic trajectory. Bistability combined with stochastic fluctuations allows for switch-like responses to signals, which may help a T cell to make committed cell-fate decisions.
2008-03-12, godz. 16:15, s. 5840
Mats Bodin (IM PAN)
Modeling copying behavior in animals - informal discussion
Decision-making plays a central role in animal groups, and ultimately results from the natural selection. We look at the evolution of copying behavior, where individuals gain information by watching others, as part of decision-making.
2008-03-05, godz. 16:15, s. 5840
Jacek Miękisz (Uniwersytet Warszawski)
How to model delays - informal discussion
2008-01-23, godz. 16:15, s. 5840
Kazimierz Sobczyk (IPPT i UW)
Złożone mikrostruktury materialne; modelowanie i propagacja fal stochastycznych. Complex material microstructures; modelling and stochastic wave propagations
2008-01-16, godz. 16:15, s. 5840
Kazimierz Sobczyk (Uniwersytet Warszawski i IPPT)
Dynamika stochastyczna i niezawodność układów z degradacją
2008-01-09, godz. 16:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Delays
2007-12-19, godz. 16:15, s. 5840
Urszuka Foryś (Uniwersytet Warszawski)
Discrete Marchuk's model with time delay
2007-12-05, godz. 16:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Discrete models with time delays: logistic equation and Marchuk's model
Tydzień temu seminarium zostało odwołane. We will consider discrete dynamical systems with time delays. The basic properties will be explain on two examples:discrete logistic equation and the discrete version of Marchuk's model of an immune system. We introduce time delay to the discrete logistic equation in different possible ways and compare the properties of such models with the classic model. It occurs that these properties depend strongly on the way of introducing delay. We also study a discrete version of Marchuk's model where time delay is introduced like in the original model. We compare the properties of discrete and continuous models.
2007-11-21, godz. 16:15, s. 5840
Cristian Morales-Rodrigo (Uniwersytet Warszawski)
Cell-matrix interactions in cancer invasion
SEMINARIUM WSPÓLNE Z SEMINARIUM RTN
2007-10-10, godz. 16:15, s. 5840
Monika Joanna Piotrowska (Uniwersytet Warszawski)
A Calibrated Cellular Automata Model of in vitro Cultivated Multicellular Tumour Spheroids
SEMINARIUM WSPÓLNE Z SEMINARIUM RTN. In this paper we present a quasi-two dmimensional Cellular Automata (CA)model describing the dynamics of the in vitro cultivated multicellular spheroid obtained from EMT6/Ro (mammary carcinoma) cell line. The CA model is coupled to the experimental data and shows extremely good agreement across a wide range of outputs including bulk growth kinetics, necrotic core formation and kinetics and response to environmental glucose levels.
2007-06-06, godz. 16:15, s. 5840
Marek Bodnar and Remigiusz Kowalczyk (Uniwersytet Warszawski)
Stochastic models in genetics
2007-05-30, godz. 16:15, s. 5840
Monika Joanna Piotrowska (Uniwersytet Warszawski)
Homeorhesis in the motion of an individual
Połączone seminarium RTN i Zakładu Biomatematyki i Teorii Gier Homeorhesis is an inherent dynamical feature of any living system. Homeorhesis is a peculiar qualitative and quantitative independence of the exogenous signals acting on the system and varying within a certain, system-relevant range. Nonliving systems do not perform homeorhesis. Mathematically, homeorhesis is the asymptotic convergence (in the infinite-time limit) of certain dynamic equilibria of the dynamical model that describes a living system (see [1], [2]). Preliminary results on the homeorhesis modelling in terms of ordinary differential equations (ODEs) are developed in [3, Appendix], [1], and [2]. In this case, both the actual mode and creode of the system are two dynamic equilibria [2] which correspond to the actual exogenous signal and the most favoured exogenous signal, respectively. In these terms, homeorhesis is the property of a living system that its actual mode in the course of time tends to its creode for any actual exogenous signal (within a certain, system-relevant range). Homeorhesis is a fundamental notion in theoretical biology (and in a more general field, theory of living matter). It is an inherent feature of any living system. Nevertheless, the literature on dynamical modelling in sociology does not include works on homeorhesis. The purpose of the present work is to fill this gap. Report [4] suggests the simplest model for homeorhesis. The present work specifies this model in the case of the motion of a single individual and illustrates the treatment with numerical-simulation results. [1] Mamontov, E., 2007, Modelling homeorhesis by ordinary differential equations, Mathl Comput. Modelling 45(5-6), pp. 694-707. [2] Mamontov, E., 2007, Dynamic-equilibrium solutions of ordinary differential equations and their role in applied problems, Appl. Math. Lett., accepted (paper AML5947). [3] E. Mamontov, K. Psiuk-Maksymowicz and A. Koptioug, Stochastic mechanics in the context of the properties of living systems, Mathl Computer Modelling 44(7-8): 595-607 (2006) [4] E. Mamontov, Homeorhesis and evolutionary properties of living systems: From ordinary differential equations to the active-particle generalized kinetics theory, In: 10th Evolutionary Biology Meeting at Marseilles, September 20-22, 2006 (Association pour l'Etude de l'Evolution Biologique, Centre Regional de Documentation Pedagogique,Marseille, France, 2006), pp. 28-29, abstract; the 13-page PDF file for the full oral presentation can be downloaded from http://www.up.univ-mrs.fr/evol-cgr/home_page/meeting2006.php
2007-05-23, godz. 16:00, s. 5840
1. Marek Bodnar and Remigiusz Kowalczyk 2. Ting Liu (Uniwersytet Warszawski)
1. Stochastic models in genetics. 2. From von Foerster to delay equations in the model of cell cycle.
2007-05-16, godz. 16:00, s. 5840
U. Foryś (UW), P. Rybka (UW), N. Kalev-Kronik i Y. Kogan (Inst. for Medical Biomath. Israel) T. Liu (UW) (Uniwersytet Warszawski)
Workshop on Mathematical Modelling of Tumour Growth
1. Apomixis, just questions. 2. Interactions between GBM brain tumour and immune system. 3. Interactions between GBM brain tumour and immune system - mathematical modelling. 4. From von Foerster to delay equations in the model of cell cycle. Workshop wspólny Zakładu Biomatematyki i Teorii Gier i RTN
2007-04-25, godz. 16:15, s. 5840
Janusz A. Hołyst (Faculty of Physics, Center of Excellence for Complex Systems Research, Warsaw University of Technology)
Universal scaling of distances in complex networks
Universal scaling of distances between vertices of Erdos-Renyi random graphs, scale-free Barabasi-Albert models, science collaboration networks, biological networks, Internet Autonomous Systems and public transport networks are observed. A mean distance distance between two nodes of degrees k_i and k_j equals to = A - B log(k_i*k_j ). The scaling is valid over several decades. A simple theory for the appearance of this scaling is presented. Parameters A and B depend on the mean value of a node degree _nn calculated for the nearest neighbors, on network clustering coefficients and degree-degree correlations. We can also explain log- periodic oscillations around this law by the discrete character of inter-node distances. References 1. Janusz A. Ho�yst, Julian Sienkiewicz, Agata Fronczak, Piotr Fronczak and Krzysztof Suchecki, Universal scaling of distances in complex networks, Phys. Rev. E 72,026108 (2005) 2. Julian Sienkiewicz, Piotr Fronczak, Janusz A. Holyst, Log-periodic oscillations due to discrete effects in complex networks, arXiv:cond-mat/0608273, Phys. Rev. E inprint, 2007.
2007-04-18, godz. 16:15, s. 5840
Agnieszka Wiszniewska-Matyszkiel (Uniwersytet Warszawski)
A new kind of equilibrium in dynamic games with beliefs
2007-04-04, godz. 16:15, s. 5840
Agnieszka Wiszniewska-Matyszkiel (Uniwersytet Warszawski)
A new kind of equilibrium in dynamic games with beliefs
We shall introduce a new notion of equilibrium -- belief-distorted Nash equilibrium (BDNE) -- in discrete time dynamic games in which players do not have perfect information about strategies chosen by the other players and form some expectations about them. A game in the strategic form is generally a triple of objects: the set of players (with, possibly, some structure imposed), their strategy sets and payoff functions defined on the set of profiles of strategies.The usual concept of Nash equilibrium requires that at an equilibrium profile every player (almost every in the case of infinitely many players represented as a measure space) maximizes his payoff as the function of his strategy given the strategies of the remaining players. In the concepts of belief-distorted Nash equilibrium we assume that at each stage of the game players maximize another functions related both to the original payoff functions and players' beliefs about future behaviour of the system. Various notions of self- verification are introduced. Equivalence between a BDNE for perfect foresight beliefs and Nash equilibrium is proven. The concepts are illustrated by examples of exploitation of a common renewable resource and a repeated minority game.
2007-03-14, godz. 16:15, s. 5840
Adam Lipowski (Dept. of Physics, A.Mickiewicz University, Poznań)
On the ecological and evolutionary dynamics of prey-predator systems
I will discuss some aspects of the dynamics of interacting populations.In particular, I will talk about oscillatory behaviour and environmentally induced large-scale synchronization (Moran effect) in lattice models of prey-predator systems. More complicated models with co-evolution of many interacting species will be also examined. In such models the number of species and some other characteristics show long-time periodic behaviour. Such a scenario is confronted with a possibility of periodicity of mass extinctions in the Earth ecosystem, as suggested by Raup and Sepkoski.
2007-03-08, godz. 12:30, s. 2100
Antoni Leon Dawidowicz (Institute of Mathematics, Jagiellonian University)
The Lasota equation and its properties
Połączone seminarium Zakładów: Równań Fizyki Matematycznej, Biomatematyki i Teorii Gier oraz Research Training Network Modeling, Mathematical Methods and Computer Simulations of Tumour Growth and Therapy
2007-01-24, godz. 16:15, s. 5840
Daniel K. Wójcik (Instytut Biologii Doświadczalnej im. M. Nenckiego)
Introduction to nonequilibrium work relations
Nonequilibrium work relations discovered by Chris Jarzynski are an example of symmetries found recently for several classes of nonequilibrium problems. These symmetries typically connect probability of a given process with that of its "reverse". I will briefly introduce Jarzynski relations, sketch a proof in a special case and illustrate them with a recent experiment on stretching DNA.
2006-12-20, godz. 14:15, s. 5840
Michalina Błażkiewicz
An attempt of mathematical description of kinematic parameters of hurdles, especially of the hurdle stride
The main objective of this presentation is utilization of mathematical apparatus for description of some kinematic parameters of hurdle race and analysis of position of limbs and of body centre of gravity during hurdle stride. The detailed objectives concern description of technique of hurdles of 100/110m and analysis of different aspects of efficiency of hurdle stride. Using mathematical analysis and calculus of variations (methods of Lagrange) I could answer a following question: What should be the athletes speed on a data distance L to have minimum time of run? On the other hand, the utilization of basic kinematics and dynamics laws allows to answer the following question: Who has major predisposition for hurdles: short or high athletes? For analysis of hurdle stride it was used the APAS (Ariel Performance Analysis System). We analysed the international champion 100m hurdles for women.
2006-12-06, godz. 16:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Generalization of the Greenspan MCS model to n-dimensional case
In tumour dynamics there is a well known model of spherical growth underthe diffusion of nutrient. The basic concept of multicellular spheroid (MCS) in 3-D was proposed by Greenspan. I'll show the formal generalization of this concept in n-dimensional case. I'll focus on the dependence of the model dynamics on the dimension parameter n, especially in comparison of two standard cases - for n=3 and n=2.
2006-11-29, godz. 16:15, s. 5840
Tadeusz Płatkowski (Uniwersytet Warszawski)
Cooperation in Multiperson Prisoner's Dilemma Games - Social Loafing
Social loafing is the tendency for individuals to expend less effort when working collectively compared with when working individually. Such reduction of commonly appearing prosocial behavior was observed in the past by psychologists. Generally, the experiments on social loafing confronted individual's efforton task in participation of two persons and several persons. It was observed that decrease of motivation to perform collective tasks changed when number of men had increased,however, not in the linear proportion, but rather exponential. We develop a mathematical model which takes into account various aspects of the phenomena, using multiperson Prisoner's Dilemma (PD) Games. The model - a 2-D system of nonlinear ODE takes into account evolution of the aspiration level in the population and mean frequency of the cooperative behavior. In particular we study limiting properties of the cooperation level in function of the order of the PD game.
2006-11-22, godz. 14:15, s. 5840
Radosław Wieczorek (IMPAN Katowice)
From individual behavior of phytoplankton to evolution equation
Phytoplankton, a mass of small plants (mostly algae) living in the water, comes out to be the main source of nutrient in oceans. That is why understanding of its bahaviour becomes so important and has been widely investigated. An individual based model, that describes spatial movement and processes of fragmentation and coagulation of phytoplankton structures, will be presented. The individuals in this model are phytoplankton aggregates (a number of cells joined by some organic glue) that move in the space, may split into smaller ones and have the possibility of joining together. After the formulation of the model in the setting of measure-valued stochastic process, we investigate its behaviour in the limit where the number of individuals grows to infinity. As a result of something like central limit theorem for our processes, we obtain the evolution equation on the density of the mass--spatial distribution of phytoplankton population.
2006-11-15, godz. 16:15, s. 5840
Ting Liu (Uniwersytet Warszawski)
Nonlinear impulse partial differential equations with delay
2006-11-08, godz. 16:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Review on Delay Differential Equations, a continuation
2006-10-25, godz. 16:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Review on Delay Differential Equations, continuation but in fact the beginning
2006-10-18, godz. 16:15, s. 5840
Urszula Foryś (Uniwersytet Warszawski)
Review on Delay Differential Equations
We introduce the basic notation used in the theory of DDE, formulate the basic theorems and show some examples of the infuence of delays on the dynamics of the system.
2006-05-24, godz. 16:15, s. 5840
Dorin Marinescu (Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Bucharest)
Simulation methods for Boltzmann-type equations
We report on the numerical approximation of the solution of a general class of nonlinear Boltzmann like equations. We provide a discretized version of the equations approximating the above class of equations. The numerical implementation of this scheme it is not possible due to the power-like growth of the computational numerical effort. For this reason we introduce stochastic techniques to diminish the numerical effort. Finally, one obtains a probabilistic convergent scheme solving the initial equations.
2006-05-17, godz. 16:15, s. 5840
Krzysztof Argasiński (Uniwersytet Jagielloński)
Sex ratio evolution from the point of view of dynamic large games
On the seminar will be presented new model of sex ratio evolution related to so called dynamic evolutionary large games. This is a new technique of modelling of multipopulaton problems. The classic approach (e.g. Shaw-Moller or Sex Ratio Game) relies on assumption that proper fitness measure is the number of grandsons. A new model shows that this is a mathematical artifact, and self-regulation of population sex ratio is not the product of 0.5 female phenotype selection. In stable state population may be heterogenic (consisting of different individual strategies) but population sex ratio is equal to 0.5. These stable states are elements of Evolutionarily Stable Set.
2006-04-05, godz. 16:15, s. 5840
Tomasz Lipniacki (IPPT)
Stochastic regulation of NF-kappaB pathway
Living cells may be considered noisy or stochastic biochemical reactors. In eukaryotic cells, in which the number of protein or mRNA molecules is relatively large, the stochastic effects may originate in regulation of gene activity or receptors activation. The stochasticity of activator binding and dissociation is amplified by transcription and translation, since target gene activation results in a burst of mRNAs molecules, and each copy of mRNA then serves as a template for numerous protein molecules. Similarly the single receptor activation can be amplified by the transduction cascade. In the present action I expand our model of the NF-kappaB (Nuclear Factor kappa B) regulatory module (Lipniacki et al., 2006. Biophys. J. 90, pp. 725-742) in order to analyze cell activation at low dose of TNF (Tumor Necrosis Factor). The considered regime of activation is important in analysis of cell-to-cell signaling. Ordinary differential equations, used for description of fast reaction channels of processes involving a large number of molecules, are combined with a stochastic switches to account for activity of genes and TNF receptors. The stochasticity in receptor activation and in gene transcription causes simulated cells to exhibit large variability. Moreover none of them behave like an "average" cell.
2006-03-28, godz. 10:15, s. 5081
Adam Bobrowski (IM PAN)
On limitations and insufficiency of the Trotter-Kato theorem with applications to a model of stochastic gene expression II (wspólne seminarium z RTN)
2006-03-21, godz. 10:00, s. 5081
Adam Bobrowski (IM PAN)
On limitations and insufficiency of the Trotter-Kato theorem, with applications to a model of stochastic gene expression
Wspólne seminarium z RTN Modeling, Mathematical Methods and Computer Simulation of Tumour Growth and Therapy. Abstract: Motivation for the talk comes from a recent model of stochastic gene expression introduced by Lipniacki et al. (J. Theor. Biol. 238: 348-367, 2006). The model involves a family of Feller processes, solutions to systems of stochastic differential equations driven by Markov chains with state-dependent jump intensities, which naturally converge to a certain deterministic process. It turns out that convergence of related semigroups of operators cannot be proved by means of the classical Trotter--Kato theorem, and the difficulty lies in a somewhat unexpected place. Before we deal with this difficulty, to explain the source of the problem, we exhibit simple examples of convergence of equibounded semigroups that cannot be captured by means of the Trotter-Kato theorem. In this context we discuss the need for semigroup-theoretical tools that would supplement this theorem in dealing with convergence problems.
2006-03-15, godz. 16:15, s. 5840
Jacek Miękisz (Uniwersytet Warszawski)
Stochastic models of genetic regulatory networks
Regulation of gene expression is a chemical process involving many coupled elementary chemical reactions modeled usually by systems of differential equations describing time evolution of molecular concentrations. However, due to low numbers of molecules involved in gene expression processes, random fluctuations may play a significant role. We will present an elementary introduction to stochastic models of such processes. Master, Fokker-Planck, and Langevin equations will be discussed. We will review a simple model of protein production which can be completely solved, that is one can obtain analytical expressions for the expected value and the variance of the number of protein molecules (Thattai and Oudenaarden, PNAS, 8614-8619, 2001). Then we will discuss specific models of mRNA- and protein-regulated networks, present some partial results and open problems.
2006-03-08, godz. 16:15, s. 5840
Mirosław Lachowicz (Uniwersytet Warszawski)
Amplification-deamplification process - odwołane